AP Physics C: Mechanics - Unit 5 Progress Check MCQ

The following four objects roll without slipping along a horizontal surface toward an incline. The velocity of the center of mass of each of the objects is the same at the bottom of the incline. Which of the objects will roll the greatest distance up the incline before coming to rest?

A. Shape: Ring Mass: M Radius: 2R

A solid sphere and a hollow sphere are released at the same time from the top of the same inclined plane. Each object has mass m and radius r. They both roll without slipping down the incline. Which of the following statements about their motion must be true?

A. The solid sphere will reach the bottom first.

Two identical uniform disks are connected by a belt of negligible mass. The rotational inertia of each disk is I. Each disk is initially rotating in a counterclockwise direction at angular speed ω0 around an axis passing through the center of the disk, as shown in Figure 1 above. A constant external counterclockwise torque τ acts on the left disk. It increases the angular speed of each disk to ωf, as shown in Figure 2 above. Which of the following equations is correct for the change in angular momentum of the two disks?

A. ΔL = 2I(ωf−ω0)

A solid disk of mass M and radius R is rotating around a central axle with a constant angular speed of ω. A thin ring also of mass M and radius R that is also around the axle but not rotating is dropped on the solid disk. The ring and disk eventually come to a common final angular speed. The value M=0.50 kg, R=0.25 m, and ω=2.0rad/s. The final angular speed of the disk-ring system is most nearly

B. 0.67 rad/s

Students run an experiment to determine the rotational inertia of a large spherically shaped object around its center. Through experimental data, the students determine that the mass of the object is distributed radially. They determine that the radius of the object as a function of its mass is given by the equation r=km^2, where k=3m/kg^2. Which of the following is a correct expression for the rotational inertia of the object?

B. 1.8 m^5

A disk is rotating with an angular velocity function given by ω=Kt+L. What is the angular acceleration of the disk at t=T ?

B. K

A child of mass mm is at the edge of a merry-go-round of diameter d. When the merry-go-round is rotating with angular acceleration α, the torque on the child is τ. The child moves to a position half way between the center and edge of the merry-go-round, and the angular acceleration increases to 2α. The torque on the child is now

B. τ/2

A machine is applying a torque to rotationally accelerate a metal disk during a manufacturing process. An engineer is using a graph of torque as a function of time to determine how much the disk's angular speed increases during the process. The engineer determines that the machine increased at a constant rate the disk's angular speed from 100 rad/s to 300 rad/s over a time of 5 seconds. What was the angular displacement of the disk during those 5 seconds?

C. 1,000 radians

Two identical spheres of mass M are fastened to opposite ends of a rod of length 2L. The radii of the spheres are negligible when compared to the length 2L, and the rod has negligible mass. This system is initially at rest with the rod horizontal and is free to rotate about a frictionless axis through the center of the rod. The axis is horizontal and perpendicular to the plane of the page. A bug of mass 3M lands gently on the sphere on the left, as shown in the figure above. Assume that the size of the bug is small compared with the length of the rod. What is the rotational inertia of the bug-rod-spheres system about the horizontal axis through the center of the rod?

C. 5ML^2

A figure skater, spinning with her arms outstretched, pulls her arms toward her and experiences an increase in angular speed. Which of the following principles explains this increase in speed?

C. The conservation of angular momentum, because as her arms move in, her overall rotational inertia is lowered, therefore increasing her angular speed

The figure shows a cylinder on an incline plane of negligible friction. The cylinder is released from rest. Which of the following correctly describes the change in the motion of the cylinder if the incline is a rough surface?

C. The cylinder slides down the incline of negligible friction and rolls down the incline with a rough surface.

A student with arms outstretched stands on a platform that is rotating with a constant angular speed. The student then pulls his arms inward. This will result in

C. an increase in the angular speed due to the conservation of angular momentum principle

A merry-go-round initially at rest at an amusement park begins to rotate at time t=0. The angle through which it rotates is described by θ(t)=π/k(t+ke^(−t/k)), where k is a positive constant, t is in seconds, and θ is in radians. The angular velocity of the merry-go-round at t=T is

C. π/k(1−e^(−Tk))

In Figure 1 above, two uniform disks of radius R are each rotating counterclockwise around an axis through their center. The top disk has mass 2m and is rotating with angular speed ω1. The bottom disk has mass m and is rotating with angular speed ω2. The top disk is slowly lowered on top of the bottom disk. The friction between the two disks brings the disks to the same final angular speed ωf, as shown in Figure 2 above. Which of the following is a correct equation for ωf ?

C. ωf = (2ω1 + ω2)/3

A solid disk of mass M and radius R is rotating around a central axle with a constant angular speed of ω. A thin ring also of mass M and radius R that is also around the axle but not rotating is dropped on the solid disk. The ring and disk eventually come to a common final angular speed. The value M=0.50 kg, R=0.25 m, and ω=2.0rad/s. At a later time, the disk-ring system is rotating at angular speed 0.25 rad/s. A frictional torque 0.050 N⋅m is applied to the system. If the rotational inertia of the disk-ring system is 0.047 kg⋅m^2, the amount of time that the torque must be applied to bring the system to rest is most nearly

D. 0.24 s

An object is rotating about a fixed axis such that its rotational inertia about the fixed axis is 10 kg⋅m^2. The object has an angular velocity ωω as a function of time t given by ω(t)=αt3−ω0, where α=2.0 rad/s^4 and ω0=4.0 rad/s. The change in angular displacement of the object from t=1 s to t=3 s is most nearly

D. 32 rad

An object is rotating about a fixed axis such that its rotational inertia about the fixed axis is 10 kg⋅m^2. The object has an angular velocity ωω as a function of time t given by ω(t)=αt3−ω0, where α=2.0 rad/s^4 and ω0=4.0 rad/s. The change in angular velocity for the object from t=1 s to t=3 s is most nearly

D. 52 rad/s

A machine is applying a torque to rotationally accelerate a metal disk during a manufacturing process. An engineer is using a graph of torque as a function of time to determine how much the disk's angular speed increases during the process. The graph of torque as a function of time starts at an initial torque value and is a straight line with positive slope. What aspect of the graph and possibly other quantities must be used to calculate how much the disk's angular speed increases during the process?

D. The area under the graph divided by the disk's rotational inertia will equal the change in angular speed.

A 60kg board that is 6 m long is placed at the edge of a platform, with 4 m of its length extending over the edge. The board is held in place by blocks of masses M1 and M2 placed with their centers of mass on either end. If M2=30kg, what is the minimum value of M1 needed to keep the board from falling off the platform?

E. 90 kg

Two identical spheres of mass M are fastened to opposite ends of a rod of length 2L. The radii of the spheres are negligible when compared to the length 2L, and the rod has negligible mass. This system is initially at rest with the rod horizontal and is free to rotate about a frictionless axis through the center of the rod. The axis is horizontal and perpendicular to the plane of the page. A bug of mass 3M lands gently on the sphere on the left, as shown in the figure above. Assume that the size of the bug is small compared with the length of the rod. After the bug lands, the rod begins to rotate. Which of the following correctly describes the change in the magnitude of the angular momentum of the bug-rod-spheres system and the change in gravitational potential energy of the bug-rod-spheres-Earth system as the rod rotates but before the rod becomes vertical?

E. Angular Momentum: Increases Gravitational Potential Energy: Decreases

In an experiment, a varying force is applied tangentially for a period of time to a solid disk mounted on a frictionless axle. Initially, the disk is at rest. Which type of graph should be created so that the final kinetic energy of the disk is represented by the area under the graph?

E. Applied torque as a function of angular displacement

A puck slides on a frictionless table toward a rod that is free to rotate about a fixed pivot point at its center. The incident puck's path makes an angle θ with the rod as shown in the overhead view. In which situation below will the angular momentum of the puck-rod system about the center of the rod not be conserved?

E. In all these situations, angular momentum is conserved.

A disk begins rotating from rest with an angular acceleration of α in rad/s. Which of the following equations is correct for the time in seconds it takes for the disk to complete one full rotation?

E. sqrt(4π/α)