Problem Bank

An archer fires an arrow from the ground so that it passes through two hoops, which are both a height h above the ground. The arrow passes through the first hoop one second after the arrow is launched, and through the second hoop another second later. What is the value of h?

An amusement park ride consists of a circular, horizontal room. A rider leans against its frictionless outer walls, which are angled back at 30◦ with respect to the vertical, so that the rider’s center of mass is 5.0 m from the center of the room. When the room begins to spin about its center, at what angular velocity will the rider’s feet first lift off the floor?

A simple bridge is made of five thin rods rigidly connected at four vertices. A B C D F The ground is frictionless, so that it can only exert vertical normal forces at B and D. The weight of the bridge is negligible, but a person stands at its middle, exerting a downward force F at vertex C. In static equilibrium, each rod can be experiencing either tension or compression. Which of the following is true?

A bouncy ball is thrown vertically upward from the ground. Air resistance is negligible, and the ball’s collisions with the ground are perfectly elastic. Which of the following shows the kinetic energy of the ball as a function of time? Assume the collisions are too quick for their duration to be seen in the plot. K K K (B)

A massless inclined plane with angle 30◦ to the horizontal is fixed to a scale. A block of mass m is released from the top of the plane, which is frictionless. m 30◦ As the block slides down the plane, what is the reading on the scale?

A pendulum is made with a string and a bucket full of water. When the string is vertical, the bottom of the bucket is near the ground. Then, the pendulum is set swinging with a small amplitude, and a very small hole is opened at the bottom of the bucket, which leaks water at a constant rate. After a few full swings, which of the following best shows the amount of water that has landed on the ground as a function of position?

A particle travels in a straight line. Its velocity as a function of time is shown below. v(t) t Which of the following shows the velocity as a function of distance x from its initial position? v(x) v(x) v(x) (B)

A rod of length L is sliding down a frictionless wall. L v When the rod makes an angle of 45◦ to the horizontal, the bottom of the rod has speed v. At this moment, what is the speed of the middle of the rod?

When a car’s brakes are fully engaged, it takes 100 m to stop on a dry road, which has coefficient of kinetic friction µk = 0.8 with the tires. Now suppose only the first 50 m of the road is dry, and the rest is covered with ice, with µk = 0.2. What total distance does the car need to stop?

A block of mass m is connected to the walls of a frictionless box by two massless springs with relaxed lengths ℓ and 2ℓ, and spring constants k and 2k respectively. The length of the box is 3ℓ. The system rotates with a constant angular velocity ω about one of its walls. ω m ℓ, k 2ℓ, 2k Suppose the block stays at a constant distance r from the axis of rotation, without touching either of the walls. What is the value of r?

Two hemispherical shells can be pressed together to form a airtight sphere of radius 40 cm. Suppose the shells are pressed together at a high altitude, where the air pressure is half its value at sea level. The sphere is then returned to sea level, where the air pressure is 105 Pa. What force F , applied directly outward to each hemisphere, is required to pull them apart?

A space probe with mass m at point P traverses through a cluster of three asteroids, at points A, B, and C. The masses and locations of the asteroids are shown below. 3M C d m P d A d M d B 2M What is the torque on the probe about point C? 1 GM m

Two frictionless blocks of mass m are connected by a massless string which passes through a fixed massless pulley, which is a height h above the ground. Suppose the blocks are initially held with horizontal separation x, and the length of the string is chosen so that the right block hangs in the air as shown. h x If the blocks are relased, the tension in the string immediately afterward will be T . Which of the following shows a plot of T versus x? T T mg mg x

A bead of mass m can slide frictionlessly on a vertical circular wire hoop of radius 20 cm. m m The hoop is attached to a stand of mass m, which can slide frictionlessly on the ground. Initially, the bead is at the bottom of the hoop, the stand is at rest, and the bead has velocity 2 m/s to the right. At some point, the bead will stop moving with respect to the hoop. At that moment, through what angle along the hoop has the bead traveled?

The viscous force between two plates of area A, with relative speed v and separation d, is F = ηAv/d, where η is the viscosity. In fluid mechanics, the Ohnesorge number is a dimensionless number proportional to η which characterizes the importance of viscous forces, in a drop of fluid of density ρ, surface tension γ, and length scale ℓ. Which of the following could be the definition of the Ohnesorge number? ηℓ

A child of mass m holds onto the end of a massless rope of length ℓ, which is attached to a pivot a height H above the ground. The child is released from rest when the rope is straight and horizontal. m ℓ H d At some point, the child lets go of the rope, flies through the air, and lands on the ground a horizontal distance d from the pivot. On Earth, the maximum possible value of d is dE . If the setup is moved to the Moon, which has 1/6 the gravitational acceleration, what is the new maximum possible value of d?

Consider the following system of massless and frictionless pulleys, ropes, and springs. k 2k 3k m Initially, a block of mass m is attached to the end of a rope, and the system is in equilibrium. Next the block is doubled in mass, and the system is allowed to come to equilibrium again. During the transition between these equilibria, how far does the end of the rope (where the block is suspended) move?

A satellite is initially in a circular orbit of radius R around a planet of mass M . It fires its rockets to instantaneously increase its speed by ∆v, keeping the direction of its velocity the same, so that it enters an elliptical orbit whose maximum distance from the planet is 2R. start R 2R end What is the value of ∆v? (Hint: when the satellite is in an elliptical orbit with semimajor axis a, its total energy per unit mass is −GM/2a.) r

A wheel of radius R has a thin rim and four spokes, each of which have uniform density. m m m m 3m The entire rim has mass m, three of the spokes each have mass m, and the fourth spoke has mass 3m. The wheel is suspended on a horizontal frictionless axle passing through its center. If the wheel is slightly rotated from its equilibrium position, what is the angular frequency of small oscillations? r

Four massless rigid rods are connected into a quadrilateral by four hinges. The hinges have mass m, and allow the rods to freely rotate. A spring of spring constant k is connected across each of the diagonals, so that the springs are at their relaxed length when the rods form a square. m k m m k m Assume the springs do not interfere with each other. If the square is slightly compressed along one of its diagonals, its shape will oscillate over time. What is the period of these oscillations? r

A syringe is filled with water of density ρ and negligible viscosity. Its body is a cylinder of cross-sectional area A1 , which gradually tapers into a needle with cross-sectional area A2 ≪ A1 . The syringe is held in place and its end is slowly pushed inward by a force F , so that it moves with constant speed v. Water shoots straight out of the needle’s tip. What is the approximate value of F ?

A spherical shell is made from a thin sheet of material with a mass per area of σ. Consider two points, P1 and P2 , which are close to each other, but just inside and outside the sphere, respectively. If the accelerations due to gravity at these points are g1 and g2 , respectively, what is the value of |g1 − g2 |?

Collisions between ping pong balls and paddles are not perfectly elastic. Suppose that if a player holds a paddle still and drops a ball on top of it from any height h, it will bounce back up to height h/2. To keep the ball bouncing steadily, the player moves the paddle up and down, so that it is moving upward with speed 1.0 m/s whenever the ball hits it. What is the height to which the ball is bouncing?

When a projectile falls through a fluid, it experiences a drag force proportional to the product of its cross-sectional area, the fluid density ρf , and the square of its speed. Suppose a sphere of density ρs ≫ ρf of radius R is dropped in the fluid from rest. When the projectile has reached half of its terminal velocity, which of the following is its displacement proportional to?

A yo-yo consists of two massive uniform disks of radius R connected by a thin axle. A thick string is wrapped many times around the axle, so that the end of the string is initially a distance R from the axle. Then, the end of the string is held in place and the yo-yo is dropped from rest. Assume that energy losses are negligible, and that the string has negligible mass and always remains vertical. Below, we show a cross-section of the yo-yo partway through its descent. Between the moment the yo-yo is released and the moment the string completely unwinds, which of the following is true regarding the yo-yo’s acceleration?

A particle is moving on a plane at a constant speed of 1 m/s, but not necessarily in a straight line. Which of the following plot pairs could describe the particle’s position over time, in rectilinear coordinates? 5 5 y (m) 10 x (m) I. 10 0 −5 −5 −10 −10 0 10 5 5 y (m) 10 0 −5 −10 −10 5 t (s) 10 5 5 y (m) 10 x (m) 10 0 −5 −10 −10 5 t (s) 10 5 t (s) 10 0 5 t (s) 10 0 5 t (s) 10 0 −5 0 0 0 −5 0 III. 5 t (s) 10 x (m) II. 0

Assume that all three disks collide simultaneously. What is the final velocity of the third disk?

Assume that there is a little imperfection in disks’ initial alignment so when the disks collide two collisions happen one at a time, rather than all three disks colliding simultaneously. What is the final speed of the third disk?

A mouse M is running from A to A′ with constant speed u1 . A cat C is chasing the mouse with constant speed u2 and direction always toward the mouse. At a certain time M C ⊥ AA′ and the length of M C = L. What is the magnitude of the acceleration of the cat C? A M u1 A′ u2 C

Three identical cylinders are used in this setup. Two of them are placed side by side on a horizontal surface, with a negligible distance between their surfaces so they do not touch. The third identical cylinder is placed on top of the first two, such that their centers form an equilateral triangle, as shown in the figure below. The coefficients of friction are: • µ1 : the coefficient of friction between the cylinders, and • µ2 : the coefficient of friction between the cylinders and the ground. For which of the
following1 pairs
(µ1 , µ2 ) will
the system remain in equilibrium? 1 1 Pair 1: 21 , 12 . Pair 2: 3 , 10 . Pair 3: 14 , 18 . µ1 µ1 µ2 µ2

A ball rolls without slipping down a ramp, which turns horizontal at the bottom; at the bottom of the ramp, the ball falls through the air, as in the diagram. If the ball starts from the position marked O, it lands 10 cm away from the bottom of the ramp. Which starting position will get the ball to land closest to 25 cm away? A B C D E O

A mechanism consists of three point masses, each of mass m, connected by two massless rods of length l and a torsion spring acting as a hinge. The potential energy of the torsion spring is given by Us . This system is designed to “walk” down a set of stairs, as shown in the figure. The angles θ1 and θ2 (see figure) represent the orientation of the rods, and their rates of change, ω1 and ω2 , are the corresponding angular velocities. Assume that the mass on the surface is instantaneously at rest. Which equation correctly describes the total energy of the system? m m m θ2 θ1

A symmetric spinning top, rotating clockwise at an angular frequency ω, is placed upright in the center of a frictionless circular plate. The plate then begins to rotate counterclockwise at a constant angular velocity ω. Assume the top’s axis remains perfectly vertical and stable without any precession. From the perspective of an observer rotating with the plate, how does the top appear to rotate?

N circles in a plane, Ci , each rotate with frequency ω relative to an inertial frame. The center of C1 is fixed in the inertial frame, and the center of Ci is fixed on Ci−1 (for i = 2, . . . , N ), as shown in the figure. Each circle has radius ri = λri−1 , where 0 < λ < 1. A mass is fixed on CN . The position of the mass relative to the center of C1 is R(t). For the N = 4 case shown, which of the following statements is true? ω ω ω C4 ω C3 C2 C1 During the time interval from 0 to 2π/ω, the magnitude of acceleration of mass on C4

If the lighter object is initially at rest, and the heavier object collides elastically with it, what is the approximate maximum fraction of the heavier object’s kinetic energy that could be transferred to the lighter object?

Now suppose that instead, the heavier object is initially at rest, and the lighter object collides elastically with it. What is the approximate maximum fraction of the lighter object’s kinetic energy that could be transferred to the heavier object?

A 50 g piece of clay is thrown horizontally with a velocity of 20 m/s striking the bob of a stationary pendulum with length l = 1 m and a bob mass of 200 g. Upon impact, the clay sticks to the pendulum weight and the pendulum starts to swing. What is the maximum change in angle of the pendulum?

Which of the following plot pairs could show the linear speed v and rotational speed ω of one of the balls over time? Assume the floor has a constant roughness.

There are three kinds of balls that can be launched in this set-up, all having the same radius R: I. a regular tennis ball (a thin spherical shell of rubber) of mass m1 II. a solid wooden ball of mass m2 III. a solid rubber ball of mass m3 where m1 < m2 < m3 . All three types of ball emerge from the launcher with the same velocity. For which ball will the final velocity be highest?

A uniform rigid rod of mass M and length 2L is attached to a massless rod of length L, which is fixed at one end to the ceiling and free to rotate in a vertical plane. The massive rod is connected to the free end of the massless rod. Suppose an impulse J is applied horizontally to the bottom of the massive rod. Determine the relationship between the magnitudes of the angular velocity ω of the massless rod and Ω of the massive rod immediately after the impulse is applied. The moment of inertia of a uniform rod of 1 md2 . length d and mass m about its center of mass is given by I = 12 L 2L

Two soap bubbles of radii R1 = 1 cm and R2 = 2 cm conjoin together in the air, such that a narrow bridge forms between them. Assuming the system starts in equilibrium, the bubbles are extremely thin, and that air can flow freely between the bubbles through the bridge, describe the evolution and final state of the bubbles.

A particle of mass m moves in the xy plane with potential energy U (x, y) = −k x2 + y 2 . 2 The closest point to the origin (x = 0, y = 0) during its motion was at a distance d, and the particle’s speed at that point was v ̸= 0. Which of the following statements is true regarding the path of the particle after a long time t (t ≫ d/v)?

A particle of mass m moves in the xy plane with potential energy U (x, y) = kxy/2. If the particle begins at the origin, then it is possible to displace it slightly in some direction, so that the particle subsequently oscillates periodically. What is the period of this motion?

Near the ground, wind speed can be modeled as proportional to height above the ground. (This is a reasonable assumption for small heights.) A wind turbine converts a constant fraction of the available kinetic energy into electricity. The conditions are such that when operating at 10 m above the ground, the turbine delivers 15 kW of power. How much power would the same windmill deliver if it were operating at 20 m above the ground?

The International Space Station orbits the Earth in a circular orbit 400 km above the surface, and a full revolution takes 93 minutes. An astronaut on a space walk neglects safety precautions and tosses away a spanner at a speed of 1 m/s directly towards the Earth. You may assume that the Earth is a sphere of uniform density. At which of the following five times will the spanner be closest to the astronaut?

What is the difference in the speed of water between the two pipes?

To measure this difference, two graduated cylinders are connected to the top of the pipe (one in the broad section and the other in the narrowed section). Water then flows up each pipe and the height the water reaches is measured. Estimate the difference in height between the two cylinders.

A student conducted an experiment to determine the spring constant of a spring using a ruler and two different weighing scales. The measured elongation of the spring was 1.5 cm, and the smallest division on the ruler was 1 mm. The mass of the attached weight was measured using two different scales in the school laboratory, yielding values of 198 g and 210 g. The student also found that the local acceleration due to gravity in her city is given as (9.806 ± 0.001) m/s2 . Calculate the percent error in measuring the spring constant.

A massive bead is attached to the end of a massless rigid rod of length L. The other end of the rod is attached to an ideal pivot, which allows it to rotate frictionlessly in any direction. The rod is initially at angle θ to the horizontal, and there is no gravitational force. θ Next, the bead receives an impulse directly into the page, giving it a speed v. How long does it take for the bead to return to its original position?

A puck of mass m can slide on a frictionless inclined plane (prism). The prism has a much greater mass compared to √ the puck and is itself sliding without friction on a horizontal surface. The velocity of the prism is v = 2gh, where g is the acceleration due to gravity and h is the height from which the puck starts sliding on the prism. The transition from the prism to the horizontal surface is smooth. The puck starts from rest relative to the prism. Find the final velocity of the puck once it begins sliding on the horizontal surface. h v

In astronomy, some galactic objects appear to sweep across the sky faster than light speed, c. This effect, called superluminal motion, comes purely from geometry and the finite travel time of light, and it has nothing to do with special relativity. A jet moves from point A to B at speed v = βc. The jet emits a pulse of light at A, and a second pulse a time δt later at B. An observer sees these pulses at point O. The angle between the jet and the line of sight is θ. Assume the angle ϕ is small, so the distances from O to points B and C can be treated as equal. B v ϕ θ O C A Find the apparent transverse velocity, vT , along CB as measured by the observer, in terms of β and θ. Express your answer as βT ≡ vcT β sin θ

A series of dominoes are stood upright in a line. The dominoes have mass density ρ, height h, and spacing d between them. When the first domino is knocked over, a wave of falling dominoes propagates down the line with speed v. Now, a second set of dominoes is set up identically, but with all dimensions (h and d) scaled by λ. The mass density is the same. Let v ′ denote the wave speed of the scaled system. Which of the following best describes how v ′ depends on λ?

A point moves in the xy plane, and its trajectory is shown in the figure below. y x One of the following pairs of graphs shows the time dependence of vx and vy for this motion. Select the correct pair. vy vx t t

Two small balls are launched simultaneously from the same point at some height above horizontal ground. One ball is launched vertically upward with speed 3 m/s, while the other is launched horizontally with speed 4 m/s and lands on the ground at a horizontal distance of 20 m from the launch point. Neglect air resistance. At the moment the second ball lands, how far is it from the first ball?

A helicopter flies at constant height h along a circular trajectory of radius R with constant angular velocity ω, continuously dropping sand. Ignore air resistance. What is the radius of the curve traced by the sand on the ground?

Newton’s Impact Depth Approximation provides an estimate on how far a projectile will penetrate a material at high speeds. If a projectile of length L and density ρA impacts a wall of density ρB , Newton’s approximation states that the projectile will penetrate to a depth of D = L ρρBA . Assuming this is true for all velocities, how does the average force the wall exerts on the projectile F scale with the initial projectile speed v?

The James Webb telescope lies at a point called the Lagrange Point 2 or L2. This is a point where the telescope would remain stationary in a frame rotating with Earth’s orbit around the sun, so a telescope placed exactly at L2 would be in an equilibrium position relative to Earth’s orbit. What is the stability of this equilibrium point with respect to radial and tangential perturbations?

A puck of mass m moves with speed v toward a heavy bump of mass M , where M ≫ m. Both the puck v and provides a and the bump slide without friction. The bump moves toward the puck with speed 2 smooth transition from the horizontal surface onto the bump. The bump is tall enough that the puck slides back down. Find the maximum height h the puck reaches. v 2 m 2

A vertical cylindrical pipe with a sealed bottom contains no air and is tall enough that its top is open to outer space (no atmosphere). A heavy plate is mounted at the bottom of the cylinder and vibrates vertically. A small ball inside the pipe undergoes elastic collisions with the plate. The plate vibrates in such a way that, during each collision, it is moving upward with a constant speed v equal to , where v is the orbital speed at zero altitude above the Earth. The plate moves downward 1000 sufficiently fast that the ball collides with the plate only while the plate is moving upward. The ball is initially dropped from rest from a height equal to the Earth’s radius above the plate. After how many collisions with the plate will the ball escape the Earth’s gravitational field?

Consider the following three solid objects, each of mass M and uniform density: (i) a half-ball of radius a; (ii) a cylinder of radius a and height a; (iii) a cylinder of radius a and height 2a. For each object, consider the gravitational acceleration at the center of its flat circular face. Let gball , ga , and g2a denote these accelerations for the half-ball, the cylinder of height a, and the cylinder of height 2a, respectively. Which of the following is correct?

A circular track of mass m lies flat horizontally on a frictionless table and is free to move on it. A bead also of mass m can slide without friction on the track. The track is initially at rest, and it is labeled every 45° with tick marks. The bead is placed on tick 1 and is given an initial velocity ⃗v0 tangential to the track as shown below. 5 4 6 3 7 O 2 8 1 bead ⃗v0 What is the speed of the track’s center O when the bead reaches tick 3?

A wake surfer of total mass M (surfer plus board) is being towed at a constant horizontal velocity v across a flat lake. The wakeboard has a specific geometry such that, at this speed, it is partially submerged and provides a static buoyant force FB (where FB < M g). To support the remainder of the weight, the board moves with an angle of attack θ relative to the horizontal water surface. Assume that the water exerts a reaction force strictly normal to the bottom surface of the board. Which of the following expressions represents the horizontal tension T in the tow rope required to maintain this constant velocity?

A block of mass m slides along a surface with a coefficient of kinetic friction µk . The block is confined to move between two walls separated by a distance L. Attached to each wall is an ideal, perfectly elastic spring with a very large force constant k. The block is launched from the midpoint between the walls with an initial speed v0 directed toward one of the springs. It bounces back and forth between the springs, losing speed only due to friction with the surface. Assume the time spent in contact with the springs is negligible compared to the travel time between them. Which of the following expressions correctly gives the total time T required for the block to come to a permanent rest?

In the following swing, all lines are rigid rods and their lengths satisfy the relation l12 + l22 = a2 + b2 . What is the period of small oscillation of the mass at point C? a A b l1 B l2 C

A ball launcher fires balls along the floor at some initial speed, applying no rotation to them. The balls initially slip along the floor, then start rolling without slipping. Ignore the potential deformation of the ball and flooring during this process, as well as air resistance. How does the final speed of the rolling ball depend on the coefficient of friction µ between the ball and the floor?

A small puck of mass m slides without friction inside a rigid, uniform triangle rack of mass 5m, whose inner boundary is an equilateral triangle ABC. The rack is initially at rest and can move freely on a frictionless horizontal table. All collisions between the puck and the rack are perfectly elastic. At some instant, the puck strikes the midpoint of side AB from inside the rack. Just before this collision, the puck’s velocity is parallel to side BC. After the collision at the midpoint of AB, the puck collides with the rack again, and then collides once more. At which part of the triangle does this second collision after the initial one occur? (Some answer choices refer to the midpoint D of side AC.) C D ⃗v A B

A ladder is leaning against a vertical wall. The ladder is a uniform rod of mass M and length L, and both the wall and the ground are frictionless. The ladder is released from rest from an almost-vertical position and begins to slide. What √ is the speed of the point of the ladder that is in contact with the floor when it is a horizontal distance 23L away from the wall? √

Three springs and a block of mass m are connected as shown in the figure below. The spring constants are k1 = k, k2 = k/5, and k3 = k/3. All pulleys and ropes are massless, and there is no friction anywhere in the system. Find the ratio of the period of oscillations of this system, T , to the period T0 of a simple T mass–spring system with mass m and spring constant k. The value of is: T0 k1 k2 k3 m The value of TT0 is:

A point mass m1 is attached to a box of mass m2 by a massless, inextensible rope of fixed length L, attached to the center of the box. The box can slide without friction on a horizontal surface. The mass m1 swings freely under gravity, and all motion occurs in the plane of the picture. Neglect air resistance. Find the angular frequency ω of small oscillations about the stable equilibrium. m2 L m1 In case you need the small-angle approximation, you may use 1 cos θ = 1 − θ2 + · · · , 2 r m1 + m2 g

Two uniform rigid rods lie in a vertical plane. The lower rod rests on a rough horizontal table at a single point O and makes angle θ ̸= 0 with the vertical. The rod can pivot about point O without slipping. The second rod is rigidly attached at its center of mass to the top of the lower rod at point P . A motor causes the second rod to rotate in the plane about P . The motor is rigidly attached to the lower rod and does not exert forces or torques on the table. P θ O g table Which of the following types of motion of the second rod could keep the angle θ fixed in time?

A student stands on a large horizontal merry-go-round (R = 2.0 m) at an initial radius of r0 = 1.0 m. Both rotate at constant angular speed ω = 1.2 rad/s. The student wants to get off without walking and performs a sequence of identical vertical jumps of height h = 0.31 m. Ignore air resistance. What is the minimum number of jumps needed for the student to land off the platform? You may assume that the merry-go-round is much more massive than the student and that friction instantly brings the student back into co-rotation with the platform.

A tricycle of mass m = 100 kg is traveling north on a horizontal surface. The geometry of the tricycle is defined as follows: • Wheelbase (distance from front axle to rear axle): L = 1.0 m • Rear track width (distance between rear wheels): d = 1.0 m • Center of mass (CM) location: on the longitudinal symmetry axis, a distance b = 0.4 m forward of the rear axle • Radius of gyration about the CM: k = 0.5 m (The radius of gyration k is defined by ICM = mk 2 , where ICM is the moment of inertia of the tricycle about a vertical axis through its center of mass.) Seeing a patch of ice on their right side ahead, the rider panics and slams on the brakes. The front wheel and the left rear wheel slide on dry pavement with a coefficient of kinetic friction µk = 0.5, while the right rear wheel slides on frictionless ice (µ = 0). This asymmetry in friction forces produces a net torque about the center of mass, causing the tricycle to begin rotating. Calculate the initial angular acceleration α of the tricycle.

Consider the infinitely repeating hexagonal sequence where the largest hexagon has side length 3a and is filled with mass. Inside it, a smaller hexagon √ of side length a is rotated 30◦ clockwise and cut out. Then, an even smaller hexagon of side length a/ 3 is rotated again by 30◦ clockwise and filled. This pattern continues infinitely. Let I1 be the moment of inertia of this 2D figure about an axis through its center of √ mass, perpendicular to its plane P∞and nI2 be1the moment of inertia of a regular hexagon with side length 3a. What is I1 /I2 equal to? ( n=0 r = 1−r for |r| < 1) √a 3 √

A thin sheet of air with density ρ is blown at a uniform speed v toward a smooth, convex cylindrical surface of radius R. The sheet has uniform thickness t ≪ R and remains everywhere tangent to the cylinder while it stays attached. Neglect gravity and viscosity. The pressure on the outer side of the air sheet is constant and equal to P . The pressure on the side adjacent to the cylinder is lower by an amount ∆P . The air sheet follows the curvature of the cylinder until it can no longer remain attached. Given the parameters below, what is the maximum integer speed vmax for which the sheet can remain attached to the surface? P P − ∆P t R v ρ = 1.2 kg/m3 ,

The five vases below are filled with water to the same initial volume and initial height H. Each vase has an identical drainage hole at the bottom. H a) H b) H c) Which vase will drain completely in the shortest amount of time?

A bead on a circular hoop with radius 2 m travels counterclockwise for 10 s and completes 2.25 rotations, at which point it reaches the position shown. 2m In the past 10 s, what were its average speed and the direction of its average velocity? √

A mass on an ideal pendulum is released from rest at point I. When it reaches point II, which of the following shows the direction of its acceleration? I (E)

A soccer ball is kicked up a hill with a flat top, as shown. The ball bounces twice on the hill, at the points shown, then lands on the top and begins rolling horizontally. Which of the following shows the vertical component of its velocity as a function of time? vy vy t

A box of mass m is at the bottom of an inclined plane with angle θ to the horizontal, and height h. h m θ A person drags the box very slowly up the plane, by applying a force parallel to the plane. The coefficient of kinetic friction between the box and plane is µk . When the box reaches the top of the plane, how much work has the person done?

Two blocks of mass m and a block of mass 3m are attached to a system of massless fixed pulleys and massless string, as shown. 3m m m Assume all surfaces are frictionless. What is the acceleration of each mass m?

A ball at the end of a rope of length 0.5 m is swung in a horizontal circle, with a speed of 15 m/s. The other end of the rope is fixed in place. What is the height difference between the ends of the rope?

Two boxes are stacked side-by-side on top of a larger box as shown. m m m F All three boxes have mass m, the coefficient of static friction between the left box and the bottom box is µs , and all other surfaces are frictionless. A rightward force F is applied to the bottom box. What is the minimum value of µs so that the upper boxes don’t slide?

Two stars α and β, with masses satisfying mα /mβ = 10, are in circular orbits around each other. In the rest frame of this system, find the ratio of the speeds vα /vβ .

A helium balloon is released from the floor in a room at rest, then slowly rises and comes to rest touching the ceiling. During this process, the gravitational potential energy of the balloon has increased. Since energy is conserved, the energy of something else must have decreased during this process. Which of the following is the main contribution to this decrease?

An archer takes aim at a target that is 100 m away. Assuming she holds the bow at the same height as the center of the target and shoots an arrow with velocity v = 100 m/s, at what angle above the horizontal should she aim the bow so that the arrow hits the center of the target?

A projectile is thrown from a horizontal surface, and reaches a maximum height h and also lands a distance h from the launch point. Neglecting air resistance, what is the maximum height for a projectile thrown directly upward with the same initial speed?

A block of mass m is initially held in place by two massless strings, as shown. m The tension in the diagonal string is T1 . Next, the horizontal string is cut, and immediately afterward the tension in the diagonal string is T2 . Which of the following is true?

A uniform box with mass m is at rest on a horizontal surface, and the coefficient of static friction between them is µs . A force directed at an angle of 85◦ above the horizontal is applied to the center of the box, with a linearly increasing magnitude F = βt. F The box will eventually slide or lift off the ground. Which of the following is correct?

The object shown below is made of three rigidly connected, identical rods with uniform density. When it stands upright on a horizontal table, what fraction of its weight rests on the left leg?

The following plot shows the speed v(r) at which stars orbit about the center of a galaxy, as a function of their distance r from the center. v(r) r Assuming the galaxy has a spherically symmetric mass distribution, which plot best shows the mass of the galaxy M (r) enclosed within radius r? M (r) M (r)

A bead attached to a string of length ℓ = 10 m is released from a very small angle θ0 to the vertical. A wall is placed in the path of the bead such that the bead collides elastically with the wall when the string is at an angle θ0 /2 to the vertical, as shown. θ0 ℓ m What is the time interval between the bead’s collisions with the wall?

Three physical pendulums are built as shown. The first is a typical pendulum with a massless rope, and the second and third are made of uniform rods. ℓ ℓ m m 2ℓ m What is the correct ranking of the moments of inertia I1 , I2 , and I3 about the pivot points?

Alice, Bob, and Carol are each given identical airtight bags containing identical rocks, and a large tub of water with a scale sitting on the bottom. Each of them measures the weight of their bag and rock by putting the bag on the scale, using three slightly different procedures. • Alice closes the bag carefully, so that there is no air inside. • Bob fills the rest of the bag with water before closing it. • Carol closes the bag loosely, so that it contains some air. What is the correct ranking of their measured weights WA , WB , and WC ?

A student sets up a simple pendulum, measures its length to be (0.50 ± 0.01) m, and observes a period of oscillation of (1.4 ± 0.1) s. Using this data, the student computes g = 10.1 m/s2 . What is the uncertainty of this measurement?

A mass attached to the end of a string oscillates like a pendulum with small amplitude. The mass has horizontal velocity vx (t) and vertical velocity vy (t). Which of the following could be a graph of the curve (vx (t), vy (t)) over a complete oscillation? vy (t)

A smooth ring of radius R and mass m lies on a frictionless surface. A point mass, also of mass m, is placed just inside the ring and given a speed v tangent to the inner surface of the ring. How long does it take for the point mass to return to its initial position relative to the ring? v πR

Two springs with different spring constants are connected in three ways, as shown. kA kB kC In the second case, the springs are connected to opposite ends of a string, which runs under a massless frictionless pulley. In each case, the two springs act like a single spring with an effective spring constant kA , kB , or kC . Which of the following is correct?

An astronomer on Earth, which is a distance L⊙ from the Sun, observes a star and galaxy. The star is a distance Ls ≫ L⊙ away, and the galaxy is much further away than the star. Throughout the year, the angular distance between the star and galaxy appears to vary, reaching a minimum of 2 arcseconds on January 1st and a maximum of 2.5 arcseconds on July 1st. (One degree is equal to 3600 arcseconds.) Assume the Sun, star, and galaxy do not move relative to each other, and that the Earth’s orbit lies within their plane. What is the ratio Ls /L⊙ ? 2 arcsecond 2.5 arcsecond galaxy star star galaxy

A very heavy plate continuously moves up and down with a small speed v, switching directions after each time t. At a random time, a ball is dropped from rest far above the plate, then bounces elastically off of it. How does the ball’s speed change during this collision?

A skier slides from rest along a frictionless slope with incline angle θ = 30◦ for 150 m, then smoothly transitions into a horizontal jumping ramp of height hr = 5 m. Neglecting air resistance, after the skier leaves the ramp, about how far will they travel horizontally before landing? 150 m hr = 5 m θ=3 ◦ 0

A projectile is thrown upward with speed v. By the time its speed has decreased to v/2, it has risen a height h. Neglecting air resistance, what is the maximum height reached by the projectile?

A car is moving at 60 miles per hour (mph), when the driver notices an obstacle ahead. Hitting the brakes, the driver decelerates at a constant rate, and manages to come to a stop just barely before hitting the obstacle. If the car had instead been moving at 70 mph, and started decelerating at the same place and at the same rate, with what speed would it have hit the obstacle?

Two blocks of mass m have an inelastic one-dimensional collision. Initially, the first block is moving with speed 5 m/s, and the second is at rest. After the collision, the first block is moving with speed 2 m/s. What percentage of the system’s original kinetic energy was lost during the collision?

A mass on an ideal pendulum is released from rest at point I. It swings over to point II, at which point the string suddenly breaks. Which of the following shows the trajectory of the mass? I II

A uniform solid ball with mass m = 1 kg and radius R = 10 cm rolls without slipping on a horizontal plane, so that its center of mass has velocity v = 1 m/s. What is the ball’s total kinetic energy?

A bob of mass m hangs from a rigid, massless rod, forming an ideal pendulum. The rod is held horizontally and released from rest. What is its maximum tension during its swing?

The following graph shows the results of measurements of two physical quantities, y and x. What is the following best describes the functional dependence of y on x? Below, A and B are positive constants. 100 y 10 1

A block of mass m is placed on a wedge of mass m, inclined at an angle θ to the horizontal. m m θ The coefficients of friction between the block and wedge, and the wedge and ground, are high enough for both the block and the wedge to remain static. What is the magnitude of the friction force of the ground on the wedge?

A person is holding a massless rope, on which hangs a mass m, as shown at left. To pull the end of the rope with constant upward velocity v, the person must exert a force Fv . To pull the end of the rope with constant upward acceleration a, the person must exert a force Fa . Now the rope is wrapped around a fixed, massless pulley, and the mass is doubled to 2m, as shown at right. F F m 2m Compared to the original setup, how do the forces Fv and Fa needed to pull the end of the rope with a given upward velocity and acceleration change? In both cases, ignore friction and air resistance.

The two ends of a uniform rod of length 2L are hung on massless strings of length L. L L 2L If the strings are attached to the ceiling, and the rod is pulled a small distance horizontally and released as shown, what is the period of oscillation? r

Two identical spherically symmetric planets, each of mass M , are somehow held at rest with respect to each other. Each planet has radius R, and the distance between the centers of the planets is 4R. If a rocket is launched from the surface of one planet with speed v, what is the minimum speed v so that the rocket can reach the other planet? r

A pulley is constructed by attaching two concentric cylinders, with the larger cylinder having twice the radius. Ropes are wrapped around both cylinders, a mass m is hung from each rope, and the system is released from rest. m m Neglect the masses of the cylinders and ropes. Each mass experiences both a gravitational and a tension force. If the net force experienced by the left mass is F1 , and the net force experienced by the right mass is F2 , what is the ratio F2 /F1 ?

Consider a laptop made of two identical uniform plates, each of mass m/2, connected by a hinge. The hinge is locked when the screen makes an angle θ to the vertical, as shown, fixing the angle between the two pieces. θ Assuming the laptop does not slip, what is the minimum force that can be exerted on the top of the laptop, in the plane of the page, to cause the bottom of the laptop to lift off the ground? mg(1 − sin θ) 2 mg(1 + sin θ) (D) 4

A small block is released from rest on the rim of a fixed, frictionless hemispherical bowl. From the time the block is released, until it reaches the bottom of the bowl, which of the following is true? I. The speed of the block never decreases. II. The magnitude of the horizontal component of the velocity of the block never decreases. III. The magnitude of the vertical component of the velocity of the block never decreases.

An egg is launched with speed v from the ground, a distance d from a vertical wall. v h θ d If v is high enough for the egg to hit the wall, which of the following could describe the angle θ that maximizes the height h at which the egg hits the wall?

A hexagonal pencil of uniform density lies at rest on a horizontal table. It is pushed horizontally with a steadily increasing force halfway up its height, as shown. F What is the minimum value of the coefficient of static friction between the floor and pencil, so that the pencil will eventually begin to roll?

A thin rod has a nonuniform density. It is mounted on an axle passing perpendicular to it, through its center of mass, as shown, and is then rotated about the axle. CM ω The axle divides the rod into two parts, one on each side of it. Which of the following must be true, no matter how the mass in the rod is distributed?

A cylindrical piece of cork of density ρc , height hc , and cross-sectional area Ac is in a larger empty cylindrical container of cross-sectional area Aw . Water of density ρw > ρc is slowly poured into the empty container. What is the height of the water in the container when the cork starts to float?

A toy elephant is standing on the bottom of a fish tank. The fish tank is filled with water to a depth of 10 cm, completely covering the toy. The elephant’s legs are perfectly polished, so that there is no water between the bottom of the legs and the tank’s floor, and the total area of contact is 0.16 cm2 . The water has density ρ = 103 kg/m3 , the toy has uniform density 2ρ, the atmospheric pressure is Patm = 105 Pa, and the toy has total mass 120 g. What is the total hydrostatic force that the water exerts on the toy? 10 cm

A bead is threaded on a frictionless wire and launched horizontally from height h with speed v0 , as shown. If the shape of the wire is steep, as in curve I, then the normal force from the wire on the bead will point inward. If it is shallow, as in curve II, then the normal force will point outward. v0 N h N I II d There is exactly one possible shape of wire, shown as a dotted line, for which the normal force of the wire on the bead is always equal to zero. What is the horizontal displacement d of the bead when it travels along this wire? r

A cork floating in a cup filled with a viscous fluid is placed in an elevator. Below is a plot of the velocity v of the elevator as a function of time t. Which of the following plots best describes the height h of the cork in the cup as a function of time? Assume that the fluid is viscous enough to dampen all oscillations, that the fluid does not slosh as the elevator accelerates, and that both the cork and fluid are incompressible. v t h

A block of mass 2m is placed symmetrically on two identical wedges of mass m, as shown. 2m θ m θ m All surfaces are frictionless, and the wedges have angle θ to the vertical. If the system is released from rest, what is the downward acceleration of the block?

For objects moving through air, the force of air resistance can be modeled as proportional to the speed (“linear drag”) or proportional to the square of the speed (“quadratic drag”), depending on the circumstances. Two identical objects, A and B, are dropped from the same height h simultaneously, but object A is given an initial horizontal velocity v. The objects hit the ground at times tA and tB . Accounting for air resistance, which of the following is true?

A satellite is in orbit around a planet of mass M . q Its maximum distance from the center of the planet is d, and at this point, it is traveling at a speed of 21 8

A cylinder is placed with its axis vertical, and a rubber band of mass m and tension T is wrapped horizontally around it. What is the minimum coefficient of static friction µ between the rubber band and the cylinder such that the band will not slide down the cylinder?

A ball is held a height h above a slope, which is at an angle 45◦ from the horizontal. The ball is dropped from rest. Assume the ball bounces off the slope perfectly elastically. What is the distance between its first and second impact points?

A massless wheel of radius R has four small masses M placed evenly along its rim. Let the moment of inertia for rotations about the center of wheel, in the plane of the wheel, be I0 . I1 I2 m 45° m m m Now consider rotations about the two axes shown, with corresponding moments of inertia I1 and I2 . Which of the following is true?

Four evenly spaced points are marked on a massless seesaw, as shown. 7 kg 5 kg I II III IV Two blocks, of masses 5 kg and 7 kg, are placed on the seesaw, so that it balances when the fulcrum is at point IV. (The diagram is not drawn to scale.) Now suppose the fulcrum is moved to point II. How much mass should be placed at point I so that the seesaw again balances?

A firework explodes, sending shells in all directions in a vertical plane, as shown. g Suppose the shells are all launched with the same speed, and ignore air resistance, but not gravity. A long time later, but before any of the shells hit the ground, what shape do the shells make?

A swimmer swims at speed v relative to still water. A river flows from a pier to a lake, with speed u. If the swimmer swims downstream from the pier to the lake, then back upstream, what was their average speed during the trip?

A block of mass m is at rest on a frictionless table. It is pushed horizontally by a constant force F , and has total momentum p when it reaches the end of the table. If a block of mass 2m is pushed across the table in the same way, also starting from rest, what is its momentum when it reaches the end of the table?

Two blocks of unequal mass M and m are hung from the ends of a massless, frictionless pulley, as shown. The blocks are held in place, and the entire pulley is mounted on a sensitive scale. scale M m After the blocks are released from rest, but before either has fallen off the pulley, what is the scale reading?

A ball is launched with speed v at an angle θ to a fixed ramp, which itself makes a 45◦ angle with the horizontal. v θ 45° Assume that when the ball hits the ramp, it bounces elastically and frictionlessly. For what value of θ will the ball bounce off the ramp two times, and then return to its original launch point?

Billy is leaning on a box of mass 30.0 kg, exerting a force 35.0◦ below the horizontal. If the coefficient of static friction of the box on the ground is µs = 0.400, what is the minimum force needed for the box to slide?

Two springs with spring constants 2 N/cm and 4 N/cm are connected in parallel. They are both connected in series to a spring of constant 3 N/cm, as shown. 2 N/cm 3 N/cm 4 N/cm What force must be exerted at the end to extend the system by 1 cm from its relaxed length?

Water with total mass M is poured into a cup of cross-sectional area A. A block of mass m, whose density is half that of water, is tied to a thin string. The string is attached to the bottom of the cup, and the block floats in the water as shown. The atmospheric pressure is Patm . What is the total pressure force that the water exerts on the bottom of the cup? M m

Two identical bricks of mass m are attached to two frictionless pulleys by a massless string, on a fixed slope of angle θ to the horizontal, as shown. m m θ Suppose that the coefficient of static friction between the lower block and the slope is µ, and all other surfaces are frictionless. What is the minimum value of µ so that the blocks can stay static?

A ballistic pendulum is designed as shown below. ℓ m v m r A uniform ball of mass m and radius r is attached to a massless rigid rod, so that its center is a distance ℓ from the ceiling. The ball has a small tunnel hollowed out. A small block of mass m and speed v goes into the tunnel and collides at the center of the ball, perfectly inelastically. Subsequently, to what maximum height does the block rise?

A cylinder of water is suspended in a space station. Under the influence of surface tension, the cylinder splits into droplets. After a short time, viscosity causes the droplets to settle into static shapes. Neglect the effect of evaporation. Compared to the original cylinder, the final set of droplets will have

Five astronauts are orbiting Earth in a low circular orbit. They decided to go around the planet following the same orbit, but much faster, to set a new record for orbiting Earth. Four of them thought that they could quickly increase the velocity, then use a jet engine to maintain an orbit of the same shape; in the first four choices below, the red arrow denotes the direction that fuel will flow out of the jet nozzle. The fifth astronaut was skeptical about such a possibility. Who is right?

A uniform rectangular block is in static equilibrium on an inclined plane, and experiences gravity, static friction, and the normal force from the plane. Though gravity acts on the entire volume of the block, for the purposes of torque balance, it is equivalent to a single resultant force acting at the block’s center of mass. Similarly, while the normal force acts on the entire bottom surface of the block, its torque is equivalent to a resultant force acting at a single point. Which of the following best shows this point? E B C D A

A box of mass M is sliding to the right with velocity v on a frictionless table. A small puck of mass M/2 slides frictionlessly inside the box. Initially, the puck is at the left wall of the box, with a rightward velocity of 2v with respect to the table. After a time T , the puck collides elastically with the right wall of the box. How much longer will it take until the puck hits the left wall of the box again? M v M 2

The following information applies to problems 18 and 19. A uniform rod of length L and mass M is placed with its ends resting on two identical springs of spring constant k, as shown. L k M k The rod is initially in equilibrium. If the rod is uniformly displaced downward and released from rest, what is the frequency f of its oscillations? 1

Next, the rod is brought back to equilibrium. It is slightly rotated about its center of mass, then released from rest. L M k k What is the frequency f of its oscillations? 1

A uniform, hollow sphere is initially at rest on a horizontal plane. The plane begins to accelerate horizontally to the right, with acceleration a. If friction is high enough to prevent the sphere from slipping, what is its translational acceleration?

Amora and Bronko are given a long, thin rectangle of sheet metal. (It has been machined very precisely, so they can assume it is perfectly rectangular.) Using calipers, Amora measures the width of the rectangle as 1 cm with 1% uncertainty. Using a tape measure, Bronko independently measures its length as 100 cm with 0.1% uncertainty. Which of the following are closest to the relative uncertainties they should report for the area and the perimeter of the rectangle, respectively?

A cannon just above of a spherical planet with mass M and radius R launches a particle with p the surface p speed v, where GM/R < v < 2GM/R. The initial velocity of the particle makes an angle θ with the vertical direction. Neglect drag. For what θ will the particle never collide with the planet?

A mass attached to a spring is performing simple harmonic motion, with velocity v(t) and acceleration a(t). Which of the following could be a graph of the curve (v(t), a(t)) over a complete oscillation? a(t) a(t) v(t)

A ramp with height h is moving with fixed, uniform speed v to the right. A small block of mass m is placed at the top of the ramp, and is released at rest with respect to the ramp. m h v The block slides smoothly to the bottom of the ramp and onto the floor. How much kinetic energy does it gain in this process? Neglect friction.

Two masses are initially at rest, separated by a distance r, and attract each other gravitationally. If their masses are m and 2m, then they will collide after a time T . How long would they take to collide if they both had mass 2m?  3/2 2

The ball starts at point A, turns around at point B, and returns to point A. Which of the following shows the speed of the ball as a function of time?

Which of the following shows the horizontal velocity of the ball as a function of time?

Two massless rods are attached to frictionless pivots, with their ends touching. The distances between the pivot points and the endpoints of the rods are shown below. F 0.5 m 2m 1m 2m F0 Neglecting friction between the rods, if a force F is applied at the left end of the left rod, what force F 0 must be applied at the right end of the right rod to keep the system in equilibrium?

Alice and Bethany stand side by side on the Earth’s equator. If Alice jumps directly upward, in her frame of reference, to a small height h much less than the radius of the Earth, she will land a distance D to the west of Bethany. If Alice had instead jumped to a height 2h, how far to the west of Bethany would she land? Neglect air resistance. √

A train starts from city A and stops in city B. The distance between the cities is s. The train’s maximal acceleration is a1 and its maximal deceleration is a2 (in absolute value). What is the shortest time in which the train can travel between A and B? r s

A cylindrical bucket of negligible mass has radius R and height h, and is open at the top. It is submerged in water of density ρ, with its top a distance H below the surface. How much work is needed to pull the bucket slowly up so that its bottom is just above the lake surface? H h R

A point mass slides with speed v on a frictionless horizontal surface between two fixed parallel walls, initially a distance L apart. It bounces between the walls perfectly elastically. You move one of the walls towards the other by a distance 0.01L, with speed 0.0001v. What is the final speed of the point mass?

A mint produces 100,000 coins. Upon weighing some of them with a precise scale, officials find that the coins vary slightly in weight, with an independent uncertainty of 1%. About how many coins must be randomly sampled and weighed in order to determine the total weight of the coins to within 0.1% uncertainty? Assume there are no sources of systematic uncertainty.

NASA trains astronauts to experience weightlessness with an airplane which flies in a parabolic arc with constant acceleration g toward the ground. The plane can remain on this trajectory for at most 25 seconds, due to the large change in altitude required. If instead of simulating weightlessness, NASA wanted to fly a trajectory that would simulate the gravitational acceleration of Mars 3.7 m/s2 , for what length of time can the plane simulate Mars gravity? Assume that the maximum change in altitude is the same for both trajectories.

A uniform solid circular disk of mass m is on a flat, frictionless horizontal table. The center of mass of the disk is at rest and the disk is spinning with angular frequency ω0 . A stone, modeled as a point object also of mass m, is placed on the edge of the disk, with zero initial velocity relative to the table. A rim built into the disk constrains the stone to slide, with friction, along the disk’s edge. After the stone stops sliding with respect to the disk, what is the angular frequency of rotation of the disk and stone together?

Projectiles A, B, and C are simultaneously thrown off a cliff, and take the trajectories shown. A B C Neglecting air resistance, rank the times tA , tB , and tC they take to hit the ground.

An object of mass m = 1 kg is attached to a platform of mass M = 4 kg with a spring of spring constant k = 400 N/m. There is no friction between the object and the platform, and the coefficient of static friction between the platform and the ground is µ = 0.1. The object is placed at its equilibrium position, and then given a horizontal velocity v. v k m M For what v will the platform never slip on the ground?

A car is driving on a semicircular racetrack. Its velocity at several points along the track is shown below. Which of the following could depict its acceleration at the corresponding points?

A bacterium cell swims by rotating its bundle of flagella to counter a viscous drag force in the medium. The drag force F (R, v, η) only depends on the typical length scale of the cell R, its speed v, and the viscosity of the fluid η, which has units of kg/(m · s). It is observed under a microscope that a cell of length 1 µm swims at about 20 µm/s. Estimate the speed of a cell of length 0.5 µm, assuming cells of all sizes generate the same amount of force from their flagella.

A block is released from rest at the top of a fixed, frictionless ramp with horizontal length L and inclination θ. θ L For a fixed value of L, which value of θ minimizes the time needed for the block to reach the bottom of the ramp?

A particle begins at the point x = 0 and moves along the x-axis with initial rightward velocity v0 . Later, the particle reaches the point x = L. The velocity as a function of position during this time interval is shown below. v v0 x L Consider the following three statements. I. The particle instantaneously stops at x = L. II. The particle is uniformly accelerated. III. The particle could be performing simple harmonic motion. Which of these statements are true?

In a certain country, the short hand of a clock is exactly half as long as the long hand, and rotates twice for each rotation of the long hand. A B C The three points shown on the clock hands have accelerations of magnitude aA , aB , and aC . The point B is at the midpoint of the long hand. Which of the following is true?

A factory’s smokestack continually produces smoke. The smoke always rises with a constant speed relative to the air around it. Suppose that the air was initially still, then abruptly started blowing to the right, then abruptly became still again for some time. Which of the following could show the resulting shape of the smoke plume?

A cavity of radius R/2 is dug out of a spherical planet with uniform mass density of mass M and radius R. What is the magnitude of the gravitational field at point P in the diagram below? P R R/2

The line between day and night on a planet or moon is called the terminator. How fast does the terminator of Earth’s moon move across its surface at the equator? The radius of the moon is 1.74 × 106 m.

A solid sphere sits at the top of a ramp of height h inclined at angle θ to the horizontal. Both the static and kinetic coefficients of friction between the sphere and incline are µk = µs = 0.2. The sphere is released from rest at the top of the incline. For which of the following values of θ is the total translational plus rotational kinetic energy of the sphere greatest when it reaches the bottom of the incline?

A ship rams into a hunk of ice floating in the sea. “Icebreakers” are ships designed so that when this happens, the ice is pushed down beneath the ship. If the coefficient of static friction between the ice and the ship is µs , what condition applies to the angle θ, as shown in the figure, so that the icebreaker functions as intended? ship θ ice

An imperfect spring has a restoring force F that depends on the displacement x from equilibrium as in the graph shown below. F x −d d The slope of the curve for x < −d and x > d is a constant −k. A mass m is attached to the spring and released from rest at the position x = A, where A > d. What is the period of the subsequent motion? r   m 2d

Two satellites are in circular orbits around a star with equal radius r, speed v, and period T . The satellites are initially diametrically opposite each other. In order to meet the second satellite in time T /2, the first satellite should decrease its speed to approximately

Spaceman Fred’s trusty pellet sprayer is held at rest a distance h away from the center of Planet Orb, which has radius R h. The pellet sprayer ejects pellets radially outward, uniformly in the plane of the page. These pellets are all launched with the same speed v, so that a pellet launched directly away from Orb by the pellet sprayer can just barely escape it. What fraction of the pellets eventually lands on Orb? (Hint: you may use the small angle approximation, sin θ ≈ θ for θ 1, where θ is in radians.) v h R Left: the pellet sprayer relative to Orb Right: a close-up of the pellet sprayer 1

A ball is launched straight toward the ground from height h. When it bounces off the ground, it loses half of its kinetic energy. It reaches a maximum height of 2h before falling back to the ground again. What was the initial speed of the ball?

A rigid ball of radius R is rolling without slipping along the rib of a right-angle chute, as shown at left. A cross section of the ball, taken perpendicular to the ball’s direction of travel, is shown at right. Which of the marked point(s) of the ball have the highest speed? Q O A side (3d) view P B cross-section (2d) view

When an axe is swung with kinetic energy E directly at a piece of wood, the edge of the axe is buried a depth L into the wood. If the axe is swung with kinetic energy 2E, how deep will it be buried into the wood? Assume that the axe is wedge-shaped with a constant angle and that the force per unit contact area between the axe and the wood during the impact is proportional to the depth.

Four identical rods, each of mass m and length 2d, are joined together to form a square. The square is then spun around its center, as shown in the figure, at an angular frequency of ω. What is the magnitude of the force that the joints between the rods (at the corners of the square) must bear? ω

A pendulum of length L oscillates inside a box. A person picks up the box and gently shakes it horizontally with frequency ω and a fixed amplitude for a fixed time. The final amplitude can be maximized if ω satisfies p

A planet is orbiting a star in a circular orbit of radius r0 . Over a very long period of time, much greater than the period of the orbit, the star slowly and steadily loses 1% of its mass. Throughout the process, the planet’s orbit remains approximately circular. The final orbit radius is closest to

An astronaut standing on the exterior of the international space station wants to dispose of three pieces of trash. They face the station’s direction of travel with the Earth to their left. From the astronaut’s perspective, the three pieces are thrown (I) left, (II) right, and (III) up. To the astronaut’s frustration, some of the pieces of trash return to the space station after several hours. They are

The velocity versus position plot of a particle is shown below. Which following choices is the correct acceleration vs. position plot of the particle? v x a a

A block of mass m is attached to a massless string. The string is passed over a massless pulley and the end of the string is fixed in place. The horizontal part of the string has length L. Now a small mass m is hung from the horizontal part of the string, and the system comes to equilibrium. (Diagram not necessarily to scale.) −→ Neglecting friction everywhere, the tension at the end of the string is

During this process, the block has been raised by approximately a height

The maximal tension per area a material can sustain without failure is called its tensile strength. Plain steel has a tensile strength of 415 MPa. What is the maximal mass one can hang on a vertical steel rod of negligible mass and a diameter of 2 cm?

A point mass m is glued inside a massless hollow rod of length L at an unknown location. When the rod is pivoted at one end, the period of small oscillations is T . When the rod is pivoted at the other end, the period of small oscillations is 2T . How far is the mass from the center?

A ballerina with moment of inertia I is quickly twirling with angular velocity ω. In her hand she has a pen of mass m at a radius R from her axis of rotation. The ballerina releases the pen. Afterward, what happens to the vertical component of the angular momentum of the system consisting of the ballerina and the pen? You may ignore all friction, but not gravity or normal forces.

Two blocks of mass m are placed on top of each other, and the bottom block is placed on the ground. The ground is frictionless. The static and kinetic coefficients of friction between the two blocks are µs and µk , with µs < µk . The blocks are at rest initially. When a constant horizontal force F is then applied to the bottom block, which of the following graphs could show its acceleration as a function of F ? m m F a a

As shown in the figure, a vessel contains two types of liquid: the liquid with density ρ1 on top and ρ2 on the bottom. The depth of the top liquid is h1 , and the interface area between the top and the bottom liquid is s1 . The bottom liquid has a depth of h2 . The area of the bottom of the vessel is s2 . What is the gauge pressure (i.e. pressure in excess of atmospheric pressure) at the bottom of the vessel? h1 s1 h2 s2

Liquid droplets store a given amount of potential energy per unit surface area, due to their surface tension. When two identical, nearly spherical liquid droplets coalesce on a certain type of surface, part of this energy can be converted into upward kinetic energy, causing the coalesced droplet to jump. Assuming the conversion is 100% efficient, how does the maximum height h depend on the radius r of the initial droplets?

Paul the Giant stands outside on a force-meter calibrated in Newtons, which reads 5000 N. Paul is wearing a large cowboy hat, which has horizontal cross-sectional area A = 1 m2 and completely covers both him and the scale when seen from directly above. At time t = 0, rain begins to fall vertically downward on Paul, and any rain that hits his hat is collected in the hat’s brim. The raindrops have a constant downward speed of 1 m/s, and the rain accumulates on the ground at a rate of 1 mm/s. What is the reading (in N) on the scale as a function of the time t > 0 (in s)? The density of water is 1

A massless rigid rod is pivoted at one end, and a mass M is at the other end. Originally, the rod rotates frictionlessly about the pivot with a uniform angular velocity such that the mass M has speed v. The rotating rod collides with another mass M at its midpoint, which then sticks to the rod. After the collision, what is the kinetic energy of the system? v M M 1 M v2 4 1 (B) M v 2 3 7 (C) M v2 18 2 (D) M v 2 5 1 (E) M v 2 2

A system of cylinders and plates is set up as shown. The cylinders all have radius r, and roll without slipping to the right with angular velocity ω. What is the speed of the top plate? plate plate ω r plate floor

A car is driving against the wind at a constant speed v0 relative to the ground. The wind direction is always opposite to the car’s velocity, but its speed fluctuates about an average speed of v relative 2 , where A is a constant and v to the ground. The air drag force is Avrel rel is the relative speed between the car and the wind. What is the average rate P of energy dissipation due to the air resistance?

A circular table has radius R and N > 2 equally spaced legs of length h attached to its perimeter. Suppose the table has a uniform mass density with total mass m, and neglect the mass of the legs. Assuming the table does not slip, the minimum horizontal force needed to tip over the table is mgR h   mgR N −2 (B) sin π h 2N π mgR (C) cos h N  mgR N −2 (D) tan π h 2N  π  mgR sin (E) h 2N

A collision occurs between two masses. In each inertial reference frame, one can compute the change in total momentum ∆P and the change in total kinetic energy ∆K due to the collision. Which of the following is true?

Steve determines the spring constant k of a spring by applying a force F to it and measuring the change in length ∆x. The tools he uses to measure F and ∆x both have a constant absolute uncertainty, leading to an uncertainty in k of δkS . If Tiffany measures the same spring constant with the same tools but by using a force that is five times larger, what will her uncertainty in k be in terms of δkS ?

A mass m is connected to one end of a zero-length spring with spring constant k. The other end of the spring is connected to a frictionless bearing mounted around a horizontal pole so that the mass can swing in a vertical circle of radius R around the pole. The setup is shown in the figure below. What is the vertical distance h between the center of the circular orbit and the axis of the pole? Assume that both the diameter of the pole and the rest length of the spring are negligible compared to R. h p mgR/k p (B) R (R + mg/k)/(R − mg/k)

A ball of negligible radius and mass m is connected to two ideal springs. Each spring has rest length `0 . The springs are connected to the ball inside a box of height 2`0 , and the ball is allowed to come to equilibrium, as shown. Under what condition is this equilibrium point stable with respect to small horizontal displacements? k1 m k2

A ball is bouncing vertically between a floor and ceiling, which are both horizontal and separated by 4 m. All collisions are perfectly elastic, and when the ball hits the floor, it has a speed of 12 m/s. How long does a complete up-down cycle take?

A uniform rod of mass m and length ` has moment of inertia m`2 /12 about an axis that is perpendicular to the rod and passes through its center. What is the moment of inertia of a uniform square plate with mass M and side length L about the axis along its diagonal?

A conical pendulum of length d swings in a horizontal circle of radius r, as shown. If ω is the angular frequency of this motion, what is ω 2 ? d r

In “zero-g” airplane rides, passengers can float around the cabin, as if they were weightless. A flight trajectory for such a ride is shown below, with a few points during the journey labelled. Which of the following statements is correct? Take a to be the acceleration of the plane during the “zero-g” part of the flight. a b c d time e

A mote of dust is initially located at distance R from the sun, which has mass M . At this point, the mote has a small tangential velocity v. Which of the following is a good approximation for the distance of closest subsequent approach between the mote and the sun? M R v R3 v 4 G2 M 2 R3 v 4 (B) 2G2 M 2 R2 v 2 (C) 2GM R2 v 2 (D) GM 2R2 v 2 (E) GM

A three-legged table is shaped like a uniform equilateral triangle, and has identical legs at each corner. When the mass of an object placed on the center of the table exceeds mmax , the table’s legs will all simultaneously break. Which of the following shaded regions shows the area within which an object of 2mmax /3 can be placed without breaking any legs of the table?

Two satellites are initially in identical circular orbits around the Sun, with orbital speed v = 1 × 104 m/s. The first satellite fires its thrusters toward the Sun, and quickly obtains a radial velocity of ∆vr = 1 m/s. The second satellite instead fires its thrusters behind it, and quickly increases its tangential velocity by ∆vt . If the two satellites subsequently perform orbits with the same period, approximately what was ∆vt ?

A conveyor belt is moving with velocity v to the east. A block with velocity v to the south slides from the ground onto the conveyor belt. The coefficient of friction between the block and the belt is µ. The block stops slipping after a time v

Two equal masses m are connected by an elastic string that acts like an ideal spring with spring constant k and unstretched length l. The two masses are hung over a frictionless pulley. What is the total length of the string at equilibrium? (Diagram not necessarily to scale.) m m

The two masses are both displaced downward by a small vertical distance x and simultaneously released from rest. What is the period of oscillation? p

An escalator can carry passengers up a vertical distance of 10 m in 30 s. A mischievous person of mass 50 kg walks down the up-escalator so that they stay in place with respect to the building. If the child does this for 30 s, the total work the child performs on the escalator, in the frame of the building, is

A platform juts out horizontally from the edge of a building. If the platform is modeled as a uniform metal rod, which one of the following statements about the tensile and compressive stress is correct?

A uniform disk of mass m and radius r is attached at its edge to a flexible pivot on the ceiling. It is given a small displacement perpendicular to the plane of the disk, so that it begins to oscillate perpendicular to the plane of the disk. What is the period of oscillation? The moment of inertia of a disk about the axis going through its center and perpendicular to the plane it’s in is Idisk = 21 mr2 p 2r/5g p (B) π 5r/g p (C) π 6r/g p (D) 2π r/g p (E) 2π 2r/g

A large block of mass 5 kg is moving to the right at a velocity of 10 m/s. Spaced out every meter are smaller, initially stationary blocks of mass 1 kg. All collisions are perfectly inelastic. Neglecting friction, how far will the large block travel before its velocity has decreased to below 3 m/s?

In the device shown, blocks of various masses are placed on pistons so that the device is in equilibrium. (The fluid in the drawing is to scale.) There are valves that are both initially open at locations A and B. One of the valves is closed, and the system is allowed to come to equilibrium. How will this affect the height of the mass m2 ? m1 m2 A m3 B

A mass M sits on top of a vertical spring of spring constant k, in equilibrium. A mass m is held a height h above it. The mass M is then pushed downward by a distance ∆x, and both masses are released from rest simultaneously. For what value of h will the two masses first collide when M first returns to its equilibrium position? πM 2 g 2 4k 2 ∆x 8M g (B) 2 π k Mg (C) k π2M g (D) 8k πk∆x2 (E) 4M g

Assume that h takes the value found in the previous question, and that the collision between the two masses is perfectly elastic. For what value of ∆x will m rebound to a maximum height that is exactly equal to its original height? r πg m3

An extendable arm is made from rigid beams free to pivot around the dots shown. Spring 1, with equilibrium length L1 , is attached between points A and C while spring 2, with equilibrium length L2 is attached between B and D. The system is allowed to come to equilibrium. In equilibrium, what is the ratio of the tension in spring 1 to the tension in spring 2? spring 2 A B C D A B C spring 1

Consider an axially symmetric object that experiences no external forces and initially rotates about its symmetry axis. The object then changes its shape while remaining axially symmetric. Afterward, it is found that its moment of inertia about the symmetry axis has increased. How have its kinetic energy T and angular velocity ω changed?

A well-calibrated scale reads zero when nothing is placed on it. When a deflated balloon is placed on the scale, it reads mg. When a non-airtight box is placed on the scale, it reads M g, where M > m. The balloon is inflated with helium, so that it floats upward in air. The balloon is placed in the box, and the box is placed on the scale. If the scale reads W , which of the following is true?

A person stands on the seat of a swing and squats down, so that the distance between their center of mass (CM) and the swing’s pivot is `. As the swing gets to the lowest point, the speed of their CM is v. At this moment, they quickly stand up, and thus decrease the distance from their CM to the swing’s pivot to `0 . Immediately after they finish standing up, their CM speed is v 0 . Which of the following statements is correct? You may neglect friction, the change in moment of inertia of the person about their CM, and the time taken to stand up.

A point mass m sits on a long block, also of mass m, which rests on the floor. The coefficient of static and kinetic friction between the mass and the block is µ, and the coefficient of static and kinetic friction between the block and the floor is µ/3. An impulse gives a horizontal momentum p to the point mass. After a long time, how far has the point mass moved relative to the block? Assume the mass does not fall off the block. 3p2 8m2 µg 15p2 (B) 32m2 µg 9p2 (C) 16m2 µg

Assume that the drag force for a fish in water depends only on the typical length scale of the fish R, its velocity v, and the density of water ρ. A pufferfish is about 10 cm in length and swims at about 5 m/s. How fast does a clown fish, about 1 cm in length, need to swim such that it experiences the same drag force as the pufferfish?

Three boxes A, B and C lie along a straight line on a horizontal frictionless surface, as shown. Box A is initially moving to the right with speed v while the other two boxes are initially at rest. If all collisions are elastic, and the masses of the boxes can be chosen freely, which of the following is closest to the maximum possible final speed of box C? A B C

If a certain radioactive decay process happens n times per hour on average, then in any given hour, √ one expects to observe n decays with an uncertainty of n. Each hour can be assumed independent of the previous one, and n can be assumed constant over time. How many hours do you need to conduct the observation so that you can determine n within an uncertainty of 1%?

A car drives anticlockwise (counterclockwise) around a flat, circular curve at constant speed, so that the left, front wheel traces out a circular path of radius R = 9.60 m. If the width of the car is 1.74 m, what is the ratio of the angular velocity about its axle of the left, front wheel to that of the right, front wheel, of the car as it moves through the curve? Assume the wheels roll without s

A 3.0 cm thick layer of oil with density ρ = 800 kg/m3 is floating above water that has density o ρ = 1000 kg/m3. A solid cylinder is floating so that 1/3 is in the water, 1/3 is in the oil, and w 1/3 is in the air. Additional oil is added until the cylinder is floating only in oil. What fraction of the cylinder is in the oil? Before After Oil Layer Water

An introductory physics student, elated by a first semester grade, celebrates by dropping a text- book from a balcony into a deep layer of soft snow which is 3.00 m below. Upon hitting the snow the book sinks a further 1.00 m into it before coming to a stop. The mass of the book is 5.0 kg. Assuming a constant retarding force, what is the force from the snow on the book?

A small bead slides from rest along a wire that is shaped like a vertical uniform helix (spring). Which graph below shows the magnitude of the acceleration a as a function of time? a noitarelecca

In the frame of the box, which of the following is a possible path followed by the particle?

If the magnitude of the acceleration of the box is chosen correctly, the launched particle will follow a path that returns to the point that it was launched. In the frame of the box, which path is followed by the particle?

A mass on a frictionless table is attached to the midpoint of an originally unstretched spring fixed at the ends. If the mass is displaced a distance A parallel to the table surface but perpendicular to the spring, it exhibits oscillations. The period T of the oscillations

Kepler’s Laws state that I. the orbits of planets are elliptical with one focus at the sun, II. a line connecting the sun and a planet sweeps out equal areas in equal times, and III. the square the period of a planet’s orbit is proportional to the cube of its semimajor axis. Which of these laws would remain true if the force of gravity were proportional to 1/r3 rather than 1/r2?

A small bead is placed on the top of a frictionless glass sphere of diameter D as shown. The bead is given a slight push and starts sliding down along the sphere. Find the speed v of the bead at the point at which the which the bead leaves the sphere. D √

Two blocks are suspended by two massless elastic strings to the ceiling as shown in the figure. The masses of the upper and lower block are m = 2 kg and m = 4 kg respectively. If the upper 1 2 string is suddenly cut just above the top block what are the accelerations of the two blocks at the moment when the top block begins to fall? m 1 m 2

The power output from a certain experimental car design to be shaped like a cube is proportional to the mass m of the car. The force of air friction on the car is proportional to Av2, where v is the speed of the car and A the cross sectional area. On a level surface the car has a maximum speed v . Assuming that all versions of this design have the same density, then which of the following max is t

A block floats partially submerged in a container of liquid. When the entire container is accel- erated upward, which of the following happens? Assume that both the liquid and the block are incompressible.

An object of mass m initially moving at speed v collides with an originally stationary object of 1 0 mass m = αm , where α < 1. The collision could be completely elastic, completely inelastic, or 2 1 partially inelastic. After the collision the two objects move at speeds v and v . Assume that the 1 2 collision is one dimensional, and that object one cannot pass through object two. After the collis

The spring attachment is moved to the midpoint of the rod, and the post is moved so the spring remains unstretched and perpendicular to the rod. The system is again set into small oscillations. What is the new frequency of oscillation? l Pivot L Post

The spring attachment is moved back to the end of the rod; the post is moved so that it is in line with the rod and the pivot and the spring is unstretched. The post is then moved away from the pivot by an additional amount l. What is the new frequency of oscillation? L+l l Post Pivot

A ball rolls from the back of a large truck traveling 10.0 m/s to the right. The ball is traveling horizontally at 8.0 m/s to the left relative to an observer in the truck. The ball lands on the roadway 1.25 m below its starting level. How far behind the truck does it land?

As shown in the figure, a ping-pong ball with mass m with initial horizontal velocity v and angular velocity ω comes into contact with the ground. Friction is not negligible, so both the velocity and angular velocity of the ping-pong ball changes. What is the critical velocity v such that the c ping-pong will stop and remain stopped? Treat the ping-pong ball as a hollow sphere. v ω

A spinning object begins from rest and accelerates to an angular velocity of ω = π/15 rad/s with an angular acceleration of α = π/75 rad/s2. It remains spinning at that constant angular velocity and then stops with an angular acceleration of the same magnitude as it previously accelerated. The object made a total of 3 complete rotations during the entire motion. How much time did the motion take?

A semicircular wire of radius R is oriented vertically. A small bead is released from rest at the top of the wire, it slides without friction under the influence of gravity to the bottom, where is then leaves the wire horizontally and falls a distance H to the ground. The bead lands a horizontal distance D away from where it was launched. R H D Which of the following is a correct graph of RH again

A uniform solid right prism whose cross section is an isosceles right triangle with height h and width w = 2h is placed on an incline that has a variable angle with the horizontal θ. What is the minimum coefficient of static friction so that the prism topples before it begins sliding as θ is slowly increased from zero? w h θ

Determine the horizontal displacement x of the second ball just before it hits the ground.

Determine the speed v of the second (originally top) ball just before it hits the ground. √

A uniform thin circular rubber band of mass M and spring constant k has an original radius R. Now it is tossed into the air. Assume it remains circular when stabilized in air and rotates at angular speed ω about its center uniformly. Which of the following gives the new radius of the rubber band?

The moment of inertia of a uniform equilateral triangle with mass m and side length a about an axis through one of its sides and parallel to that side is (1/8)ma2. What is the moment of inertia of a uniform regular hexagon of mass m and side length a about an axis through two opposite vertices?

Three students make measurements of the length of a 1.50 meter rod. Each student reports an uncertainty estimate representing an independent random error applicable to the measurement. Alice: A single measurement using a 2.0 meter long tape measure, to within ±2 mm. Bob: Two measurements using a wooden meter stick, each to within ±2 mm, which he adds together. Christina: Two measurements using a m

A motorcycle rides on the vertical walls around the perimeter of a large circular room. The friction coefficient between the motorcycle tires and the walls is µ. How does the minimum µ needed to prevent the motorcycle from slipping downwards change with the motorcycle’s speed, s?

A mass m hangs from a massless spring connected to the roof of a box of mass M. When the box is held stationary, the mass-spring system oscillates vertically with angular frequency ω. If the box is dropped falls freely under gravity, how will the angular frequency change?

A ball of radius R and mass m is magically put inside a thin shell of the same mass and radius 2R. The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface? m m 2R

Several identical cars are standing at a red light on a one-lane road, one behind the other, with negligible (and equal) distance between adjacent cars. When the green light comes up, the first car takes off to the right with constant acceleration. The driver in the second car reacts and does the same 0.2 s later. The third driver starts moving 0.2 s after the second one and so on. All cars accele

A projectile is launched with speed v off the edge of a cliff of height h, at an angle θ from the horizontal. 0 Air friction is negligible. To maximize the horizontal range of the projectile, θ should satisfy

In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are M = 400 g, M = 200 g, and M = 500 g. What is the value of M for the system to be in 1 2 4 3 static equilibrium? 0.4 m 0.5 m 0.1 m 0.2 m 0.1 m 0.2 m M 1 M 2 M M 3 4

The rate at which the kinetic energy of the train and snow increases is

The minimum power required from the engine to keep the train traveling at a constant speed v is

Flasks A, B, and C each have a circular base with a radius of 2 cm. An equal volume of water is poured into each flask, and none overflow. Rank the force of water F on the base of the flask from greatest to least. A B C

The handle of a gallon of milk is plugged by a manufacturing defect. After removing the cap and pouring out some milk, the level of milk in the main part of the jug is lower than in the handle, as shown in the figure. Which statement is true of the gauge pressure P of the milk at the bottom of the jug? ρ is the density of the milk. Plug H h

In order for the rod not to rotate after the collision, the second sphere should hit the thin rod at

In order for the rod not to rotate after the collision, the second sphere should have a mass M given by

A massless rope passes over a frictionless pulley. Particles of mass M and M +m are suspended from the two different ends of the rope. If m = 0, the tension T in the pulley rope is Mg. If instead the value m increases to infinity, the value of the tension

Which of the following four objects, each with mass M and radius R, would have the largest acceleration down the incline?

Whichofthefollowingfourobjects, eachauniformsolidspherereleasedfromrest, wouldhavethelargest speed after the center of mass has moved through a vertical distance h?

A rod moves freely between the horizontal floor and the slanted wall. When the end in contact with the floor is moving at v, what is the speed of the end in contact with the wall? θ α sinθ

An object is thrown directly downward from the top of a 180 meter tall building. It takes 1.0 seconds for the object to fall the last 60 meters. With what initial downward speed was the object thrown from the roof?

A uniform disk is being pulled by a force F through a string attached to its center of mass. Assume that the disk is rolling smoothly without slipping. At a certain instant of time, in which region of the disk (if any) is there a point moving with zero total acceleration? II I F III IV

A puck is kicked up a ramp, which makes an angle of 30◦ with the horizontal. The graph below depicts the speed of the puck versus time. What is the coefficient of friction between the puck and the ramp? v t

If the collision is completely inelastic, under what condition will the fractional momentum transfer be- tween the two objects be a maximum?

If the collision is perfectly elastic, what is the maximum possible fractional momentum transfer, f ? max 1

The fractional energy transfer is the absolute value of the final kinetic energy of M divided by the initial kinetic energy of m. If the collision is perfectly elastic, under what condition will the fractional energy transfer between the two objects be a maximum?

A spring has a length of 1.0 meter when there is no tension on it. The spring is then stretched between two points 10 meters apart. A wave pulse travels between the two end points in the spring in a time of 1.0 seconds. The spring is now stretched between two points that are 20 meters apart. The new time it takes for a wave pulse to travel between the ends of the spring is closest to

A ball of mass m moving at speed v collides with a massless spring of spring constant k mounted on a stationary box of mass M in free space. No mechanical energy is lost in the collision. If the system does not rotate, what is the maximum compression x of the spring? (cid:113)

A planet orbits around a star S, as shown in the figure. The semi-major axis of the orbit is a. The perigee, namely the shortest distance between the planet and the star is 0.5a. When the planet passes point P (on the line through the star and perpendicular to the major axis), its speed is v . What is its 1 speed v when it passes the perigee? 2 v 1 P S a v 2

Which of the following graphs best shows the velocity versus time of an object originally moving upward in the presence of air friction?

A 3.0 kg mass moving at 30 m/s to the right collides elastically with a 2.0 kg mass traveling at 20 m/s to the left. After the collision, the center of mass of the system is moving at a speed of

Ball 1 traveling in the positive x direction strikes an equal mass ball 2 that is originally at rest. All of the following must be true after the collision, except

AsatelliteisfollowinganellipticalorbitaroundtheEarth. Itsenginesarecapableofprovidingaone-time impulse of a fixed magnitude. In order to maximize the energy of the satellite, the impulse should be perigee Earth apogee

Two masses are attached with pulleys by a massless rope on an inclined plane as shown. All surfaces are frictionless. If the masses are released from rest, then the inclined plane m 2 m 1

A packing crate with mass m = 115 kg is slid up a 5.00 m long ramp which makes an angle of 20.0◦ with respect to the horizontal by an applied force of F = 1.00×103 N directed parallel to the ramp’s incline. A frictional force of magnitude f = 4.00×102 N resists the motion. If the crate starts from rest, what is its speed at the top of the ramp?

A car has a maximum acceleration of a and a minimum acceleration of −a . The shortest possible time 0 0 for the car to begin at rest, then arrive at rest at a point a distance d away is (cid:112)

A disk of radius r rolls uniformly without slipping around the inside of a fixed hoop of radius R. If the period of the disc’s motion around the hoop is T, what is the instantaneous speed of the point on the disk opposite to the point of contact? R r

A uniform stick of mass m is originally on a horizontal surface. One end is attached to a vertical rope, which pulls up with a constant tension force F so that the center of the mass of the stick moves upward with acceleration a < g. The normal force N of the ground on the other end of the stick shortly after the right end of the stick leaves the surface satisfies F m

Which of the following graphs best shows the acceleration versus time of an object originally moving upward in the presence of air friction?

A light, uniform, ideal spring is fixed at one end. If a mass is attached to the other end, the system oscillates with angular frequency ω. Now suppose the spring is fixed at the other end, then cut in half. The mass is attached between the two half springs. The new angular frequency of oscillations is

A group of students wish to measure the acceleration of gravity with a simple pendulum. They take one length measurement of the pendulum to be L = 1.00±0.05 m. They then measure the period of a single swing to be T = 2.00±0.10 s. Assume that all uncertainties are Gaussian. The computed acceleration of gravity from this experiment illustrating the range of possible values should be recorded as

A massless cable of diameter 2.54 cm (1 inch) is tied horizontally between two trees 18.0 m apart. A tightrope walker stands at the center of the cable, giving it a tension of 7300 N. The cable stretches and makes an angle of 1.50◦ with the horizontal. 1.5o The Young’s modulus is defined as the ratio of stress to strain, where stress is the force applied per unit area and strain is the fractional

Three identical masses are connected with identical rigid rods and pivoted at point A. If the lowest mass receives a small horizontal push to the left, it oscillates with period T . If it instead receives a small push 1 into the page, it oscillates with period T . The ratio T /T is 2 1 2 A

A satellite is in a circular orbit about the Earth. Over a long period of time, the effects of air resistance decrease the satellite’s total energy by 1J. The kinetic energy of the satellite

A cylindrical space station produces ‘artificial gravity’ by rotating with angular frequency ω. Consider working in the reference frame rotating with the space station. In this frame, an astronaut is initially at rest standing on the floor, facing in the direction that the space station is rotating. The astronaut jumps up vertically relative to the floor of the space station, with an initial speed

A stream of sand is dropped out of a helicopter initially moving at a constant speed v to the right. The helicopter suddenly turns and begins moving a constant speed v to the left. Neglecting air resistance on the sand, what is the shape of the stream of sand, as viewed from the ground? The black dot represents the helicopter.

A mass m is attached to a thin rod of length (cid:96) so that it can freely spin in a vertical circle with period T. The difference in the tensions in the rod when the mass is at the top and the bottom of the circle is

Raindrops with a number density of n drops per cubic meter and radius r hit the ground with a speed 0 v . The resulting pressure on the ground from the rain is P . If the number density is doubled, the drop 0 0 radius is halved, and the speed is halved, the new pressure will be

A spring stretched to double its unstretched length has a potential energy U . If the spring is cut in half, 0 and each half spring is stretched to double its unstretched length, then the total potential energy stored in the two half springs will be

A ping-pong ball (a hollow spherical shell) with mass m is placed on the ground with initial velocity v 0 and zero angular velocity at time t = 0. The coefficient of friction between the ping-pong ball and the ground is µ = µ = µ. The time the ping-pong ball begins to roll without slipping is s k v 0 v(cid:48) ω = 0

A small hole is punched into the bottom of a rectangular boat, allowing water to enter the boat. As the boat sinks into the water, which of the following graphs best shows how the rate water flows through the hole varies with time? Assume that the boat remains horizontal as it sinks. water level outside boat water level inside hole

The coefficients of static and kinetic friction between a ball and an ramp are µ = µ = µ. The ball s k is released from rest at the top of the ramp. Which of the following graphs best shows the rotational acceleration of the ball about its center of mass as a function of the angle of the ramp?

A mass is attached to one end of a rigid rod, while the other end of the rod is attached to a fixed horizontal axle. Initially the mass hangs at the end of the rod and the rod is vertical. The mass is given an initial kinetic energy K. If K is very small, the mass behaves like a pendulum, performing small-angle oscillations with period T . As K is increased, the period of the motion for the mass 0

AliceandBobareworkingonalabreport. Alicemeasurestheperiodofapendulumtobe1.013±0.008s, while Bob independently measures the period to be 0.997±0.016 s. Alice and Bob can combine their measurements in several ways. 1: Keep Alice’s result and ignore Bob’s 2: Average Alice’s and Bob’s results 3: Perform a weighted average of Alice’s and Bob’s results, with Alice’s result weighted 4 times more than Bob

A large lump of clay is dropped off of a wall and lands on the ground. Which graph best represents the acceleration of the center of mass of the clay as a function of time?

A uniform block of mass 10 kg is released at rest from the top of an incline with length 10 m and inclination 30◦, and slides to the bottom. The coefficients of static and kinetic friction are µ = µ = 0.1. s k How much energy is dissipated due to friction?

A 3.0 kg mass moving at 40 m/s to the right collides with and sticks to a 2.0 kg mass traveling at 20 m/s to the right. After the collision, the kinetic energy of the system is closest to

A basketball is released from rest and bounces on the ground. Considering only the ball just before and just after the bounce, which of the following statements must be true?

The hard disk in a computer will spin up to speed within 10 rotations, but when turned off will spin through 50 rotations before coming to a stop. Assuming the hard disk has constant angular acceleration α and angular deceleration α , the ratio α /α is 1 2 1 2

A massless beam of length L is fixed on one end. A downward force F is applied to the free end of the beam, deflecting the beam downward by a distance x. The deflection x is linear in F and is inversely proportional to the cross-section moment I, which has units m4. The deflection is also dependent on Young’s modulus E, which has units N/m2. Then x depends on L according to √

A pendulum of length L oscillates inside a box. A person picks up the box and gently shakes it vertically withfrequencyωandafixedamplitudeforafixedtime. Tomaximizethefinalamplitudeofthependulum, ω should satisfy (cid:112)

The coefficients of static and kinetic friction between a ball and a horizontal plane are µ = µ = µ. The s k ball is given a horizontal speed but no rotational velocity about its center of mass. Which of the following graphs best shows the rotational velocity of the ball about its center of mass as a function of time?

A 3.0 kg ball moving at 10 m/s east collides elastically with a 2.0 kg ball moving 15 m/s west. Which of the following statements could be true after the collision?

A balloon filled with air submerged in water at a depth h experiences a buoyant force B . The balloon is 0 moved to a depth of 2h, where it experiences a buoyant force B. Assuming the water is incompressible and the balloon and air are compressible, the buoyant force B satisfies

A circle of rope is spinning in outer space with an angular velocity ω . Transverse waves on the rope have 0 speed v , as measured in a rotating reference frame where the rope is at rest. If the angular velocity of 0 the rope is doubled, the new speed of transverse waves, as measured in a rotating reference frame where the rope is at rest, will be

Achildinacircular, rotatingspacestationtossesaballinsuchawaysothatoncethestationhasrotated through one half rotation, the child catches the ball. From the child’s point of view, which plot shows the trajectory of the ball? The child is at the bottom of the space station in the diagrams below, but only the initial location of the ball is shown.

Two blocks of masses m = 2.0 kg and m = 1.0 kg are stacked together on top of a frictionless table as 1 2 shown. The coefficient of static friction between the blocks is µ = 0.20. What is the minimum horizontal s force that must be applied to the top block to make it slide across the bottom block? m 1 m 2

A spool is made of a cylinder with a thin disc attached to either end of the cylinder, as shown. The cylinder has radius r = 0.75 cm and the discs each have radius R = 1.00 cm. A string is attached to the cylinder and wound around the cylinder a few times. At what angle above the horizontal can the string be pulled so that the spool will slip without rotating? side−view 2R 2r θ

You are standing on a weight scale that reads 700 Newtons while holding a large physics textbook that is originally at rest. At time t = 1 seconds you begin moving the textbook upward so that by time t = 2 seconds the textbook is now half a meter higher and once again at rest. Which of the following graphs best illustrates how the reading on the scale might vary with time?

A plane can fly by tilting the trailing edge of the wings downward by a small angle θ, called the angle of attack. In still air, a plane with ground speed v will have a lift force proportional to v2θ and a drag force is proportional to v2. Consider a plane initially in level flight in still air with constant ground speed v. If the plane enters a region with a tailwind with speed w < v (the wind is

A pogo stick is modeled as a massless spring of spring constant k attached to the bottom of a block of mass m. The pogo stick is dropped with the spring pointing downward and hits the ground with speed v. At the moment of the collision, the free end of the spring sticks permanently to the ground. m v k During the subsequent oscillations, the maximum speed of the block is

A spring of relaxed length (cid:96) and spring constant k is placed ‘in parallel’ with a spring of relaxed length 1 1 (cid:96) and spring constant k . A force F is applied to each end. 2 2 Spring 1 F F Spring 2 The combination of the springs acts like a single spring with spring constant k and relaxed length (cid:96) where

In an experiment to determine the speed of sound, a student measured the distance that a sound wave traveled to be 75.0±2.0 cm, and found the time it took the sound wave to travel this distance to be 2.15±0.10 ms. Assume the uncertainties are Gaussian. The computed speed of sound should be recorded as

A massive, uniform, flexible string of length L is placed on a horizontal table of length L/3 that has a coefficient of friction µ = 1/7, so equal lengths L/3 of string hang freely from both sides of the table. s The string passes over the edges of the table on smooth, frictionless, curved surfaces. Now suppose that one of the hanging ends of the string is pulled a distance x downward, then releas

A uniform bar of length L and mass M is supported by a fixed pivot a distance x from its center. The bar is released from rest from a horizontal position. The period of the resulting oscillations is minimal when

Two particles of mass m are connected by pulleys as shown. m m v The mass on the left is given a small horizontal velocity, and oscillates back and forth. The mass on the right

Two particles with mass m and m are connected by a massless rigid rod of length L and placed on a 1 2 horizontal frictionless table. At time t = 0, the first mass receives an impulse perpendicular to the rod, giving it speed v. At this moment, the second mass is at rest. The next time the second mass is at rest is

A particle of mass m is placed at the center of a hemispherical shell of radius R and mass density σ, where σ has dimensions of kg/m2. m σ R The gravitational force of the shell on the particle is

A student is measuring the surface area of a cylindrical wire. The student measures the radius of the wire to be 1±0.1 cm using a ruler and the length of the wire to be 1.00±0.01 m using a meter-stick. The precision of their result can be increased in several ways. 1: Upgrade the ruler to a caliper with uncertainty 0.01 cm. 2: Upgrade the meter-stick to a tape measure with uncertainty 0.001 m. 3:

A coin of mass m is dropped straight down from the top of a very tall building. As the coin approaches terminal speed, which is true of the net force on the coin?

A mass of 3M moving at a speed v collides with a mass of M moving directly toward it, also with a speed v. If the collision is completely elastic, the total kinetic energy after the collision is K . If the two e masses stick together, the total kinetic energy after the collision is K . What is the ratio K /K ? s e s

A block of mass m is launched horizontally onto a curved wedge of mass M at a velocity v. What is the maximum height reached by the block after it shoots off the vertical segment of the wedge? Assume all surfaces are frictionless; both the block and the curved wedge are free to move. The curved wedge does not tilt or topple. M m v v2

Amasslessspringhangsfromtheceiling,andamassishungfromthebottomofit. Themassissupported so that initially the tension in the spring is zero. The mass is then suddenly released. At the bottom of its trajectory, the mass is 5 centimeters from its original position. Find its oscillation period.

A cylinder has a radius R and weight G. You try to roll it over a step of height h < R. The minimum force needed to roll the cylinder over is: R h √ 2Rh−h2

A ball is released from rest above an inclined plane and bounces elastically down the plane. As the ball progresses down the plane, the time and the distance between each collision will:

An object of mass m is attached to the end of a massless rod of length L. The other end of the rod is attached to a frictionless pivot. The object is raised so that its height is 0.8L above the pivot, as shown in the figure. After the object is released from rest, what is the tension in the rod when it is horizontal? L 0.8L

The mass and the radius of the Earth are M and R. If an object starting at a distance of R from the (cid:113) Earth’s surface is moving at a velocity v = 2GM tangentially, what is its trajectory? 0 3R

A wheel of radius R is rolling without slipping with angular velocity ω. ω θ A For point A on the wheel at an angle θ with respect to the vertical, shown in the figure, what is the magnitude of its velocity with respect to the ground?

A flat uniform disk of radius 2R has a hole of radius R removed from the center. The resulting annulus is then cut in half along the diameter. The remaining shape has mass M. What is the moment of inertia of this shape, about the axis of rotational symmetry of the original disk? 45

To test the speed of a model car, you time the car with a stopwatch as it travels a distance of 100m. You record a time of 5.0s, and your measurement has an uncertainty of 0.2s. What is the uncertainty in your estimate of the car’s speed? Assume that the car travels at a constant speed and the distance of 100m is known very precisely.

A steel ball bearing bounces vertically on a steel plate. If the speed of the ball just before a bounce is v , i the speed of the ball immediately afterward is v = αv , with α < 1. Which one of the following graphs f i best shows the time between successive bounces, τ, as a function of time?

A juggler juggles N identical balls, catching and tossing one ball at a time. Assuming that the juggler requires a minimum time T between ball tosses, the minimum possible power required for the juggler to continue juggling is proportional to

A man standing at 30◦ latitude fires a bullet northward at a speed of 200 m/s. The radius of the Earth is 6371 km. What is the sideway deflection of the bullet after traveling 100 m?

An upright rod of length (cid:96) is launched into the air with vertical velocity v . It is given enough angular y momentum so that the rod rotates by an angle of 2π before landing. Find the initial horizontal velocity of the bottom of the rod. 2(cid:96)g

The depth of a well, d, is measured by dropping a stone into it and measuring the time t until the splash is heard at the bottom. What is the smallest value of d for which ignoring the time for the sound to travel gives less than a 5% error in the depth measurement? The speed of sound in air is 330 m/s.

A hard disc is pushed into the launcher between the boards until the boards make contact with it a distance (cid:96) from the hinge and are open to an angle θ, as shown in the figure. What is the minimum force necessary to hold the disc in this position? m θ τ (cid:96)

The disc is removed and replaced with a pie-shaped wedge of the same mass m, so that the hinge is still initially held open at an angle θ, as shown in the following figure. If the wedge is released from rest, what is its speed after it exits the launcher? m θ τ (cid:114) τ

A small rock is tied to a massless string of length 5 m. The density of the rock is twice the density of the water. The rock is lowered into the water, while the other end of the string is attached to a pivot. Neglect any resistive forces from the water. The rock oscillates like a pendulum with angular frequency of which of the following?

A uniform rod of mass M and length L is hinged on a horizontal surface at the bottom. Its top end is connected to two springs, both with spring constant k. What relation must M, k, and L satisfy such that the position shown in the figure is a stable equilibrium? k k C

A spherical cloud of dust has uniform mass density ρ and radius R. Satellite A of negligible mass is orbiting the cloud at its edge, in a circular orbit of radius R, and satellite B is orbiting the cloud just inside the cloud, in a circular orbit of radius r, with r < R. If v is the speed of satellite i and T is the i i period of satellite i, which of the following is true? Neglect any drag forces

A vertical pole has two massless strings, both of length L, attached a distance L apart. The other ends of the strings are attached to a mass M. The mass is rotated around the pole with an angular speed ω. Which of the following graphs best gives the ratio of the tension in the bottom string to the tension in the top string as a function of ω?

A rectangular slab sits on a frictionless surface. A sphere sits on the slab. There is sufficient friction between the sphere and the slab such that the sphere will not slip relative to the slab. A force to the right is applied to the slab, with both the slab and the sphere initially at rest. The sphere will then:

A particle moves in the xy plane with the potential energy U(x,y) = 9kx2+16ky2. The particle can perform several different types of periodic motion. The ratio between the maximum and minimum possible periods is √

A car is turning left along a circular track of radius r at a constant speed v. A cylindrical beaker is placed vertically inside the car. The beaker has a small hole on its right side. If the water’s highest point in the beaker is a height h above the hole, at what instantaneous speed does water escape the hole, from a passenger’s perspective? √

A uniform block of mass 10 kg is released at rest from the top of an incline of length 10 m and inclination 30◦. The coefficients of static and kinetic friction between the incline and the block are µ = 0.15 and s µ = 0.1. The end of the incline is connected to a frictionless horizontal surface. After a long time, how k much energy is dissipated due to friction?

Three cubical blocks of the same volume are made out of wood, styrofoam, and plastic. When the plastic block is placed in water, half of its volume is submerged. If the wooden block is placed in water with the plastic block on top, the wooden block is just fully submerged. Similarly, if the styrofoam block is placed in oil with the plastic block on top, the styrofoam block is just fully submerged.

Two springs of spring constants k and k , respectively, are connected in series and stretched, as shown 1 2 below. What is the ratio of their potential energies, U /U ? 1 2 k 1 k 2 F

Consider two identical masses that interact only by gravitational attraction to each other. If one mass is fixed in place and the other is released from rest, then the two masses collide in time T. If both masses are released from rest, they collide in time

The density of the Earth increases gradually from around 3 g/cm3 at the crust to about 13 g/cm3 at the core. Which one of these plots could show local gravitational acceleration as a function of distance from Earth’s center?

A very long cylinder of dust is spinning about its axis with angular velocity ω at steady state. Let r be the distance from the axis. If the dust is only held together by gravity, the density of the dust is proportional to:

A trough half-filled with water is suspended from wires, as shown. The tension is initially the same in each wire. A boat is placed in the trough directly under the left wire. It floats without touching the sides of the trough or overflowing the water. How does the tension in the wires change as a result?

A scale is calibrated so that it gives a correct reading when sitting on the ground. A person holds the scale and presses it on both sides with their hands, pushing up on the bottom with the left hand and pushing down on the top with the right. The scale has a mass of 5 kg, and their left hand exerts a force of 200 N. The reading on the scale is:

A light board of length (cid:96) is hinged to a vertical wall. A solid ball of weight G and radius R is held between the wall and the board by a force F applied perpendicular at the end of the board, as shown in the figure below. Both the wall and the board are frictionless. The angle between the board and the wall is 60◦. What is the magnitude of F? F R ℓ 60◦ 2

A ball of mass m rolls in a bowl bolted on a cart. The bowl and cart move together and have a combined mass M , as shown in the figure. Initially, the cart moves to the right with speed v with respect to the c 0 ground, while the ball is at rest with respect to the cart. The ball then slides down to the bottom of the bowl. When it reaches the bottom of the bowl, it moves with velocity v with respe

Consider a flat uniform square of mass M and side length L. Cut a circle out of the square that has a diameter equal to the length of the side of the square, with the same center as the square. Determine the moment of inertia of the remaining shape about an axis through the center and perpendicular to the plane of the square.

Two planets A and B have masses m = 2m . They orbit a star in circular orbits of radius r = 3r . A B A B LetE andL bethekineticenergyandthemagnitudeoftheangularmomentumofplaneti, respectively. i i Which of the following is true?

A water hose is at ground level against the base of a large wall. By aiming the hose at some angle, and squirting water at a speed v, one wets a region on the wall, as shown below. If the speed of the water is doubled, what is the new region that can be wetted? Ignore the effect of water splashing beyond the point of contact. In the answer choices, the dotted line marks the initial wetted region.

A ball is launched at speed v at angle θ above the horizontal toward a vertical wall a distance L away. It bounces elastically off the wall and falls back to its launch point. What was its initial speed? √Lg

A mass of M = 100 g is attached to the end of a string of length R = 2 m. A person swings the mass overhead such that their hand traverses a circle of radius r = 3 cm at angular velocity ω = 10 rad/s, ahead of the mass M by an angle of π/2. Estimate the force of air resistance on the object.

How long does it take the hoop to return to its starting position? v 0

Would the answer to the previous question, namely the amount of time to return to the starting position, be the same for a uniform disk with the same coefficient of friction launched with the same initial conditions?

in the time for a car to complete a race. The car begins with an initial speed v and maintains a constant 0 acceleration a throughout the race. Both v and a have independent uncertainties of 10%. Which 0 contributes greater uncertainty to your estimate of the time for the car to complete the race?

A physical pendulum consists of a mass on one end of a massless rigid rod that can pivot about the opposite end. Assuming the oscillations have an amplitude of 90◦, which of the following graphs best shows the total acceleration of the mass as a function of the angular position θ, measured between the pendulum and the vertical direction?

A block on a ramp is given an initial velocity v upward along the ramp. It slides upward for a time t , 0 u traveling some distance, and then slides downward for a time t until it returns to its original position. d What is t in terms of t ? d u The height of the incline is 0.6 times its length and the coefficient of kinetic friction between the block and the incline is 0.5. 0.6(cid:96) v0 (cid:96

An empty freight car on a level railroad track has a mass M. A chute above the freight car opens and grain falls down into the car at a rate of r, measured in kilograms per second. The grain falls a vertical distance h before hitting the bed of the freight car without bouncing up. How much normal force is exerted on the freight car from the rails at a time t after the grain begins to hit the bed o

The bruise threshold for a fruit is the largest height it can be dropped from rest without bruising. The bruise threshold of a 0.2 kg apple on steel is 10 cm. Suppose that it always takes 0.1 seconds after impact for the apple to fully stop and that 4 cm2 of the apple comes into contact with the surface during the impact. What minimum average pressure is required to cause the apple to bruise?

A train travels at 360 km/hr on an almost straight track. The track is slightly sinusoidal, with a vertical amplitudeofhovera1kmdistance,asshowninthefigure. Ifthemaximumtolerableverticalacceleration of the train is set at 0.1 m/s2, what is the maximum allowable size of h? 360km/hr h 1km

A uniform rope of length L and mass M passes over a frictionless pulley, and hangs with both ends at equal heights. If one end is pulled down a distance x and the rope is released, the acceleration of that end at this instant will be:

A student makes an estimate of the acceleration due to gravity, g, by dropping a rock from a known height h and measuring the time, t, it takes to hit the ground. Neglecting air resistance, which one of the following situations will lead to the smallest value of the relative uncertainty, (∆g)/g, in the estimate?

1. Consider a dripping faucet, where the faucet is 10 cm above the sink. The time between drops is such that when one drop hits the sink, one is in the air and another is about to drop. At what height above the sink will the drop in the air be right as a drop hits the sink?

2. A cannonball is launched with initial velocity of magnitude v0 over a horizontal surface. At what minimum angle θmin above the horizontal should the cannonball be launched so that it rises to a height H which is larger than the horizontal distance R that it will travel when it returns to the ground?

3. An equilateral triangle is sitting on an inclined plane. Friction is too high for it to slide under any circumstance, but if the plane is sloped enough it can “topple” down the hill. What angle incline is necessary for it to start toppling?

4. A particle at rest explodes into three particles of equal mass in the absence of external forces. Two particles emerge at a right angle to each other with equal speed v. What is the speed of the third particle?

5. A 12 kg block moving east at 4 m/s collides head on with a 6 kg block that is moving west at 2 m/s. The two blocks move together after the collision. What is the loss in kinetic energy in this collision?

The following information applies to questions 6 and 7 Two cannons are arranged vertically, with the lower cannon pointing upward (towards the upper cannon) and the upper cannon pointing downward (towards the lower cannon), 200m above the lower cannon. Simultaneously, they both fire. The muzzle velocity of the lower cannon is 25m/s and the muzzle velocity of the upper cannon is 55m/s. 200 meters 6. How long after the cannons fire do the projectiles collide?

The following information applies to questions 6 and 7 Two cannons are arranged vertically, with the lower cannon pointing upward (towards the upper cannon) and the upper cannon pointing downward (towards the lower cannon), 200m above the lower cannon. Simultaneously, they both fire. The muzzle velocity of the lower cannon is 25m/s and the muzzle velocity of the upper cannon is 55m/s. 200 meters 7. How far beneath the top cannon do the projectiles collide?

8. A block of mass m = 3.0 kg is moving on a horizontal surface towards a massless spring with spring constant k = 80.0 N/m. The coefficient of kinetic friction between the block and the surface is µk = 0.50. The block has a speed of 2.0 m/s when it first comes in contact with the spring. How far will the spring be compressed?

9. A uniform spherical planet has radius R and the acceleration due to gravity at its surface is g. What is the escape velocity of a particle from the planet’s surface? √gR

10. Four objects are placed at rest at the top of an inclined plane and allowed to roll without slipping to the bottom in the absence of rolling resistance and air resistance. • Object A is a solid brass ball of diameter d. • Object B is a solid brass ball of diameter 2d. • Object C is a hollow brass sphere of diameter d. • Object D is a solid aluminum ball of diameter d. (Aluminum is less dense than brass.) The balls are placed so that their centers of mass all travel the same distance. In each case, the time of motion T is measured. Which of the following statements is correct?

11. As shown below, Lily is using the rope through a fixed pulley to move a box with constant speed v. The kinetic friction coefficient between the box and the ground is µ < 1; assume that the fixed pulley is massless and there is no friction between the rope and the fixed pulley. Then, while the box is moving, which of the following statements is correct?

12. A rigid hoop can rotate about the center. Two massless strings are attached to the hoop, one at A, the other at B. These strings are tied together at the center of the hoop at O, and a weight G is suspended from that point. The strings have a fixed length, regardless of the tension, and the weight G is only supported by the strings. Originally OA is horizontal. A B G O T1 T2 Now, the outer hoop will start to slowly rotate 90◦clockwise until OA will become vertical, while keeping the angle between the strings constant and keeping the object static. Which of the following statements about the tensions T1 and T2 in the two strings is correct?

13. Shown below is a graph of the x component of force versus position for a 4.0 kg cart constrained to move in one dimension on the x axis. At x = 0 the cart has a velocity of −3.0 m/s (in the negative direction). Which of the following is closest to the maximum speed of the cart? 2 4 6 −2 −4 4 2 6 8 F (Newtons) x (meters) −2

14. A uniform cylinder of radius a originally has a weight of 80 N. After an off-axis cylinder hole at 2a/5 was drilled through it, it weighs 65 N. The axes of the two cylinders are parallel and their centers are at the same height. 2a/5 a A force T is applied to the top of the cylinder horizontally. In order to keep the cylinder is at rest, the magnitude of the force is closest to:

15. A car of mass m has an engine that provides a constant power output P. Assuming no friction, what is the maximum constant speed vmax that this car can drive up a long incline that makes an angle θ with the horizontal?

16. Inside a cart that is accelerating horizontally at acceleration⃗a, there is a block of mass M connected to two light springs of force constants k1 and k2. The block can move without friction horizontally. Find the vibration frequency of the block. q

17. Shown below is a log/log plot for the data collected of amplitude and period of oscillation for certain non-linear oscillator. 0 1 2 log (A) log (T) 4 2 8 6 According to the data, the relationship between period T and amplitude A is best given by

18. A mass hangs from the ceiling of a box by an ideal spring. With the box held fixed, the mass is given an initial velocity and oscillates with purely vertical motion. When the mass reaches the lowest point of its motion, the box is released and allowed to fall. To an observer inside the box, which of the following quantities does not change when the box is released? Ignore air resistance.

19. A 1,500 Watt motor is used to pump water a vertical height of 2.0 meters out of a flooded basement through a cylindrical pipe. The water is ejected though the end of the pipe at a speed of 2.5 m/s. Ignoring friction and assuming that all of the energy of the motor goes to the water, which of the following is the closest to the radius of the pipe? The density of water is ρ = 1000 kg/m3.

20. A container of water is sitting on a scale. Originally, the scale reads M1 = 45 kg. A block of wood is suspended from a second scale; originally the scale read M2 = 12 kg. The density of wood is 0.60 g/cm3; the density of the water is 1.00 g/cm3. The block of wood is lowered into the water until half of the block is beneath the surface. What is the resulting reading on the scales? 45.0 kg ??? kg ??? kg 12.0 kg

21. A spring system is set up as follows: a platform with a weight of 10 N is on top of two springs, each with spring constant 75N/m. On top of the platform is a third spring with spring constant 75N/m. If a ball with a weight of 5.0 N is then fastened to the top of the third spring and then slowly lowered, by how much does the height of the spring system change? Original height The ball is placed here

22. The softest audible sound has an intensity of I0 = 10−12 W/m2. In terms of the fundamental units of kilograms, meters, and seconds, this is equivalent to

23. Which of the following sets of equipment cannot be used to measure the local value of the acceleration due to gravity (g)?

24. Three point masses m are attached together by identical springs. When placed at rest on a horizontal surface the masses form a triangle with side length l. When the assembly is rotated about its center at angular velocity ω, the masses form a triangle with side length 2l. What is the spring constant k of the springs?

25. Consider the two orbits around the sun shown below. Orbit P is circular with radius R, orbit Q is elliptical such that the farthest point b is between 2R and 3R, and the nearest point a is between R/3 and R/2. Consider the magnitudes of the velocity of the circular orbit vc, the velocity of the comet in the elliptical orbit at the farthest point vb, and the velocity of the comet in the elliptical orbit at the nearest point va. Which of the following rankings is correct? vb va vc b a c orbit P orbit Q R

1. An observer stands on the side of the front of a stationary train. When the train starts moving with constant acceleration, it takes 5 seconds for the first car to pass the observer. How long will it take for the 10th car to pass?

2. Jordi stands 20 m from a wall and Diego stands 10 m from the same wall. Jordi throws a ball at an angle of 30◦ above the horizontal, and it collides elastically with the wall. How fast does Jordi need to throw the ball so that Diego will catch it? Consider Jordi and Diego to be the same height, and both are on the same perpendicular line from the wall.

3. Tom throws a football to Wes, who is a distance l away. Tom can control the time of flight t of the ball by choosing any speed up to vmax and any launch angle between 0◦and 90◦. Ignore air resistance and assume Tom and Wes are at the same height. Which of the following statements is incorrect? (A) If vmax < √gl, the ball cannot reach Wes at all.

4. The sign shown below consists of two uniform legs attached by a frictionless hinge. The coefficient of friction between the ground and the legs is µ. Which of the following gives the maximum value of θ such that the sign will not collapse? θ L

The following information applies to questions 5 and 6 A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 0 20 40 60 80 100 120 Time (s) Scale reading (kg) 5. At what time(s) does the student have maximum downward velocity?

The following information applies to questions 5 and 6 A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 0 20 40 60 80 100 120 Time (s) Scale reading (kg) 6. How tall is the building?

7. A light car and a heavy truck have the same momentum. The truck weighs ten times as much as the car. How do their kinetic energies compare?

The following information applies to questions 8 and 9 A truck is initially moving at velocity v. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force F to the truck. The truck rolls a distance x before coming to a stop, and the time it takes to stop is t. 8. Which of the following expressions is equal the initial kinetic energy of the truck (i.e. the kinetic energy before the driver starts braking)?

The following information applies to questions 8 and 9 A truck is initially moving at velocity v. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force F to the truck. The truck rolls a distance x before coming to a stop, and the time it takes to stop is t. 9. Which of the following expressions is equal the initial momentum of the truck (i.e. the momentum before the driver starts braking)?

10. Which of the following can be used to distinguish a solid ball from a hollow sphere of the same radius and mass?

11. A right-triangular wooden block of mass M is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass m, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger block while the block remains at rest. What is the normal force from the system to the table? α β 90 m m M

12. A spherical shell of mass M and radius R is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle θ with the horizontal. What will be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is g. The moment of inertia of a thin shell of radius r and mass m about the center of mass is I = 2 3mr2; the moment of inertia of a solid sphere of radius r and mass m about the center of mass is I = 2 5mr2.

13. There is a ring outside of Saturn. In order to distinguish if the ring is actually a part of Saturn or is instead part of the satellites of Saturn, we need to know the relation between the velocity v of each layer in the ring and the distance R of the layer to the center of Saturn. Which of the following statements is correct?

14. A cart of mass m moving at 12 m/s to the right collides elastically with a cart of mass 4.0 kg that is originally at rest. After the collision, the cart of mass m moves to the left with a velocity of 6.0 m/s. Assuming an elastic collision in one dimension only, what is the velocity of the center of mass (vcm) of the two carts before the collision?

15. A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is 5/9 that of water. At equilibrium, what portion of the rod is above water?

16. Inspired by a problem from the 2012 International Physics Olympiad, Estonia. A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume ρ0, and occupy a region of radius r0. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses fully? (The constant 0.5427 is actually q 3π 32 .)

17. Two small, equal masses are attached by a lightweight rod. This object orbits a planet; the length of the rod is smaller than the radius of the orbit, but not negligible. The rod rotates about its axis in such a way that it remains vertical with respect to the planet. • Is there a force in the rod? If so, is it tension or compression? • Is the equilibrium stable, unstable, or neutral with respect to a small perturbation in the angle of the rod? (Assume this perturbation maintains the rate of rotation, so that in the co-rotating frame the rod is still stationary but at an angle to the vertical.)

18. Two point particles, each of mass 1 kg, begin in the state shown below. −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 1.0 m/s 1.0 m/s x (m) y (m) The system evolves through internal forces only. Which of the following could be the state after some time has passed?

The following information applies to questions 19, 20, and 21. A simple pendulum experiment is constructed from a point mass m attached to a pivot by a massless rod of length L in a constant gravitational field. The rod is released from an angle θ0 < π/2 at rest and the period of motion is found to be T0. Ignore air resistance and friction. 19. At what angle θg during the swing is the tension in the rod the greatest?

The following information applies to questions 19, 20, and 21. A simple pendulum experiment is constructed from a point mass m attached to a pivot by a massless rod of length L in a constant gravitational field. The rod is released from an angle θ0 < π/2 at rest and the period of motion is found to be T0. Ignore air resistance and friction. 20. What is the maximum value of the tension in the rod?

The following information applies to questions 19, 20, and 21. A simple pendulum experiment is constructed from a point mass m attached to a pivot by a massless rod of length L in a constant gravitational field. The rod is released from an angle θ0 < π/2 at rest and the period of motion is found to be T0. Ignore air resistance and friction. 21. The experiment is repeated with a new pendulum with a rod of length 4L, using the same angle θ0, and the period of motion is found to be T. Which of the following statements is correct?

22. A simplified model on the foot is shown. When a student of mass m = 60 kg stands on a single toe, the tension T in the Achilles Tendon is closest to 0 4 8 12 16 20 24 28 32 36 40 Leg Bone Approximation of Foot Bone Toe Outline of Foot Ankle Joint Centimeter Rule Tendon Achilles

The following information applies to questions 23 and 24 A man with mass m jumps offof a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance H at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke’s force law with a spring constant k, and stretches from an original length of L0 to a final length L = L0 + h. The maximum tension in the Bungee cord is four times the weight of the man. 23. Determine the spring constant k.

The following information applies to questions 23 and 24 A man with mass m jumps offof a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance H at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke’s force law with a spring constant k, and stretches from an original length of L0 to a final length L = L0 + h. The maximum tension in the Bungee cord is four times the weight of the man. 24. Find the maximum extension of the bungee cord h.

25. A box with weight W will slide down a 30◦incline at constant speed under the influence of gravity and friction alone. If instead a horizontal force P is applied to the box, the box can be made to move up the ramp at constant speed. What is the magnitude of P?

1. A car turning to the right is traveling at constant speed in a circle. From the driver’s perspective, the angular momentum vector about the center of the circle points X and the acceleration vector of the car points Y where

2. A ball rolls without slipping down an inclined plane as shown in the diagram. Which of the following vectors best represents the direction of the total force that the ball exerts on the plane?

3. An object of uniform density floats partially submerged so that 20% of the object is above the water. A 3 N force presses down on the top of the object so that the object becomes fully submerged. What is the volume of the object? The density of water is ρH2O = 1000 kg/m3.

4. What are the correct values of the numbers in the following statements? Assume there are no external forces, and take N = 1 to mean that the statement cannot be made for any meaningful number of particles. • If a particle at rest explodes into N1 or fewer particles with known masses, and the total kinetic energy of the new particles is known, the kinetic energy of each of the new particles is completely determined. • If a particle at rest explodes into N2 or fewer particles, the velocities of the new particles must lie in a line. • If a particle at rest explodes into N3 or fewer particles, the velocities of the new particles must lie in a plane.

5. A unicyclist goes around a circular track of radius 30 m at a (amazingly fast!) constant speed of 10 m/s. At what angle to the left (or right) of vertical must the unicyclist lean to avoid falling? Assume that the height of the unicyclist is much smaller than the radius of the track.

6. A cubical box of mass 10 kg with edge length 5 m is free to move on a frictionless horizontal surface. Inside is a small block of mass 2 kg, which moves without friction inside the box. At time t = 0, the block is moving with velocity 5 m/s directly towards one of the faces of the box, while the box is initially at rest. The coefficient of restitution for any collision between the block and box is 90%, meaning that the relative speed between the box and block immediately after a collision is 90% of the relative speed between the box and block immediately before the collision. 2 kg block 10 kg cube frictionless surface After 1 minute, the block is a displacement x from the original position. Which of the following is closest to x?

7. A 1.00 m long stick with uniform density is allowed to rotate about a point 30.0 cm from its end. The stick is perfectly balanced when a 50.0 g mass is placed on the stick 20.0 cm from the same end. What is the mass of the stick?

8. An object of mass M is hung on a vertical spring of spring constant k and is set into vertical oscillations. The period of this oscillation is T0. The spring is then cut in half and the same mass is attached and the system is set up to oscillate on a frictionless inclined plane making an angle θ to the horizontal. Determine the period of the oscillations on the inclined plane in terms of T0.

9. A 5.0 kg object undergoes a time-varying force as shown in the graph below. If the velocity at t = 0.0 s is +1.0 m/s, what is the velocity of the object at t = 7 s? 0 1 2 3 4 5 6 7 8 time (s) 1 2 3 4 Force (N)

10. A radio controlled car is attached to a stake in the ground by a 3.00 m long piece of string, and is forced to move in a circular path. The car has an initial angular velocity of 1.00 rad/s and smoothly accelerates at a rate of 4.00 rad/s2. The string will break if the centripetal acceleration exceeds 2.43 × 102 m/s2. How long can the car accelerate at this rate before the string breaks?

11. A point mass m is connected to an ideal spring on a horizontal frictionless surface. The mass is pulled a short distance and then released. Which of the following is the most correct plot of the kinetic energy as a function of potential energy?

The following information applies to questions 12 and 13 A paper helicopter with rotor radius r and weight W is dropped from a height h in air with a density of ρ. r Assuming that the helicopter quickly reaches terminal velocity, a function for the time of flight T can be found in the form T = khαrβρδW ω. where k is an unknown dimensionless constant (actually, 1.164). α, β, δ, and ω are constant exponents to be determined. 12. Determine α.

The following information applies to questions 12 and 13 A paper helicopter with rotor radius r and weight W is dropped from a height h in air with a density of ρ. r Assuming that the helicopter quickly reaches terminal velocity, a function for the time of flight T can be found in the form T = khαrβρδW ω. where k is an unknown dimensionless constant (actually, 1.164). α, β, δ, and ω are constant exponents to be determined. 13. Determine β.

14. A disk of moment of inertia I, mass M, and radius R has a cord wrapped around it tightly as shown in the diagram. The disk is free to slide on its side as shown in the top down view. A constant force of T is applied to the end of the cord and accelerates the disk along a frictionless surface. Force T Solid disk rope After the disk has accelerated some distance, determine the ratio of the translational KE to total KE of the disk, KEtranslational/KEtotal =

15. The maximum torque output from the engine of a new experimental car of mass m is τ. The maximum rotational speed of the engine is ω. The engine is designed to provide a constant power output P. The engine is connected to the wheels via a perfect transmission that can smoothly trade torque for speed with no power loss. The wheels have a radius R, and the coefficient of static friction between the wheels and the road is µ. What is the maximum sustained speed v the car can drive up a 30 degree incline? Assume no frictional losses and assume µ is large enough so that the tires do not slip.

16. An object of mass m1 initially moving at speed v0 collides with an object of mass m2 = αm1, where α < 1, that is initially at rest. The collision could be completely elastic, completely inelastic, or partially inelastic. After the collision the two objects move at speeds v1 and v2. Assume that the collision is one dimensional, and that object one cannot pass through object two. After the collision, the speed ratio r1 = v1/v0 of object 1 is bounded by

17. A spherical cloud of dust in space has a uniform density ρ0 and a radius R0. The gravitational acceleration of free fall at the surface of the cloud due to the mass of the cloud is g0. A process occurs (heat expansion) that causes the cloud to suddenly grow to a radius 2R0, while maintaining a uniform (but not constant) density. The gravitational acceleration of free fall at a point R0 away from the center of the cloud due to the mass of the cloud is now

18. Consider the following diagram of a box and two weight scales. Scale A supports the box via a massless rope. A pulley is attached to the top of the box; a second massless rope passes over the pulley, one end is attached to the box and the other end to scale B. The two scales read indicate the tensions TA and TB in the ropes. Originally scale A reads 30 Newtons and scale B reads 20 Newtons. Box Frame Pulley Scale A Scale B TB TA If an additional force pulls down on scale B so that the reading increases to 30 Newtons, what will be the new reading on scale A?

19. A helicopter is flying horizontally at constant speed. A perfectly flexible uniform cable is suspended beneath the helicopter; air friction on the cable is not negligible. Which of the following diagrams best shows the shape of the cable as the helicopter flies through the air to the right?

20. A crew of scientists has built a new space station. The space station is shaped like a wheel of radius R, with essentially all its mass M at the rim. When the crew arrives, the station will be set rotating at a rate that causes an object at the rim to have radial acceleration g, thereby simulating Earth’s surface gravity. This is accomplished by two small rockets, each with thrust T newtons, mounted on the station’s rim. How long a time t does one need to fire the rockets to achieve the desired condition? gR3 M/(2T)

21. Two pulleys (shown in figure) are made of the same metal with density ρ. Pulley A is a uniform disk with radius R. Pulley B is identical except a circle of R/2 is removed from the center. When two boxes M = αm (α > 1) are connected over the pulleys through a massless rope and move without slipping, what is the ratio between the accelerations in system A and B? The mass of pulley A is M + m. m M m M Pulley A Pulley B

22. A body of mass M and a body of mass m ≪M are in circular orbits about their center of mass under the influence of their mutual gravitational attraction to each other. The distance between the bodies is R, which is much larger than the size of either body. A small amount of matter δm ≪m is removed from the body of mass m and transferred to the body of mass M. The transfer is done in such a way so that the orbits of the two bodies remain circular, and remain separated by a distance R. Which of the following statements is correct?

The following information applies to questions 23 and 24 A 100 kg astronaut carries a launcher loaded with a 10 kg bowling ball; the launcher and the astronaut’s spacesuit have negligible mass. The astronaut discovers that firing the launcher results in the ball moving away from her at a relative speed of 50 m/s. 23. What is the impulse delivered to the astronaut when firing the launcher?

The following information applies to questions 23 and 24 A 100 kg astronaut carries a launcher loaded with a 10 kg bowling ball; the launcher and the astronaut’s spacesuit have negligible mass. The astronaut discovers that firing the launcher results in the ball moving away from her at a relative speed of 50 m/s. 24. The astronaut in the previous situation is now moving at 10 m/s (as measured in a certain frame of reference). She wishes to fire the launcher so that her velocity turns through as large an angle as possible (in this frame of reference). What is this maximum angle? (Hint: a diagram may be useful.)

25. A block with mass m is released from rest at the top of a frictionless ramp. The block starts at a height h1 above the base of the ramp, slides down the ramp, and then up a second ramp. The coefficient of kinetic friction between the block and the second ramp is µk. If both ramps make an angle of θ with the horizontal, to what height h2 above the base of the second ramp will the block rise?

1. A 600 meter wide river flows directly south at 4.0 m/s. A small motor boat travels at 5.0 m/s in still water and points in such a direction so that it will travel directly east relative to the land. 4 m/s 600 m The time it takes to cross the river is closest to

2. A car travels directly north on a straight highway at a constant speed of 80 km/hr for a distance of 25 km. The car then continues directly north at a constant speed of 50 km/hr for a distance of 75 more kilometers. The average speed of the car for the entire journey is closest to

3. The force of friction on an airplane in level flight is given by Ff = kv2, where k is some constant, and v is the speed of the airplane. When the power output from the engines is P0, the plane is able to fly at a speed v0. If the power output of the engines is increased by 100% to 2P0, the airplane will be able to fly at a new speed given by

4. A 2.0 kg box is originally at rest on a horizontal surface where the coefficient of static friction between the box and the surface is µs and the coefficient of the kinetic friction between the box and the surface is µk = 0.90µs. An external horizontal force of magnitude P is then applied to the box. Which of the following is a graph of the acceleration of the box a versus the external force P?

5. A 470 gram lead ball is launched at a 60 degree angle above the horizontal with an initial speed of 100 m/s directly toward a target on a vertical cliffwall that is 150 meters away as shown in the figure. target 60◦ 150 meters 100 m/s Ignoring air friction, by what distance does the lead ball miss the target when it hits the cliffwall?

6. Three trolley carts are free to move on a one dimensional frictionless horizontal track. Cart A has a mass of 1.9 kg and an initial speed of 1.7 m/s to the right; Cart B has a mass of 1.1 kg and an initial speed of 2.5 m/s to the left; cart C has a mass of 1.3 kg and is originally at rest. Collisions between carts A and B are perfectly elastic; collisions between carts B and C are perfectly inelastic. Cart A Cart B Cart C 1.9 kg 1.1 kg 1.3 kg 1.7 m/s 2.5 m/s What is the velocity of the center of mass of the system of the three carts after the last collision?

The following information applies to questions 7 and 8 Cart A Cart B Cart C m 3m 9m v0 0 m 1 m 2 m Carts A, B, and C are on a long horizontal frictionless track. The masses of the carts are m, 3m, and 9m. Originally cart B is at rest at the 1.0 meter mark and cart C is at rest on the 2.0 meter mark. Cart A is originally at the zero meter mark moving toward the cart B at a speed of v0. 7. Assuming that all collisions are completely inelastic, what is the final speed of cart C?

The following information applies to questions 7 and 8 Cart A Cart B Cart C m 3m 9m v0 0 m 1 m 2 m Carts A, B, and C are on a long horizontal frictionless track. The masses of the carts are m, 3m, and 9m. Originally cart B is at rest at the 1.0 meter mark and cart C is at rest on the 2.0 meter mark. Cart A is originally at the zero meter mark moving toward the cart B at a speed of v0. 8. Assuming that all collisions are completely elastic, what is the final speed of cart C?

The following information applies to questions 9 and 10 A 0.650 kg ball moving at 5.00 m/s collides with a 0.750 kg ball that is originally at rest. After the collision, the 0.750 kg ball moves offwith a speed of 4.00 m/s, and the 0.650 kg ball moves off at a right angle to the final direction of motion of the 0.750 kg ball. 9. What is the final speed of the 0.650 kg ball?

The following information applies to questions 9 and 10 A 0.650 kg ball moving at 5.00 m/s collides with a 0.750 kg ball that is originally at rest. After the collision, the 0.750 kg ball moves offwith a speed of 4.00 m/s, and the 0.650 kg ball moves off at a right angle to the final direction of motion of the 0.750 kg ball. 10. Let the change in total kinetic energy in this collision be defined by ∆K = Kf −Ki, where Kf is the total final kinetic energy, and Ki is the total initial kinetic energy. Which of the following is true?

11. A sphere floats in water with 2/3 of the volume of the sphere submerged. The sphere is removed and placed in oil that has 3/4 the density of water. If it floats in the oil, what fraction of the sphere would be submerged in the oil?

The following information applies to questions 12 and 13 A pendulum consists of a small bob of mass m attached to a fixed point by a string of length L. The pendulum bob swings down from rest from an initial angle θmax < 90 degrees. 12. Which of the following statements about the pendulum bob’s acceleration is true?

The following information applies to questions 12 and 13 A pendulum consists of a small bob of mass m attached to a fixed point by a string of length L. The pendulum bob swings down from rest from an initial angle θmax < 90 degrees. 13. Consider the pendulum bob when it is at an angle θ = 1 2θmax on the way up (moving toward θmax). What is the direction of the acceleration vector?

The following information applies to questions 14 and 15 A 3.0 meter long massless rod is free to rotate horizontally about its center. Two 5.0 kg point objects are originally located at the ends of the rod; they are free to slide on the frictionless rod and are kept from flying offthe rod by an inflexible massless rope that connects the two objects. Originally the system is rotating at 4.0 radians per second; assume the system is completely frictionless; and ignore any concerns about instability of the system. 14. Calculate the original tension in the rope.

The following information applies to questions 14 and 15 A 3.0 meter long massless rod is free to rotate horizontally about its center. Two 5.0 kg point objects are originally located at the ends of the rod; they are free to slide on the frictionless rod and are kept from flying offthe rod by an inflexible massless rope that connects the two objects. Originally the system is rotating at 4.0 radians per second; assume the system is completely frictionless; and ignore any concerns about instability of the system. 15. The rope is slowly tightened by a small massless motor attached to one of the objects. It is done in such a way as to pull the two objects closer to the center of the rotating rod. How much work is done by the motor in pulling the two objects from the ends of the rod until they are each 0.5 meters from the center of rotation?

16. Shown below is a graph of potential energy as a function of position for a 0.50 kg object. 1 2 3 4 5 x (cm) 0 10 20 -10 E (J) 6 Which of the following statements is NOT true in the range 0 cm < x < 6 cm?

17. A flywheel can rotate in order to store kinetic energy. The flywheel is a uniform disk made of a material with a density ρ and tensile strength σ (measured in Pascals), a radius r, and a thickness h. The flywheel is rotating at the maximum possible angular velocity so that it does not break. Which of the following expression correctly gives the maximum kinetic energy per kilogram that can be stored in the flywheel? Assume that α is a dimensionless constant.

18. Shown below are three graphs of the same data. y x log(y) x log(y) log(x) Which is the correct functional relationship between the data points? Assume a and b are con- stants.

The following information applies to questions 19 and 20 A U-tube manometer consists of a uniform diameter cylindrical tube that is bent into a U shape. It is originally filled with water that has a density ρw. The total length of the column of water is L. Ignore surface tension and viscosity. 19. The water is displaced slightly so that one side moves up a distance x and the other side lowers a distance x. Find the frequency of oscillation.

The following information applies to questions 19 and 20 A U-tube manometer consists of a uniform diameter cylindrical tube that is bent into a U shape. It is originally filled with water that has a density ρw. The total length of the column of water is L. Ignore surface tension and viscosity. 20. Oil with a density half that of water is added to one side of the tube until the total length of oil is equal to the total length of water. Determine the equilibrium height difference between the two sides

21. An object launched vertically upward from the ground with a speed of 50 m/s bounces offof the ground on the return trip with a coefficient of restitution given by CR = 0.9, meaning that immediately after a bounce the upward speed is 90% of the previous downward speed. The ball continues to bounce like this; what is the total amount of time between when the ball is launched and when it finally comes to a rest? Assume the collision time is zero; the bounce is instantaneous. Treat the problem as ideally classical and ignore any quantum effects that might happen for very small bounces.

22. A solid ball is released from rest down inclines of various inclination angles θ but through a fixed vertical height h. The coefficient of static and kinetic friction are both equal to µ. Which of the following graphs best represents the total kinetic energy of the ball at the bottom of the incline as a function of the angle of the incline?

23. A 2.0 kg object falls from rest a distance of 5.0 meters onto a 6.0 kg object that is supported by a vertical massless spring with spring constant k = 72 N/m. The two objects stick together after the collision, which results in the mass/spring system oscillating. What is the maximum magnitude of the displacement of the 6.0 kg object from its original location before it is struck by the falling object?

24. The speed of a transverse wave on a long cylindrical steel string is given by v = s T M/L, where T is the tension in the string, M is the mass, and L is the length of the string. Ignore any string stiffness, and assume that it does not stretch when tightened. Consider two steel strings of the same length, the first with radius r1 and a second thicker string with radius r2 = 4r1. Each string is tightened to the maximum possible tension without breaking. What is the ratio f1/f2 of the fundamental frequencies of vibration on the two strings?

25. Two identical carts A and B each with mass m are connected via a spring with spring constant k. Two additional springs, identical to the first, connect the carts to two fixed points. The carts are free to oscillate under the effect of the springs in one dimensional frictionless motion. Cart A Cart B Under suitable initial conditions, the two carts will oscillate in phase according to xA(t) = x0 sin ω1t = xB(t) where xA and xB are the locations of carts A and B relative to their respective equilibrium positions. Under other suitable initial conditions, the two carts will oscillate exactly out of phase according to xA(t) = x0 sin ω2t = −xB(t) Determine the ratio ω2/ω1 √

1. A cyclist travels at a constant speed of 22.0 km/hr except for a 20 minute stop. The cyclist’s average speed was 17.5 km/hr. How far did the cyclist travel?

Questions 2 to 4 refer to the three graphs below which show velocity of three objects as a function of time. Each object is moving only in one dimension. 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) Object I Object II Object III 2. Rank the magnitudes of the average acceleration during the ten second interval.

Questions 2 to 4 refer to the three graphs below which show velocity of three objects as a function of time. Each object is moving only in one dimension. 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) Object I Object II Object III 3. Rank the magnitudes of the maximum velocity achieved during the ten second interval.

Questions 2 to 4 refer to the three graphs below which show velocity of three objects as a function of time. Each object is moving only in one dimension. 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) 0 2 4 6 8 10 −2 0 +2 +4 velocity (m/s) time (s) Object I Object II Object III 4. Rank the magnitudes of the distance traveled during the ten second interval.

5. A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of 150,000,000 km from the Sun. Which of the following is the closest for the acceleration of the Earth in this orbit?

6. A child is sliding out of control with velocity vc across a frozen lake. He runs head-on into another child, initially at rest, with 3 times the mass of the first child, who holds on so that the two now slide together. What is the velocity of the couple after the collision?

7. An ice skater can rotate about a vertical axis with an angular velocity ω0 by holding her arms straight out. She can then pull in her arms close to her body so that her angular velocity changes to 2ω0, without the application of any external torque. What is the ratio of her final rotational kinetic energy to her initial rotational kinetic energy? √

8. When a block of wood with a weight of 30 N is completely submerged under water the buoyant force on the block of wood from the water is 50 N. When the block is released it floats at the surface. What fraction of the block will then be visible above the surface of the water when the block is floating?

9. A spring has an equilibrium length of 2.0 meters and a spring constant of 10 newtons/meter. Alice is pulling on one end of the spring with a force of 3.0 newtons. Bob is pulling on the opposite end of the spring with a force of 3.0 newtons, in the opposite direction. What is the resulting length of the spring?

10. Which of the following changes will result in an increase in the period of a simple pendulum?

11. A large metal cylindrical cup floats in a rectangular tub half-filled with water. The tap is placed over the cup and turned on, releasing water at a constant rate. Eventually the cup sinks to the bottom and is completely submerged. Which of the following five graphs could represent the water level in the sink as a function of time?

12. You are given a large collection of identical heavy balls and lightweight rods. When two balls are placed at the ends of one rod and interact through their mutual gravitational attraction (as is shown on the left), the compressive force in the rod is F. Next, three balls and three rods are placed at the vertexes and edges of an equilateral triangle (as is shown on the right). What is the compressive force in each rod in the latter case?

13. The apparatus in the diagram consists of a solid cylinder of radius 1 cm attached at the center to two disks of radius 2 cm. It is placed on a surface where it can roll, but will not slip. A thread is wound around the central cylinder. When the thread is pulled at the angle θ = 90 to the horizontal (directly up), the apparatus rolls to the right. Which below is the largest value of θ for which it will not roll to the right when pulling on the thread? θ

14. You have 5 different strings with weights tied at various point, all hanging from the ceiling, and reaching down to the floor. The string is released at the top, allowing the weights to fall. Which one will create a regular, uniform beating sound as the weights hit the floor?

15. A vertical mass-spring oscillator is displaced 2.0 cm from equilibrium. The 100 g mass passes through the equilib- rium point with a speed of 0.75 m/s. What is the spring constant of the spring?

Questions 16 and 17 refer to the information and diagram below. Jonathan using rope to lift a box with Becky in it; the box is hanging offthe side of a bridge, Jonathan is on the top. A rope is hooked up from the box and passes a fixed railing; Jonathan holds the box up by pressing the rope against the rail- ing with a massless, frictionless physics textbook. The static friction coefficient between the rope and railing is µs; the kinetic friction coefficient between the rope and railing is µk < µs; the mass of the box and Becky combined is M; and the initial height of the bottom of the box above the ground is h. Assume a massless rope. Becky Loose rope Floor Jonathan, pushes rope fixed hard railing on book against 16. What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping?

Questions 16 and 17 refer to the information and diagram below. Jonathan using rope to lift a box with Becky in it; the box is hanging offthe side of a bridge, Jonathan is on the top. A rope is hooked up from the box and passes a fixed railing; Jonathan holds the box up by pressing the rope against the rail- ing with a massless, frictionless physics textbook. The static friction coefficient between the rope and railing is µs; the kinetic friction coefficient between the rope and railing is µk < µs; the mass of the box and Becky combined is M; and the initial height of the bottom of the box above the ground is h. Assume a massless rope. Becky Loose rope Floor Jonathan, pushes rope fixed hard railing on book against 17. Jonathan applies a normal force that is just enough to keep the rope from slipping. Becky makes a small jump, barely leaving contact with the floor of the box. Upon landing on the box, the force of the impact causes the rope to start slipping from Jonathan’s hand. At what speed does the box smash into the ground? Assume Jonathan’s normal force does not change. (A) √2gH(µk/µs)

18. A block of mass m = 3.0 kg slides down one ramp, and then up a second ramp. The coefficient of kinetic friction between the block and each ramp is µk = 0.40. The block begins at a height h1 = 1.0 m above the horizontal. Both ramps are at a 30◦incline above the horizontal. To what height above the horizontal does the block rise on the second ramp?

Questions 19 and 20 refer to the following information A particle of mass 2.00 kg moves under a force given by F = −(8.00 N/m)(xˆi + yˆj) where ˆi and ˆj are unit vectors in the x and y directions. The particle is placed at the origin with an initial velocity v = (3.00 m/s)ˆi + (4.00 m/s)ˆj. 19. After how much time will the particle first return to the origin?

Questions 19 and 20 refer to the following information A particle of mass 2.00 kg moves under a force given by F = −(8.00 N/m)(xˆi + yˆj) where ˆi and ˆj are unit vectors in the x and y directions. The particle is placed at the origin with an initial velocity v = (3.00 m/s)ˆi + (4.00 m/s)ˆj. 20. What is the maximum distance between the particle and the origin?

21. An engineer is given a fixed volume Vm of metal with which to construct a spherical pressure vessel. Interestingly, assuming the vessel has thin walls and is always pressurized to near its bursting point, the amount of gas the vessel can contain, n (measured in moles), does not depend on the radius r of the vessel; instead it depends only on Vm (measured in m3), the temperature T (measured in K), the ideal gas constant R (measured in J/(K·mol)), and the tensile strength of the metal σ (measured in N/m2). Which of the following gives n in terms of these parameters?

22. This graph depicts the torque output of a hypothetical gasoline engine as a function of rotation frequency. The engine is incapable of running outside of the graphed range. I II III Engine Revolutions per Minute 0 0 10 20 30 Output Torque (Nm) 1,000 2,000 At what engine RPM (revolutions per minute) does the engine produce maximum power?

23. A particle is launched from the surface of a uniform, stationary spherical planet at an angle to the vertical. The particle travels in the absence of air resistance and eventually falls back onto the planet. Spaceman Fred describes the path of the particle as a parabola using the laws of projectile motion. Spacewoman Kate recalls from Kepler’s laws that every bound orbit around a point mass is an ellipse (or circle), and that the gravitation due to a uniform sphere is identical to that of a point mass. Which of the following best explains the discrepancy?

24. A turntable is supported on a Teflon ring of inner radius R and outer radius R+δ (δ ≪R), as shown in the diagram. To rotate the turntable at a constant rate, power must be supplied to overcome friction. The manufacturer of the turntable wishes to reduce the power required without changing the rotation rate, the weight of the turntable, or the coefficient of friction of the Teflon surface. Engineers propose two solutions: increasing the width of the bearing (increasing δ), or increasing the radius (increasing R). What are the effects of these proposed changes? R

25. A hollow cylinder with a very thin wall (like a toilet paper tube) and a block are placed at rest at the top of a plane with inclination θ above the horizontal. The cylinder rolls down the plane without slipping and the block slides down the plane; it is found that both objects reach the bottom of the plane simultaneously. What is the coefficient of kinetic friction between the block and the plane?

Confidential Questions 1 to 3 refer to the figure below which shows a representation of the motion of a squirrel as it runs in a straight-line along a telephone wire. The letters A through E refer to the indicated times. A B C D E time 1. If the graph is a graph of POSITION vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?

Confidential Questions 1 to 3 refer to the figure below which shows a representation of the motion of a squirrel as it runs in a straight-line along a telephone wire. The letters A through E refer to the indicated times. A B C D E time 2. If, instead, the graph is a graph of VELOCITY vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?

Confidential Questions 1 to 3 refer to the figure below which shows a representation of the motion of a squirrel as it runs in a straight-line along a telephone wire. The letters A through E refer to the indicated times. A B C D E time 3. If, instead, the graph is a graph of ACCELERATION vs. TIME and the squirrel starts from rest, then the squirrel has the greatest speed at what time(s) or during what time interval?

4. Two teams of movers are lowering a piano from the window of a 10 floor apartment building. The rope breaks when the piano is 30 meters above the ground. The movers on the ground, alerted by the shouts of the movers above, first notice the piano when it is 14 meters above the ground. How long do they have to get out of the way before the piano hits the ground?

Confidential 5. Two projectiles are launched from a 35 meter ledge as shown in the diagram. One is launched from a 37 degree angle above the horizontal and the other is launched from 37 degrees below the horizontal. Both of the launches are given the same initial speed of v0 = 50 m/s. Projectile 2 Projectile 1 35 m The difference in the times of flight for these two projectiles, t1 −t2, is closest to

6. A projectile is launched across flat ground at an angle θ to the horizontal and travels in the absence of air resistance. It rises to a maximum height H and lands a horizontal distance R away. What is the ratio H/R?

7. Harry Potter is sitting 2.0 meters from the center of a merry-go-round when Draco Malfoy casts a spell that glues Harry in place and then makes the merry-go-round start spinning on its axis. Harry has a mass of 50.0 kg and can withstand 5.0 g’s of acceleration before passing out. What is the magnitude of Harry’s angular momentum when he passes out?

8. A car attempts to accelerate up a hill at an angle θ to the horizontal. The coefficient of static friction between the tires and the hill is µ > tan θ. What is the maximum acceleration the car can achieve (in the direction upwards along the hill)? Neglect the rotational inertia of the wheels.

Confidential 9. A point object of mass M hangs from the ceiling of a car from a massless string of length L. It is observed to make an angle θ from the vertical as the car accelerates uniformly from rest. Find the acceleration of the car in terms of θ, M, L, and g. L θ

10. A block of mass m1 is on top of a block of mass m2. The lower block is on a horizontal surface, and a rope can pull horizontally on the lower block. The coefficient of kinetic friction for all surfaces is µ. What is the resulting acceleration of the lower block if a force F is applied to the rope? Assume that F is sufficiently large so that the top block slips on the lower block. 2 1 F

11. The three masses shown in the accompanying diagram are equal. The pulleys are small, the string is lightweight, and friction is negligible. Assuming the system is in equilibrium, what is the ratio a/b? The figure is not drawn to scale! a b

12. A ball with mass m projected horizontally offthe end of a table with an initial kinetic energy K. At a time t after it leaves the end of the table it has kinetic energy 3K. What is t? Neglect air resistance.

13. A ball of mass M and radius R has a moment of inertia of I = 2 5MR2. The ball is released from rest and rolls down the ramp with no frictional loss of energy. The ball is projected vertically upward offa ramp as shown in the diagram, reaching a maximum height ymax above the point where it leaves the ramp. Determine the maximum height of the projectile ymax in terms of h. h

14. A 5.0 kg block with a speed of 8.0 m/s travels 2.0 m along a horizontal surface where it makes a head-on, perfectly elastic collision with a 15.0 kg block which is at rest. The coefficient of kinetic friction between both blocks and the surface is 0.35. How far does the 15.0 kg block travel before coming to rest?

15. A small block moving with initial speed v0 moves smoothly onto a sloped big block of mass M. After the small block reaches the height h on the slope, it slides down. Find the height h. (A) h = v2

16. Following the previous set up, find the speed v of the small block after it leaves the slope.

17. Four masses m are arranged at the vertices of a tetrahedron of side length a. What is the gravitational potential energy of this arrangement? (A) −2 Gm2

18. Which of the following represents the force corresponding to the given potential?

Confidential 19. Consider the following graphs of position vs. time. I. II. III. Which of the graphs could be the motion of a particle in the given potential?

20. Consider the following graph of position vs. time, which represents the motion of a certain particle in the given potential. What is the total energy of the particle?

21. The gravitational self potential energy of a solid ball of mass density ρ and radius R is E. What is the gravitational self potential energy of a ball of mass density ρ and radius 2R?

22. A balloon filled with helium gas is tied by a light string to the floor of a car; the car is sealed so that the motion of the car does not cause air from outside to affect the balloon. If the car is traveling with constant speed along a circular path, in what direction will the balloon on the string lean towards? D B C A

23. Two streams of water flow through the U-shaped tubes shown. The tube on the left has cross-sectional area A, and the speed of the water flowing through it is v; the tube on the right has cross-sectional area A′ = 1/2A. If the net force on the tube assembly is zero, what must be the speed v′ of the water flowing through the tube on the right? Neglect gravity, and assume that the speed of the water in each tube is the same upon entry and exit.

Confidential 24. A uniform circular disk of radius R begins with a mass M; about an axis through the center of the disk and perpendicular to the plane of the disk the moment of inertia is I0 = 1 2MR2. A hole is cut in the disk as shown in the diagram. In terms of the radius R and the mass M of the original disk, what is the moment of inertia of the resulting object about the axis shown? axis of rotation R R R/2

25. Spaceman Fred’s spaceship (which has negligible mass) is in an elliptical orbit about Planet Bob. The minimum distance between the spaceship and the planet is R; the maximum distance between the spaceship and the planet is 2R. At the point of maximum distance, Spaceman Fred is traveling at speed v0. He then fires his thrusters so that he enters a circular orbit of radius 2R. What is his new speed? R 2R

1. A 0.3 kg apple falls from rest through a height of 40 cm onto a flat surface. Upon impact, the apple comes to rest in 0.1 s, and 4 cm2 of the apple comes into contact with the surface during the impact. What is the average pressure exerted on the apple during the impact? Ignore air resistance.

The following information is used for questions 2 and 3. Three blocks of identical mass are placed on a frictionless table as shown. The center block is at rest, whereas the other two blocks are moving directly towards it at identical speeds v. The center block is initially closer to the left block than the right one. All motion takes place along a single horizontal line. 2. Suppose that all collisions are instantaneous and perfectly elastic. After a long time, which of the following is true?

The following information is used for questions 2 and 3. Three blocks of identical mass are placed on a frictionless table as shown. The center block is at rest, whereas the other two blocks are moving directly towards it at identical speeds v. The center block is initially closer to the left block than the right one. All motion takes place along a single horizontal line. 3. Suppose, instead, that all collisions are instantaneous and perfectly inelastic. After a long time, which of the following is true?

4. A spaceman of mass 80 kg is sitting in a spacecraft near the surface of the Earth. The spacecraft is accelerating upward at five times the acceleration due to gravity. What is the force of the spaceman on the spacecraft?

5. Three equal mass satellites A, B, and C are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are LA, LB, and LC. Which of the following statements is correct? A B C

6. An object is thrown with a fixed initial speed v0 at various angles α relative to the horizon. At some constant height h above the launch point the speed v of the object is measured as a function of the initial angle α. Which of the following best describes the dependence of v on α? (Assume that the height h is achieved, and assume that there is no air resistance.)

7. A bird is flying in a straight line initially at 10 m/s. It uniformly increases its speed to 15 m/s while covering a distance of 25 m. What is the magnitude of the acceleration of the bird?

The following information is used for questions 8 and 9. A flat disk rotates about an axis perpendicular to the plane of the disk and through the center of the disk with an angular velocity as shown in the graph below. 0 2 4 3 2 1 −2 Angular Velocity (rad/s) Time (s) 8. Determine the angular acceleration of the disk when t = 2.0 s.

The following information is used for questions 8 and 9. A flat disk rotates about an axis perpendicular to the plane of the disk and through the center of the disk with an angular velocity as shown in the graph below. 0 2 4 3 2 1 −2 Angular Velocity (rad/s) Time (s) 9. Through what net angle does the disk turn during the 3 seconds?

10. A person standing on the edge of a fire escape simultaneously launches two apples, one straight up with a speed of 7 m/s and the other straight down at the same speed. How far apart are the two apples 2 seconds after they were thrown, assuming that neither has hit the ground?

11. A 2.25kg mass undergoes an acceleration as shown below. How much work is done on the mass? 0 2 4 −2 Position (m) 2 4 6 8 10 12 Acceleration (m/s/s)

12. Batman, who has a mass of M = 100 kg, climbs to the roof of a 30 m building and then lowers one end of a massless rope to his sidekick Robin. Batman then pulls Robin, who has a mass of m = 75 kg, up the roof of the building. Approximately how much total work has Batman done after Robin is on the roof?

13. Lucy (mass 33.1 kg), Henry (mass 63.7 kg), and Mary (mass 24.3 kg) sit on a lightweight seesaw at evenly spaced 2.74 m intervals (in the order in which they are listed; Henry is between Lucy and Mary) so that the seesaw balances. Who exerts the most torque (in terms of magnitude) on the seesaw? Ignore the mass of the seesaw.

14. A wooden block (mass M) is hung from a peg by a massless rope. A speeding bullet (with mass m and initial speed v0) collides with the block at time t = 0 and embeds in it. Let S be the system consisting of the block and bullet. Which quantities are conserved between t = −10 s and t = +10 s?

15. A 22.0 kg suitcase is dragged in a straight line at a constant speed of 1.10 m/s across a level airport floor by a student on the way to the 40th IPhO in Merida, Mexico. The individual pulls with a 1.00 x 102 N force along a handle with makes an upward angle of 30.0 degrees with respect to the horizontal. What is the coefficient of kinetic friction between the suitcase and the floor?

16. Two identical objects of mass m are placed at either end of a spring of spring constant k and the whole system is placed on a horizontal frictionless surface. At what angular frequency ω does the system oscillate?

17. You are given a standard kilogram mass and a tuning fork that is calibrated in Hz. You are also provided with a complete collection of laboratory equipment, but none of it is calibrated in SI units. You do not know the values of any fundamental constants. Which of the following quantities could you measure in SI units?

18. A simple pendulum of length L is constructed from a point object of mass m suspended by a massless string attached to a fixed pivot point. A small peg is placed a distance 2L/3 directly below the fixed pivot point so that the pendulum would swing as shown in the figure below. The mass is displaced 5 degrees from the vertical and released. How long does it take to return to its starting position? Fixed Pivot Point Small Peg Point Object of mass m L q  q 2 

19. A certain football quarterback can throw a football a maximum range of 80 meters on level ground. What is the highest point reached by the football if thrown this maximum range? Ignore air friction.

20. Consider a completely inelastic collision between two lumps of space goo. Lump 1 has mass m and originally moves directly north with a speed v0. Lump 2 has mass 3m and originally moves directly east with speed v0/2. What is the final speed of the masses after the collision? Ignore gravity, and assume the two lumps stick together after the collision.

The following information is used for questions 21 and 22. Two stars orbit their common center of mass as shown in the diagram below. The masses of the two stars are 3M and M. The distance between the stars is d. 21. What is the value of the gravitational potential energy of the two star system? (A) −GM2

The following information is used for questions 21 and 22. Two stars orbit their common center of mass as shown in the diagram below. The masses of the two stars are 3M and M. The distance between the stars is d. 22. Determine the period of orbit for the star of mass 3M. q d3

23. A mass is attached to an ideal spring. At time t = 0 the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is T . At what time is the power delivered to the mass by the spring first a maximum?

24. A uniform rectangular wood block of mass M, with length b and height a, rests on an incline as shown. The incline and the wood block have a coefficient of static friction, µs. The incline is moved upwards from an angle of zero through an angle θ. At some critical angle the block will either tip over or slip down the plane. Determine the relationship between a, b, and µs such that the block will tip over (and not slip) at the critical angle. The box is rectangular, and a̸ = b. a θ b

25. Two discs are mounted on thin, lightweight rods oriented through their centers and normal to the discs. These axles are constrained to be vertical at all times, and the discs can pivot frictionlessly on the rods. The discs have identical thickness and are made of the same material, but have differing radii r1 and r2. The discs are given angular velocities of magnitudes ω1 and ω2, respectively, and brought into contact at their edges. After the discs interact via friction it is found that both discs come exactly to a halt. Which of the following must hold? Ignore effects associated with the vertical rods. r1 r2

1. A bird flying in a straight line, initially at 10 m/s, uniformly increases its speed to 18 m/s while covering a distance of 40 m. What is the magnitude of the acceleration of the bird?

2. A cockroach is crawling along the walls inside a cubical room that has an edge length of 3 m. If the cockroach starts from the back lower left hand corner of the cube and finishes at the front upper right hand corner, what is the magnitude of the displacement of the cockroach? √

3. The position vs. time graph for an object moving in a straight line is shown below. What is the instantaneous velocity at t = 2 s? 0 2 4 3 2 1 −2 Position (m) Time (s)

The information below is for the next two problems Shown below is the velocity vs. time graph for a toy car moving along a straight line. 0 2 4 3 2 1 −2 Velocity (m/s) Time (s) 4. What is the maximum displacement from start for the toy car?

The information below is for the next two problems Shown below is the velocity vs. time graph for a toy car moving along a straight line. 0 2 4 3 2 1 −2 Velocity (m/s) Time (s) 5. Which of the following acceleration vs. time graphs most closely represents the acceleration of the

6. A cannon fires projectiles on a flat range at a fixed speed but with variable angle. The maximum range of the cannon is L. What is the range of the cannon when it fires at an angle π 6 above the horizontal? Ignore air resistance. √

7. A toboggan sled is traveling at 2.0 m/s across the snow. The sled and its riders have a combined mass of 120 kg. Another child (mchild = 40 kg) headed in the opposite direction jumps on the sled from the front. She has a speed of 5.0 m/s immediately before she lands on the sled. What is the new speed of the sled? Neglect any effects of friction.

8. Riders in a carnival ride stand with their backs against the wall of a circular room of diameter 8.0 m. The room is spinning horizontally about an axis through its center at a rate of 45 rev/min when the floor drops so that it no longer provides any support for the riders. What is the minimum coefficient of static friction between the wall and the rider required so that the rider does not slide down the wall?

9. A ball of mass m1 travels along the x-axis in the positive direction with an initial speed of v0. It collides with a ball of mass m2 that is originally at rest. After the collision, the ball of mass m1 has velocity v1xˆx + v1yˆy and the ball of mass m2 has velocity v2xˆx + v2yˆy. Consider the following five statements: I) 0 = m1v1x + m1v2x II) m1v0 = m1v1y + m2v2y III) 0 = m1v1y + m2v2y IV) m1v0 = m1v1x + m1v1y V) m1v0 = m1v1x + m2v2x Of these five statements, the system must satisfy

The following information applies to the next two problems An experiment consists of pulling a heavy wooden block across a level surface with a spring force meter. The constant force for each try is recorded, as is the acceleration of the block. The data are shown below. Force F in Newtons 3.05 3.45 4.05 4.45 5.05 acceleration a in meters/second2 0.095 0.205 0.295 0.405 0.495 10. Which is the best value for the mass of the block?

The following information applies to the next two problems An experiment consists of pulling a heavy wooden block across a level surface with a spring force meter. The constant force for each try is recorded, as is the acceleration of the block. The data are shown below. Force F in Newtons 3.05 3.45 4.05 4.45 5.05 acceleration a in meters/second2 0.095 0.205 0.295 0.405 0.495 11. Which is the best value for the coefficient of friction between the block and the surface?

12. A uniform disk rotates at a fixed angular velocity on an axis through its center normal to the plane of the disk, and has kinetic energy E. If the same disk rotates at the same angular velocity about an axis on the edge of the disk (still normal to the plane of the disk), what is its kinetic energy?

13. A mass is attached to the wall by a spring of constant k. When the spring is at its natural length, the mass is given a certain initial velocity, resulting in oscillations of amplitude A. If the spring is replaced by a spring of constant 2k, and the mass is given the same initial velocity, what is the amplitude of the resulting oscillation?

14. A spaceborne energy storage device consists of two equal masses connected by a tether and rotating about their center of mass. Additional energy is stored by reeling in the tether; no external forces are applied. Initially the device has kinetic energy E and rotates at angular velocity ω. Energy is added until the device rotates at angular velocity 2ω. What is the new kinetic energy of the device? √

15. A uniform round tabletop of diameter 4.0 m and mass 50.0 kg rests on massless, evenly spaced legs of length 1.0 m and spacing 3.0 m. A carpenter sits on the edge of the table. What is the maximum mass of the carpenter such that the table remains upright? Assume that the force exerted by the carpenter on the table is vertical and at the edge of the table. 3.0 m 4.0 m 1.0 m

16. A massless spring with spring constant k is vertically mounted so that bottom end is firmly attached to the ground, and the top end free. A ball with mass m falls vertically down on the top end of the spring, becoming attached, so that the ball oscillates vertically on the spring. What equation describes the acceleration a of the ball when it is at a height y above the original position of the top end of the spring? Let down be negative, and neglect air resistance; g is the magnitude of the acceleration of free fall.