Phy
The graph shows the angular velocity ω as a function of time t for a point on a rotating disk. The magnitude of the angular acceleration of the disk at t=2s is most nearly
A 0.7rad/s20.7rad/s2 B 1.5rad/s21.5rad/s2 C 10.0rad/s210.0rad/s2 D 20.0rad/s220.0rad/s2 B
A uniform meterstick is balanced at the center, as shown above. Which of the following shows how a 0.50 kg mass and a 1.0 kg mass could be hung on the meterstick so that the stick stays balanced?
B
An object revolves around a central axis of rotation. The motion of the object is described by the following equation. ω2=(10rad/s)2−(4rad/s2)θ Which two of the following graphs correctly shows the angular motion of the object? Select two answers.
BC
Two identical clay spheres of mass m0 traveling with identical velocity v0 collide with and stick to two different vertical rods. The shorter rod has length 2R and mass 2m0, while the longer rod has length 3R and mass 3m0, as shown in the figure. One sphere collides with the end of the short rod, which begins to rotate about its pivot with angular momentum L1. The other sphere collides with the center of the long rod, which begins to rotate about its pivot with angular momentum L2. What is the ratio of L1 to L2 ?
A 4949 B 2323 C 1 D 4/3 D
For which of the following motions of an object must the acceleration always be zero? I. Any motion in a straight line II. Simple harmonic motion III. Any motion in a circle
A I only B II only C III only D Either I or III, but not II E None of these motions guarantees zero acceleration. E
The figure above represents a stick of uniform density that is attached to a pivot at the right end and has equally spaced marks along its length. Any one or a combination of the four forces shown can be exerted on the stick as indicated. Two of the four forces are exerted on the stick. Which of the following predictions is correct about the change in angular velocity of the stick per unit of time?
A When F1F1 and F2F2 are exerted on the stick, the stick will have the greatest change in angular velocity per unit of time. B When F1F1 and F3F3 are exerted on the stick, the stick will not change its angular velocity per unit of time. C When F2F2 and F3F3 are exerted on the stick, the stick's change in angular velocity per unit of time will be in the clockwise direction. D When F3F3 and F4F4 are exerted on the stick, the stick will have the smallest change in angular velocity per unit of time. a
A student conducts an experiment to determine the relationship between applied torque and change in angular velocity. The student uses the apparatus shown in the figure above, consisting of two disks that are glued together and mounted on a horizontal axle. Blocks of varying mass are hung from a string wound around the smaller disk. The blocks are released from rest, exerting different torques on the disks, and are allowed to fall a fixed distance. For each block, the time of fall t and the final angular velocity ωf of the disks are measured. There is considerable friction between the disks and the axle. Which of the following best represents a plot that can be obtained from the student's data?
A
A uniform ladder of mass M and length L rests against a smooth wall at an angle θ0, as shown in the figure. What is the torque due to the weight of the ladder about its base?
A MgL sin (θ0)MgL sin (θ0) B MgL cos (θ0)MgL cos (θ0) C MgL sin (θ0)2MgL sin (θ0)2 D MgL cos (θ0)2 d
A rod of length 2D0 and mass 2M0 is at rest on a flat, horizontal surface. One end of the rod is connected to a pivot that the rod will rotate around if acted upon by a net torque. A sphere of mass m0 is launched horizontally toward the free end of the rod with velocity v0, as shown in the figure. After the sphere collides with the rod, the sphere sticks to the rod and both objects rotate around the pivot with a common angular velocity. Which of the following predictions is correct about angular momentum and rotational kinetic energy of the sphere-rod system immediately before the collision and immediately after the collision?
A The angular momentum immediately before the collision is greater than the angular momentum immediately after the collision. The rotational kinetic energy immediately before the collision is greater than the rotational kinetic energy immediately after the collision. B The angular momentum immediately before the collision is greater than the angular momentum immediately after the collision. The rotational kinetic energy immediately before the collision is equal to the rotational kinetic energy immediately after the collision. C The angular momentum immediately before the collision is equal to the angular momentum immediately after the collision. The rotational kinetic energy immediately before the collision is greater than the rotational kinetic energy immediately after the collision. D The angular momentum immediately before the collision is equal to the angular momentum immediately after the collision. The rotational kinetic energy immediately before the collision is equal to the rotational kinetic energy immediately after the collision. C
A disk of radius 50cm rotates about a center axle. The angular position as a function of time for a point on the edge of the disk is shown. Which two of the following quantities of the point on the edge of the disk can be correctly mathematically determined from the graph using the methods described? Justify your selections. Select two answers.
A The angular velocity, because this quantity can be determined by calculating the slope of the graph. B The translational speed, because v=rωv=rω. C The angular acceleration, because this quantity can be determined by calculating the area bound by the curve and the horizontal axis from 0s0s to 5s5s. D The translation acceleration, because a=v2ra=v2r. AB
One end of a string is attached to the ceiling, and the other end of the string is attached to a cradle that has a meterstick that runs through it. The meterstick can slide through the cradle so that the horizontal position of a point on the meterstick can be changed in the horizontal direction. Students may hang objects of various masses from the meterstick, as shown in the figure. The students notice that when the meterstick-cradle-object-object system is not balanced, the meterstick will rotate. Consider the situation shown above in which the center of the meterstick is aligned with the center of the cradle, which is at a position of x=0m. Which of the following statements is correct about the torques exerted on the meterstick?
A The torque exerted by the force due to gravity is the greatest torque exerted on the meterstick. B The torque exerted by the 1.0kg1.0kg object is the greatest torque exerted on the meterstick. C The torque exerted by the 0.5kg0.5kg object is the greatest torque exerted on the meterstick. D The torque exerted by the 1.0kg1.0kg object is equal to the torque exerted by the 0.5kg0.5kg object. D
The figures below indicate forces acting on a rod in different situations. The lengths of the force vectors are proportional to the magnitudes of the forces. In which situation is the rod in both translational and rotational equilibrium?
B
A solid metal bar is at rest on a horizontal frictionless surface. It is free to rotate about a vertical axis at the left end. The figures below show forces of different magnitudes that are exerted on the bar at different locations. In which case does the bar's angular speed about the axis increase at the fastest rate?
a one f in the right
Planet X is in a stable circular orbit around a star, as shown in the figure. Which of the following graphs best predicts the angular momentum of the planet as a function of its horizontal position from point A to point B if the planet is moving counterclockwise as viewed in the figure above?
c - straight line
Planet X is in a stable circular orbit around a star, as shown in the figure. Which of the following graphs best predicts the angular momentum of the planet as a function of its horizontal position from point A to point B if the planet is moving counterclockwise as viewed in the figure above?
c straight line across
A rod of length 2D0 and mass 2M0 is at rest on a flat, horizontal surface. One end of the rod is connected to a pivot that the rod will rotate around if acted upon by a net torque. A lump of clay of mass m0 is launched horizontally toward the center of the rod with velocity v0, as shown in the figure. After the clay collides with the rod, the clay sticks to the rod and both objects rotate around the pivot with the common angular velocity ωf. The rotational inertia of the clay-rod system is Is. Which of the following equations could a student use to solve for the common angular velocity ωf immediately after the collision? Justify your selection.
A m0v0D0=Isωfm0v0D0=Isωf, because the clay hits the rod a distance D0D0 from the pivot. B 2m0v0D0=Isωf2m0v0D0=Isωf, because the length of the rod is 2D02D0. C 2M0v0D0=Isωf2M0v0D0=Isωf, because the rod's mass is 2M02M0 and the clay hits the rod a distance D0D0 from the pivot. D 4M0v0D0=Isωf4M0v0D0=Isωf, because the rod's mass is 2M02M0 and its length is 2D02D0. A
A thin rod of length d on a frictionless surface is pivoted about one end, as shown above, and can rotate freely. The rod is at rest when it is struck by a sphere with linear momentum of magnitude pi perpendicular to the rod. The sphere rebounds along its original line of motion with momentum of magnitude pf. What is the magnitude of the angular momentum of the rod immediately after the collision?
A pf - pi B pf + pi C (pf - pi )d D (pf + pi )d D
An axle passes through a pulley. Each end of the axle has a string that is tied to a support. A third string is looped many times around the edge of the pulley and the free end attached to a block of mass mb , which is held at rest. When the block is released, the block falls downward. Consider clockwise to be the positive direction of rotation, frictional effects from the axle are negligible, and the string wrapped around the disk never fully unwinds. The rotational inertia of the pulley is 12MR2 about its center of mass. Which of the following graphs, if any, shows the angular velocity ω of the pulley as a function of time t after the block is released from rest?
B x=y
Two small objects of mass m0 and a rotating platform of radius R and rotational inertia Ip about its center compose a single system. Students use the system to conduct two experiments. The objects are assumed to be point masses. Each object of mass m0 is placed a distance r1 away from the center of the platform such that both masses are on opposite sides of the platform. A constant tangential force F0 is applied to the edge of the platform for a time Δt0, as shown in Figure 1. The system is initially at rest. Each object of mass m0 is placed a distance r2 away from the center of the platform such that both masses are on opposite sides of the platform. Distance r2<r1 . A constant tangential force F0 is applied to the edge of the platform for a time Δt0 , as shown in Figure 2. The system is initially at rest. Which of the following graphs represents the angular displacement of the system as a function of time for the system in experiment 1?
C curve up towards the right
In an experiment, a student applies a net force to the edge of a disk for 10s. The student may change the magnitude and the direction of the force at any time in the experiment. A graph of the net torque exerted on the edge of the disk as a function of time is shown. After observing the graph, the student concludes that the angular momentum of the disk has not changed. Does the graph support the student's conclusion? Justify your answer.
A Yes, because the change in the net torque from 0s0s to 10s10s is zero. B Yes, because the slope of the curve of the graph from 0s0s to 5s5s has the same magnitude but opposite sign as the slope of the curve from 5s5s to 10s10s. C No, because the area bound by the curve and the horizontal axis from 0s0s to 10s10s is not zero. D No, because the graph cannot be used to determine how the angular momentum has changed. C
A satellite that is a spinning cylinder has initial rotational inertia I0 and angular velocity w0. Solar panels unfold from the satellite and are extended outward. The satellite then has rotational inertia If = aI0 and angular velocity wf = bw0, where a and b are constants. Which of the following is true about the constants a and b?
A a = 1 and b = 1 B a > 1 and b < 1 C a > 1 and b = 1 D a < 1 and b < 1 B
A disk-shaped platform has a known rotational inertia ID. The platform is mounted on a fixed axle and rotates in a horizontal plane with an initial angular velocity of ωD in the counterclockwise direction, as shown. After an unknown time interval, the disk comes to rest. A single point on the disk revolves around the center axle hundreds of times before the disk comes to rest. Frictional forces are considered to be constant. In a different experiment, the original disk is replaced with a disk for which frictional forces are considered to be negligible. The disk is set into motion such that it rotates with a constant angular speed. As the disk spins, a small sphere of clay is dropped onto the disk, and the sphere sticks to the disk. Which of the following claims is correct about the angular momentum and the total kinetic energy of the disk-sphere system immediately before and immediately after the collision?
A Angular MomentumKinetic EnergyGreater before the collisionGreater before the collision B Angular MomentumKinetic EnergyGreater before the collisionThe same before and after the collision C Angular MomentumKinetic EnergyThe same before and after the collisionGreater before the collision D Angular MomentumKinetic EnergyThe same before and after the collisionThe same before and after the collision c
A horizontal disk is at rest on top of an axle, and the friction between the disk and axle is not negligible. In experiment 1, an applied torque is exerted to the edge of the disk for 2s. At that moment, the applied torque is removed, and the disk eventually comes to rest as a result of a frictional torque. Graphs of the disk's angular momentum as a function of time are shown in Figures 1 and 2 for the two experiments. Which of the following statements is correct about the applied torque tapplied and frictional torque tfrictionexerted on the disk in experiment 1 and experiment 2 ?
A Applied TorqueFrictional Torqueτapplied 1>τapplied 2τapplied 1>τapplied 2τfriction 1>τfriction 2τfriction 1>τfriction 2 B Applied TorqueFrictional Torqueτapplied 1<τapplied 2τapplied 1<τapplied 2τfriction 1>τfriction 2τfriction 1>τfriction 2 C Applied TorqueFrictional Torqueτapplied 1>τapplied 2τapplied 1>τapplied 2τfriction 1=τfriction 2τfriction 1=τfriction 2 D Applied TorqueFrictional Torqueτapplied 1<τapplied 2τapplied 1<τapplied 2τfriction 1=τfriction 2τfriction 1=τfriction 2 D
An axle passes through a pulley. Each end of the axle has a string that is tied to a support. A third string is looped many times around the edge of the pulley and the free end attached to a block of mass mb , which is held at rest. When the block is released, the block falls downward. Consider clockwise to be the positive direction of rotation, frictional effects from the axle are negligible, and the string wrapped around the disk never fully unwinds. The rotational inertia of the pulley is 12MR2 about its center of mass. How many forces are applied to the pulley-axle system, and how many torques are applied to the pulley about its center when the block is released from rest?
A Number of ForcesNumber of Torques42 B Number of ForcesNumber of Torques41 C Number of ForcesNumber of Torques22 D Number of ForcesNumber of Torques21 B
A horizontal disk of radius 0.2m and mass 0.3kg is mounted on a central vertical axle so that a student can study the relationship between net torque and change in angular momentum of the disk. In the experiment, the student uses a force probe to collect data pertaining to the net torque exerted on the edge of the disk as a function of time, as shown in the graph. The disk is initially at rest. At what instant in time does the disk have the greatest angular momentum?
A 0.00s0.00s B 1.00s1.00s C 1.75s1.75s D 2.50s2.50s D
A horizontal, uniform board of weight 125 N and length 4 m is supported by vertical chains at each end. A person weighing 500 N is sitting on the board. The tension in the right chain is 250 N. How far from the left end of the board is the person sitting?
A 0.4 m B 1.5 m C 2 m D 2.5 m E 3 m B
A student is at rest on a stool that may freely spin about its central axis of rotation. As the stool spins, the student holds onto two dumbbells as the stool spins at an angular speed of 1.2 rad/s with the student's arms completely stretched out from the student's body. At this instant, the student-dumbbell system has rotational inertia of 6 kg⋅m2. The student then brings their arms close to their body, and rotational inertia of the student-dumbbell system is changed to 2 kg⋅m2. What is the new angular speed of the student?
A 0.4 rad/s0.4 rad/s B 1.2 rad/s1.2 rad/s C 3.6 rad/s3.6 rad/s D 7.2 rad/s7.2 rad/s C
An object weighing 120 N is set on a rigid beam of negligible mass at a distance of 3 m from a pivot, as shown above. A vertical force is to be applied to the other end of the beam a distance of 4 m from the pivot to keep the beam at rest and horizontal. What is the magnitude F of the force required?
A 10 N B 30 N C 90 N D 120 N E 160 N C
The figure above represents a stick of uniform density that is attached to a pivot at the right end and has equally spaced marks along its length. Any one or a combination of the four forces shown can be exerted on the stick as indicated. All four forces are exerted on the stick that is initially at rest. What is the angular momentum of the stick after 2.0s ?
A 150 kg⋅m2s150 kg·m2s B 450 kg⋅m2s450 kg·m2s C 650 kg⋅m2s650 kg·m2s D 750 kg⋅m2s a
A horizontal, uniform board of weight 125 N and length 4 m is supported by vertical chains at each end. A person weighing 500 N is sitting on the board. The tension in the right chain is 250 N. What is the tension in the left chain?
A 250 N B 375 N C 500 N D 625 N E 875 N B
A uniform, rigid rod of length 2m lies on a horizontal surface. One end of the rod can pivot about an axis that is perpendicular to the rod and along the plane of the page. A 10N force is applied to the rod at its midpoint at an angle of 37°. A second force F is applied to the free end of the rod so that the rod remains at rest, as shown in the figure. The magnitude of the torque produced by force F is most nearly A
A 3.0N⋅m3.0N⋅m B 6.0N⋅m6.0N⋅m C 8.0N⋅m8.0N⋅m D 12.0N⋅m B
A 3kg horizontal disk of radius 0.2m rotates about its center with an angular velocity of 50rad/s. The edge of the horizontal disk is placed in contact with a wall, and the disk comes to rest after 10s. Which of the following situations associated with linear impulse is analogous to the angular impulse that is described?
A A 3kg3kg block is initially at rest. An applied force of 3N3N is applied to the block, but the block does not move. B A 3kg3kg block is initially at rest. A net force of 3N3N is applied to the block until it has a speed of 10m/s10m/s. C A 3kg3kg block is initially traveling at 10m/s10m/s. An applied force of 3N3N is applied to the block in the direction of its velocity vector for 10s10s. D A 3kg3kg block is initially traveling at 10m/s10m/s. The block encounters a 3N3N frictional force until the block eventually stops. D
A uniform cylinder of radius R, mass M, and rotational inertia I0 is initially at rest. The cylinder is mounted so that it is free to rotate with negligible friction about an axle that is oriented through the center of the cylinder and perpendicular to the page. A light string is wrapped around the cylinder. At time t=0, the string is pulled with a constant force F0, which causes the cylinder to rotate, as shown in the figure. After time t=t0, the string is completely unwound from the cylinder and loses contact with it. If necessary, how can a student determine the change in angular momentum ΔL of the cylinder from t=0 to t=t0?
A A determination is not necessary. The cylinder does not have a change in angular momentum, because the cylinder does not change its horizontal or vertical position as it rotates. B Multiply F0F0, RR, and t0t0 to calculate the angular impulse. This quantity is equal to ΔLΔL. C Use Newton's second law of motion in the rotational form to determine the angular acceleration of the cylinder. Use the angular acceleration to determine the cylinder's change in angular velocity. Multiply the result by the mass of the cylinder. D Calculate the net torque exerted on the cylinder, then divide the net torque by the cylinder's rotational inertia to find the average angular velocity of cylinder. Multiply the result by the average angular velocity by mass MM. B
A disk-shaped platform has a known rotational inertia ID. The platform is mounted on a fixed axle and rotates in a horizontal plane with an initial angular velocity of ωD in the counterclockwise direction, as shown. After an unknown time interval, the disk comes to rest. A single point on the disk revolves around the center axle hundreds of times before the disk comes to rest. Frictional forces are considered to be constant. A student must determine the angular impulse that frictional forces exert on the disk from the moment it rotates with angular velocity ωD in the counterclockwise direction until it stops. Which of the following could the student have used in order to approximate the initial angular velocity of the rotating disk?
A A motion sensor to collect data about the disk's linear position as a function of time after the disk was set into motion B A motion sensor to collect data about the disk's angular position as a function of time after the disk was set into motion C A slow-motion camera that filmed the disk to determine the amount of time it took a particular point on the disk to make its first revolution after the disk was set into motion D A stopwatch to determine the amount of time it took a particular point on the disk to make all of its revolutions from the instant in time when the disk was set into motion to the instant in time when the disk came to rest C
The center of mass of a uniform meterstick is placed on a fulcrum. Two objects of known mass, m1 and m2, are hung at known positions on the meterstick. One end of a string is attached to one end of the meterstick, and the other end of the string is looped around a pulley and connected to hanging object X of unknown mass, as shown in the figure. The meterstick does not rotate and is level with the horizontal. Which of the following measuring devices, if any, should be used to make measurements to determine the unknown mass of object X? Justify your selection.
A A ruler, because the length of the string that is looped around the pulley should be measured. B A protractor, because the angle between the force due to gravity exerted on objects m1m1 and m2m2 and the distance between where the force is applied and the tip of the fulcrum should be measured. C A motion sensor, because the distance per unit of time of object XX should be measured. D No additional equipment or measurements are needed, because the force due to gravity exerted on objects m1m1 and m2m2 and the distance between where the force is applied and the tip of the fulcrum are already known. D
A rod of length ℓ and rotational inertia Ir about one end may freely rotate about a pivot that is attached to the ceiling and upper end of the rod. A sphere of mass M and radius R is launched horizontally with velocity v0 toward the rod. It collides with the bottom of the rod, as shown in Figure 1. After colliding with the rod, the sphere momentarily comes to rest before it falls vertically to the ground. The rod swings upward with an angular speed ω2, as shown in Figure 2. The rotational inertia of the sphere is Is. Which of the following mathematical procedures, if any, can a student use to determine ω2 immediately after the sphere collides with the rod? Justify your selection.
A Angular momentum is conserved, so calculate the initial angular momentum of the sphere as measured from the end of the rod, and set that equal to the angular momentum of the rod immediately after the collision. B Angular momentum is conserved, so calculate the initial angular momentum of the sphere as measured from the pivot, and set that equal to the angular momentum of the rod immediately after the collision. C Energy is conserved, so calculate the initial kinetic energy of the sphere immediately before the collision, and set that equal to the kinetic energy of the rod immediately after the collision. D None of the procedures can be used to determine ω2ω2 immediately after the collision. B
A light string is attached to a massive pulley of known rotational inertia IP, as shown in the figure. A student must determine the relationship between the torque exerted on the pulley and the change in the pulley's angular velocity when the torque is applied for 2.0s. In addition to a stopwatch to measure the time interval, what two measurements could the student make in order to determine the relationship? Select two answers.
A Mass of the pulley B Radius of the pulley C Entire length of the string D Force exerted on the string to turn the pulley BD
Disk X is held at rest above disk Y, which rotates with angular velocity +ω0 about its center, as shown in the figure. Disk Y is slowly lowered onto disk X until the disks remain in contact and travel together at angular velocity +ω1. Which of the following linear collisions is analogous to the rotational collision that is described?
A Block XX travels toward block YY with velocity +v1+v1. Block YY is initially at rest. After the collision, block YY travels with velocity +v1+v1, and block XX remains at rest. B Block XX travels toward block YY with velocity +v1+v1. Block YY is initially at rest. After the collision, block XX and block YY travel together with velocity +v2+v2. C Block XX travels toward block YY with velocity +v1+v1. Block YY is initially traveling toward block XX with velocity −v1−v1. After the collision, block XX and block YY travel together with velocity +v2+v2. D Block XX travels toward block YY with velocity +v1+v1. Block YY is initially traveling toward block XX with velocity −v1−v1. After the collision, block XX travels with velocity −v2−v2 and block YY travels with velocity +v3+v3. B
A Block XX travels toward block YY with velocity +v1+v1. Block YY is initially at rest. After the collision, block YY travels with velocity +v1+v1, and block XX remains at rest. B Block XX travels toward block YY with velocity +v1+v1. Block YY is initially at rest. After the collision, block XX and block YY travel together with velocity +v2+v2. C Block XX travels toward block YY with velocity +v1+v1. Block YY is initially traveling toward block XX with velocity −v1−v1. After the collision, block XX and block YY travel together with velocity +v2+v2. D Block XX travels toward block YY with velocity +v1+v1. Block YY is initially traveling toward block XX with velocity −v1−v1. After the collision, block XX travels with velocity −v2−v2 and block YY travels with velocity +v3+v3.
A Conservation of angular momentum, because the gravitational force exerted by the moon on the planet is the same as that exerted by the planet on the moon B Conservation of angular momentum, because the gravitational force exerted on the moon is always directed toward the planet C Conservation of energy, because the gravitational force exerted on the moon is always directed toward the planet D Conservation of energy, because the gravitational force exerted by the moon on the planet is the same as that exerted by the planet on the moon B
A moon is in an elliptical orbit about a planet as shown above. At point A the moon has speed μA and is at distance RA from the planet. At point B the moon has speed μB. Which of the following explains a correct method for determining the distance of the moon from the planet at point B in terms of the given quantities?
A Conservation of angular momentum, because the gravitational force exerted by the moon on the planet is the same as that exerted by the planet on the moon B Conservation of angular momentum, because the gravitational force exerted on the moon is always directed toward the planet C Conservation of energy, because the gravitational force exerted on the moon is always directed toward the planet D Conservation of energy, because the gravitational force exerted by the moon on the planet is the same as that exerted by the planet on the moon B
Two identical disks rotate about their centers in opposite directions with the same magnitude of angular speed ω0. The top disk is dropped onto the bottom disk, as shown in the figure, so they collide and stick together. Which of the following predictions is correct about the motion of each individual disk after the collision?
A Each disk will spin with the same final angular velocity ωfωf where ωf=0ωf=0. B Each disk will spin with the same final angular velocity ωfωf where ωf>ω0ωf>ω0. C Each disk will spin with the same final angular velocity ωfωf where 0<ωf<ω00<ωf<ω0. D Each disk will spin with the same final angular velocity ωfωf where ωf=ω0ωf=ω0. A
The figure above represents a stick of uniform density that is attached to a pivot at the right end and has equally spaced marks along its length. Any one or a combination of the four forces shown can be exerted on the stick as indicated. Which of the four forces, when exerted in the absence of the other three forces, will change the angular momentum of the stick at the smallest rate?
A F1F1 B F2F2 C F3F3 D F4F4 D
A system consists of a disk rotating on a frictionless axle and a piece of clay moving toward it, as shown in the figure above. The outside edge of the disk is moving at a linear speed v, and the clay is moving at speed v/2. The clay sticks to the outside edge of the disk. How does the angular momentum of the system after the clay sticks compare to the angular momentum of the system before the clay sticks, and what is an explanation for the comparison?
A It is the same because there is no external torque acting on the system. B It is greater because the rotating mass increases, which increases the rotational inertia. C It is less because the speed of the disk decreases when the clay sticks to it. D It is less because the angular momentum of the clay opposes that of the disk. A
The figure above shows a uniform beam of length L and mass M that hangs horizontally and is attached to a vertical wall. A block of mass M is suspended from the far end of the beam by a cable. A support cable runs from the wall to the outer edge of the beam. Both cables are of negligible mass. The wall exerts a force FW on the left end of the beam. For which of the following actions is the magnitude of the vertical component of FW smallest?
A Keeping the support cable and block as shown in the diagram B Moving the lower end of the support cable to the center of the beam and leaving the block at the outer end of the beam C Keeping the lower end of the support cable at the outer end of the beam and moving the block to the center of the beam D Moving both the support cable and the block to the center of the beam D
Two boxes are tied together with a string. They are thrown onto a layer of ice such that they spin around their center of mass C as they slide horizontally across the icy surface, as shown in the figure. The system has an angular momentum L⃗ 0 and an angular velocity ω⃗ 0 at a time t=0s. A graph of the two-block system's center of mass velocity as a function of time is shown. Which of the following predictions are correct about the angular momentum L⃗ 4 and angular velocity ω⃗ 4 at t=4s ?
A L0=L4L0=L4 and ω0=ω4ω0=ω4 B L0>L4L0>L4 and ω0>ω4ω0>ω4 C L0<L4L0<L4 and ω0<ω4ω0<ω4 D L0>L4L0>L4 and ω0<ω4ω0<ω4 A
A lump of clay of mass mclay with speed vclay=8 m/s travels toward various spheres that are suspended from the ceiling by lightweight strings of different lengths, as shown in the figure. For the three scenarios, the clay collides with the suspended sphere and sticks to it. Which of the following correctly relates the angular momentum L of the clay-bob system immediately after the collision for each scenario, where the angular momentum is taken about the point where the string is attached to the ceiling?
A L2>L1=L3L2>L1=L3 B L1=L2=L3L1=L2=L3 C L1>L2=L3L1>L2=L3 D L1>L2>L3 D
A uniform horizontal beam of mass M and length L0 is attached to a hinge at point P, with the opposite end supported by a cable, as shown in the figure. The angle between the beam and the cable is θ0. What is the magnitude of the torque that the cable exerts on the beam?
A MgL02MgL02 B MgLMgL C MgL0sin(θ0)2MgL0sin(θ0)2 D MgL0sin(θ0) a
A uniform disk spins about an axis that passes through the center of the disk and is perpendicular to the plane of the disk, as shown in Figure 1. The disk has an initial angular velocity of ωd and uniformly accelerates to rest over time. The angular velocity of the disk as a function of time is shown in Figure 2. A student must determine the angular displacement of a point on the edge of the disk from t=0 to the instant in time the disk comes to rest if the point's initial velocity is changed to 2ωd but its angular acceleration is the same as shown in Figure 2. How can the graph in Figure 2 be changed before the student can determine the angular displacement? Justify your selection.
A Recreate the graph with a vertical intercept that is twice the value of the intercept shown in Figure 2, because the angular velocity is increased from ωdωd to 2ωd2ωd. The horizontal intercept should be the same in both graphs, because the angular acceleration is the same in both graphs. B Recreate the graph with a vertical intercept that is twice the value of the intercept shown in Figure 2, because the angular velocity is increased from ωdωd to 2ωd2ωd. The slope of the line should be the same in both graphs, because the angular acceleration is the same in both graphs. C Recreate the graph with the same vertical intercept in both graphs, because the angular acceleration is the same in both graphs. The slope of the curve in the new graph should be twice as steep as the slope in Figure 2, because the angular velocity is increased from ωdωd to 2ωd2ωd. D Recreate the graph with the same vertical intercept in both graphs, because the angular acceleration is the same in both graphs. The horizontal intercept in the new graph should be twice the value of the horizontal intercept in Figure 2, because the angular velocity is increased from ωdωd to 2ωd2ωd. B
In an experiment, one end of a string is attached to object X, and the other end of the string is attached to object Y. The string is then placed around a pulley, as shown in the figure, and the system containing object X, object Y, the string, and the pulley remains at rest. A support string connects the pulley to the ceiling. Which two of the following measuring devices could students use to mathematically verify that the net torque exerted on the system is zero? Select two answers.
A Stopwatch B Meterstick C Protractor D Electronic balance BD
An object rotates with an angular speed that varies with time, as shown in the graph. How can the graph be used to determine the magnitude of the angular acceleration α of the object? Justify your selection.
A Subtract the greatest value of the angular speed from the smallest value of the angular speed, because α=Δωα=Δω. B Determine the slope of the line from 0s0s to 2s2s, because the slope represents ΔωΔtΔωΔt. C Determine the area bounded by the line and the horizontal axis from 0s0s to 2s2s, because α=12ωΔtα=12ωΔt. D The angular acceleration cannot be determined without knowing the rotational inertia of the object. B
A system consists of two spheres, of mass m and 2m, connected by a rod of negligible mass, as shown above. The system is held at its center of mass with the rod horizontal and released from rest near Earth's surface at time t = 0 . Which of the following best explains why the system does not rotate around its center of mass as it falls?
A The Earth exerts the same gravitational force on both spheres, causing them to accelerate at the same rate. B The Earth exerts the same gravitational force on both spheres, generating torques that cancel out. C The Earth exerts a larger gravitational force on the sphere of mass 2m, but that sphere is closer to the center of mass and the torques cancel out. D The Earth exerts a larger gravitational force on the sphere of mass 2m, but that sphere has more inertia and the torques cancel out. C
A ball of mass M swings in a horizontal circle at the end of a string of radius R at an initial tangential speed v0 as it undergoes uniform centripetal motion. A student gradually pulls the string inward such that the radius of the circle decreases, as shown in the figure. Which of the following predictions is correct regarding the angular momentum and rotational inertia of the ball about the axis of revolution as the ball is pulled inward?
A The angular momentum of the ball increases. The rotational inertia of the ball about the axis of revolution decreases. B The angular momentum of the ball increases. The rotational inertia of the ball about the axis of revolution stays the same. C The angular momentum of the ball remains constant. The rotational inertia of the ball about the axis of revolution decreases. D The angular momentum of the ball remains constant. The rotational inertia of the ball about the axis of revolution stays the same. C
The diagram above shows a top view of a child of mass M on a circular platform of mass 5M that is rotating counterclockwise. Assume the platform rotates without friction. Which of the following describes an action by the child that will result in an increase in the total angular momentum of the child-platform system?
A The child moves toward the center of the platform. B The child moves away from the center of the platform. C The child moves along a circle concentric with the platform (dashed line shown) opposite the direction of the platform's rotation. D None of the actions described will change the total angular momentum of the child-platform system. D
In experiment 1, a disk of mass M0 and radius R0 rotates about a center axle, as shown in the figure. An object of mass m0 hangs from a string that is wound around the disk. The object is released from rest and falls a vertical distance H before striking the ground. The string does not completely unwind. The disk's angular momentum is L1 when the falling object strikes the ground. The rotational inertia of the disk is Id=MR22. In experiment 2, a thin circular hoop with mass M0 and radius R0 replaces the disk, and the experiment is repeated. The rotational inertia of the hoop is Ih=MR2. How does the magnitude of the hoop's final angular momentum compare to the magnitude of the disk's final angular momentum?
A The hoop's final angular momentum is less than the disk's final angular momentum. B The hoop's final angular momentum is equal to the disk's final angular momentum. C The hoop's final angular momentum is greater than the disk's final angular momentum. D The comparison cannot be made without knowing the net torque exerted on the disk and the hoop. C
Two students, X and Y, sit on a seesaw that is in static equilibrium, as shown in the figure. A fulcrum is located at the center of the seesaw. The mass mY of Student Y is known. A third student, Z, must determine the mass of Student X. Which of the following measurements should Student Z make in order to determine the mass of Student X? Justify your selection.
A The length of the seesaw, because the entire length of the seesaw is in static equilibrium. B The distance that Student XX is from the edge of the seesaw and the distance that Student YY is from the edge of the seesaw, because the seesaw would no longer be in static equilibrium if one or both students moved closer to the edge of the seesaw. C The distance that Student XX is from the center of the seesaw and the distance that Student YY is from the center of the seesaw, because these are the distances from the fulcrum to the location of the forces exerted on the two-student-seesaw system. D The mass of the seesaw, because a force due to gravity is exerted on the seesaw from Earth. C
A rod of length 0.5m is placed on a horizontal surface. One end of the rod is connected to a pivot that will allow the rod to rotate around the pivot in the absence of frictional forces. A lump of clay is launched toward the free end of the rod at a known speed vc. When the lump of clay strikes the free end of the rod, it sticks to the rod. The equation for the rotational inertia of the rod about the pivot is I=13Mℓ2. Which of the following quantities, when used together, could a student measure in order to determine the change in angular momentum of the rod from when it was initially at rest to the instant in time when the rod has rotated 90° in the counterclockwise direction? Select two answers.
A The mass of the lump of clay B The mass of the rod C The tangential speed of the end of the rod after it has rotated 90°90° in the counterclockwise direction D The time it takes the rod to rotate 90°90° in the counterclockwise direction BC
A student conducts an experiment in which the angular velocity of a rotating object about a central axis of rotation changes as a function of time, as shown by the graph. The student makes the following claim. "The net torque responsible for the rotation of the object changes direction at approximately 5.0s." Which of the following statements is correct about the student's evaluation of the data from the graph? Justify your selection.
A The student is correct, because the angular velocity cannot change direction unless the net torque also changes direction. B The student is correct, because the net torque and direction exerted on the object before 5.0 s5.0 s is different from the net torque and direction exerted on the object after 5.0 s5.0 s. C The student is incorrect, because the angular acceleration of the object remains constant. D The student is incorrect, because the angular velocity is inversely related to the net torque exerted on the object. C
One end of a string is attached to a 0.1kg object. The string is wrapped around a pulley of known rotational inertia and radius 0.5m that may rotate about its central axis. The central axis is supported by strings that are connected to the ceiling, as shown in Figure 1. In an experiment, the 0.1kg object is released from rest, and the necessary data are collected to graph the distance fallen by the object as a function of the square of the time fallen, as shown in Figure 2. A student makes the following claim. "Figure 1 and Figure 2 can be used to determine the magnitude and direction of the net torque exerted on the pulley." Which of the following statements is correct about the student's evaluation of the data from the graph? Justify your selection.
A The student is correct, because the linear acceleration of the 0.1kg0.1kg object can be determined from the graph. The angular acceleration of the pulley can then be determined. The direction of the net torque will be in the direction of the angular acceleration. B The student is correct, because the average linear velocity of the 0.1kg0.1kg object can be determined from the graph. The average angular velocity of the pulley can then be determined. The direction of the net torque will be in the direction of the average angular velocity. C The student is incorrect, because the linear acceleration of the 0.1kg0.1kg object can be determined from the graph. However, there is no relationship between the linear acceleration of the 0.1kg0.1kg object and the net torque exerted on the pulley. D The student is incorrect, because the average linear velocity of the 0.1kg0.1kg object can be determined from the graph. However, there is no relationship between the average linear velocity of the 0.1kg0.1kg object and the net torque exerted on the pulley. A
One end of a string is attached to the ceiling, and the other end of the string is attached to a cradle that has a meterstick that runs through it. The meterstick can slide through the cradle so that the horizontal position of a point on the meterstick can be changed in the horizontal direction. Students may hang objects of various masses from the meterstick, as shown in the figure. The students notice that when the meterstick-cradle-object-object system is not balanced, the meterstick will rotate. Consider the situation shown above in which the center of the meterstick is aligned with the center of the cradle, which is at a position of x=0m. The system is released from rest. Which of the following claims is correct about the motion of the system containing the meterstick, cradle, and two objects if the system is free to rotate?
A The system will rotate in the clockwise direction with a constant angular speed. B The system will rotate in the clockwise direction with an increasing angular speed. C The system will rotate in the counterclockwise direction with a constant angular speed. D The system will rotate in the counterclockwise direction with an increasing angular speed. B
A disk-shaped platform has a known rotational inertia ID. The platform is mounted on a fixed axle and rotates in a horizontal plane with an initial angular velocity of ωD in the counterclockwise direction, as shown. After an unknown time interval, the disk comes to rest. A single point on the disk revolves around the center axle hundreds of times before the disk comes to rest. Frictional forces are considered to be constant. A student must determine the angular impulse that frictional forces exert on the disk from the moment it rotates with angular velocity in the counterclockwise direction until it stops. What additional data, if any, should a student collect to determine the angular impulse on the disk? Justify your selection.
A The time interval in which the net torque is applied, because a net torque is not exerted on the disk at a single instant in time. B The net torque exerted on the disk, because a net torque is responsible for an angular impulse. C The force of friction exerted on the disk, because a force component perpendicular to the line connecting the axis of rotation and the point of application of the force results in a torque about that axis. D No additional data are necessary, because the rotational inertia of the disk and its initial angular velocity are known. D
A disk-shaped platform has a known rotational inertia ID. The platform is mounted on a fixed axle and rotates in a horizontal plane with an initial angular velocity of ωD in the counterclockwise direction, as shown. After an unknown time interval, the disk comes to rest. A single point on the disk revolves around the center axle hundreds of times before the disk comes to rest. Frictional forces are considered to be constant. A student must determine the angular impulse that frictional forces exert on the disk from the moment it rotates with angular velocity in the counterclockwise direction until it stops. What additional data, if any, should a student collect to determine the angular impulse on the disk? Justify your selection.
A The time interval in which the net torque is applied, because a net torque is not exerted on the disk at a single instant in time. B The net torque exerted on the disk, because a net torque is responsible for an angular impulse. C The force of friction exerted on the disk, because a force component perpendicular to the line connecting the axis of rotation and the point of application of the force results in a torque about that axis. D No additional data are necessary, because the rotational inertia of the disk and its initial angular velocity are known. D
A massless rigid rod of length 3d is pivoted at a fixed point W, and two forces each of magnitude F are applied vertically upward as shown above. A third vertical force of magnitude F may be applied, either upward or downward, at one of the labeled points. With the proper choice of direction at each point, the rod can be in equilibrium if the third force of magnitude F is applied at point
A W only B Y only C V or X only D V or Y only E V, W, or X C
Two identical wheels, wheel 1 and wheel 2, initially at rest begin to rotate with constant angular accelerations α. After rotating through the same angular displacement, Δθ0 , the angular velocity of wheel 1 is ω1 and the angular velocity of wheel 2 is ω2=3ω1 . How does the angular acceleration of wheel 2, α2, compare to the angular acceleration of wheel 1, α1 ?
A α2=α1α2=α1 B α2=α13α2=α13 C α2=3α1α2=3α1 D α2=9α1α2=9α1 D
A disk is initially rotating counterclockwise around a fixed axis with angular speed w0. At time t = 0, the two forces shown in the figure above are exerted on the disk. If counterclockwise is positive, which of the following could show the angular velocity of the disk as a function of time?
C
