A student initially stands on a circular platform that is free to rotate without friction about its center. The student jumps off tangentially, setting the platform spinning. Quantities that are conserved for the student-platform system as the student jumps include which of the following?

I. Angular momentum

II. Linear momentum

III. Kinetic energy

All of the following are vector quantities EXCEPT

Two horizontal disks of mass M have the radii shown above. Disk A is attached to an axle of negligible mass spinning freely with angular velocity ω₀. Disk B, not attached to the axle and initially held at rest, is released and drops down onto disk A. When both disks spin together without slipping, the angular velocity ω_f of the disks is

A sphere starts from rest at the top of a ramp, as shown above. It rolls without slipping down the ramp. The potential energy of the sphere-Earth system is zero at the bottom of the ramp. Which of the following is true of the sphere when it reaches the bottom of the ramp?

Suppose that a spherical star spinning at an initial angular velocity ω suddenly collapses to half of its original radius without any loss of mass. Assume the star has uniform density before and after the collapse. What will the angular velocity of the star be after the collapse?

A uniform beam of weight W is attached to a wall by a pivot at one end and is held horizontal by a cable attached to the other end of the beam and to the wall, as shown above. T is the tension in the cable, which makes an angle θ with the beam. Which of the following is equal to T ?

A ball, rolling without slipping on a horizontal surface, encounters a frictionless, downwardsloping ramp, as shown above. Which of the following correctly describes the motion of the ball on the ramp?

In order to model the motion of an extinct ape, scientists measure its hand and arm bones. From shoulder to wrist, the arm bones are 0.60 m long and their mass is 4.0 kg. From wrist to the tip of the fingers, the hand bones are 0.10 m long and their mass is 1.0 kg. In the model above, each bone is assumed to have a uniform density.

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The arm is held in the horizontal position and the hand is bent at the wrist so the fingers point up, as shown in the figure above. The torque exerted by the weight of the hand with respect to the shoulder is most nearly

A horizontal uniform plank is supported by ropes I and II at points P and Q, respectively, as shown above. The two ropes have negligible mass. The tension in rope I is 150 N. The point at which rope II is attached to the plank is now moved to point R halfway between point Q and point C, the center of the plank. The plank remains horizontal. Which of the following are most nearly the new tensions in the two ropes?

The uniform thin rod shown above has mass 𝑚 and length ℓ . The moment of inertia of the rod about an axis through its center and perpendicular to the rod is (1/12) 𝑚ℓ². What is the moment of inertia of the rod about an axis perpendicular to the rod and passing through point P, which is halfway between the center and the end of the rod?

A stone falls from rest from the top of a building as shown above. Which of the following graphs best represents the stone’s angular momentum L about the point P as a function of time?

A circular platform has a radius R and rotational inertia I. The platform rotates around a fixed pivot at its center with negligible friction and an initial angular velocity ω . A child of mass m (represented by the small circle in the figure above) runs tangentially with speed v and jumps on the outer edge of the platform. When the child is standing on the outer edge of the platform, the new angular velocity is

A wheel of radius R is fixed to an axle of radius R/3 and rotates at constant angular speed ω , as shown in the figure above. A force of magnitude F₁ is applied tangent to the outer edge of the wheel. A second force of magnitude F₂ is applied tangent to the edge of the axle to keep the wheel rotating at constant angular speed ω. The magnitude F₂ is equal to

A solid disk of mass M and radius R is freely rotating horizontally in a counterclockwise direction with angular speed ω about a vertical axis through its center with negligible friction. The rotational inertia MR²/2 of the disk is . A second identical disk is at rest and suspended above the first disk with the centers of the two disks aligned, as shown in the figure above. There is no contact between the disks. The second disk is dropped onto the first disk, and after a short time they rotate counterclockwise with the same angular speed ω_f.

The interval of time Δt is the elapsed time from the moment of first contact between the two disks until they are both spinning at the same angular velocity. Which of the following expressions gives the magnitude of the average torque that the first disk exerts on the second disk?

A solid disk of mass M and radius R is freely rotating horizontally in a counterclockwise direction with angular speed ω about a vertical axis through its center with negligible friction. The rotational inertia MR²/2 of the disk is . A second identical disk is at rest and suspended above the first disk with the centers of the two disks aligned, as shown in the figure above. There is no contact between the disks. The second disk is dropped onto the first disk, and after a short time they rotate counterclockwise with the same angular speed ω_f.

The two disks are now shifted so that the axis of rotation goes through a point on the edge of the disks. The rotational inertia of the two-disk system is now

A solid disk of mass M and radius R is freely rotating horizontally in a counterclockwise direction with angular speed ω about a vertical axis through its center with negligible friction. The rotational inertia MR₂/2 of the disk is . A second identical disk is at rest and suspended above the first disk with the centers of the two disks aligned, as shown in the figure above. There is no contact between the disks. The second disk is dropped onto the first disk, and after a short time they rotate counterclockwise with the same angular speed ω_f.

Which of the following properties of the two-disk system must be conserved between the time the second disk is dropped on the first disk and the time that the two disks begin rotating with the same speed?

Two satellites of masses m₁ and m₂ orbit a planet of mass M in circular orbits. The satellites travel in opposite directions with speeds v₁ and v₂, as shown in the figure above. Their orbital radii are R₁ and R₂,respectively. Assume that M >> m₂ > m₁.

The magnitude of the angular momentum of the two-satellite system is best represented by

A uniform ladder of weight W leans without slipping against a wall making an angle 𝜽 with a floor as shown above. There is friction between the ladder and the floor, but the friction between the ladder and the wall is negligible.

The magnitude of the friction force exerted on the ladder by the floor is

A wheel of mass m, which has a rotational inertia I and a radius r, rotates with an angular speed ω about an axis through its center. A retarding force F is applied tangentially to the rim of the wheel.

The retarding force F finally stops the rotation of the wheel. Which of the following best represents the total reduction in mechanical energy in the process of stopping the wheel?

A wheel of mass m, which has a rotational inertia I and a radius r, rotates with an angular speed ω about an axis through its center. A retarding force F is applied tangentially to the rim of the wheel.

Which of the following is equal to the magnitude of the angular acceleration of the wheel?

The rotational inertia of a sphere of mass M and radius R about a diameter is  $\frac{2}{5}MR^2$  . The rotational inertia about an axis tangent to the sphere is

Two blocks of masses m₁ and m₂ are connected by a massless string that passes over a wheel of mass m, as shown above. The string does not slip on the wheel and exerts forces T₁ and T₂ on the blocks. When the wheel is released from rest in the position shown, it undergoes an angular acceleration and rotates clockwise. Which of the following statements about T₁ and T₂ is correct?

A particle of mass m moves counterclockwise around a horizontal circle of radius r, as shown above. The angular speed of the particle is given as a function of time t by ω (t) = bt , where b is a positive constant and t ≥ 0.

What is the magnitude of the angular momentum of the particle about the center of the circle as a function of time?

The bar shown above is pivoted about one end and is initially at rest in a vertical position. The bar is displaced slightly and as it falls it makes an angle 𝜽 with the vertical at any given time, as shown above.

Which of the following graphs best represents the bar’s angular velocity ω as a function of time?

The bar shown above is pivoted about one end and is initially at rest in a vertical position. The bar is displaced slightly and as it falls it makes an angle 𝜽 with the vertical at any given time, as shown above.

Which of the following graphs best represents the bar’s angular acceleration 𝜶 as a function of angle 𝜽 ?