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?

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 graph of the angular velocity ω as a function of time t is shown for an object that rotates about an axis. Three time intervals, 1–3, are shown. Which of the following correctly compares the angular displacement Δθ of the object during each time interval?

A uniform horizontal beam of mass M and length  $L_0$   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  $\theta_0$  . What is the magnitude of the torque that the cable exerts on the beam?

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 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 disk is initially rotating counterclockwise around a fixed axis with angular speed ω₀.

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?

A uniform, rigid rod of length 2 m 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 10 N 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 rod of length  $2D_0$  and mass  $2M_0$  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  $m_0$  is launched horizontally toward the free end of the rod with velocity  $v_0$  , 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?

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  $m_b$  , 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  $\frac{1}{2}MR^2$  about its center of mass.
The block falls for a time  $t_0$  , but the string does not completely unwind. What is the change in angular momentum of the pulley-block system from the instant that the block is released from rest until time  $t_0$  ?

A satellite that is a spinning cylinder has initial rotational inertia  $I_0$  and angular velocity  $\omega_0$ . Solar panels unfold from the satellite and are extended outward. The satellite then has rotational inertia  $I_f=aI_0$  and angular velocity  $\omega_f=b\omega_0$  , where  $a$  and  $b$  are constants. Which of the following is true about the constants  $a$  and  $b$  ?

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 rod of length 0.5 m 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  $v_c$  . 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  $I=\frac{1}{3}Ml^2$ .  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 horizontal disk of radius 0.2 m and mass 0.3 kg 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 rod is at rest on a flat, horizontal surface. One end of the rod is attached to a pivot, and the rod may freely rotate around the pivot if acted upon by a net external torque, as shown in Figure 1. In an experiment, the rod is initially at rest and student exerts a net torque on the rod. Data are collected to create a graph of the rod’s angular acceleration as a function of time, as shown in Figure 2. Frictional forces are considered to be negligible. How can the student use the graph to determine the angular momentum of the rod at 5 s?

A lump of clay of mass  $m_{clay}$  with speed  $v_{clay}=8\ \frac{m}{s}$  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 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 5.0 s.”

Which of the following statements is correct about the student’s evaluation of the data from the graph? Justify your selection.

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  $\omega_f$   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?

Two blocks are joined by a light string that passes over the pulley shown in the diagram, which has radius  $R$  and moment of inertia  $I$   about its center.  $T_1$  and  $T_2$  are the tensions in the string on either side of the pulley
and  $\alpha$   is the angular acceleration of the pulley. Which of the following equations best describes the pulley’s
rotational motion during the time the blocks accelerate?

A solid cylinder of mass $m$  and radius  $R$  has a string wound around it. A person holding the string pulls vertically upward, as shown in the diagram, such that the cylinder is suspended in midair for a brief time interval,  $\Delta t$  , and its center of mass does not move. The tension in the string is  $T$  , and the rotational inertia of the cylinder about its axis is  $\frac{1}{2}MR^2$  . What is the net force on the cylinder during the time interval  $\Delta t$  ?

A sphere of mass  $M$  , radius  $r$  , and rotational inertia  $I$  is released from rest at the top of an inclined plane of height  $h$  , as shown. If the plane is frictionless, what is the speed,  $v_{\text{CM}}$  , of the center-of-mass of the sphere at the bottom of the incline?

Two objects are released from rest at the top of ramps with the same dimensions, as shown in the diagram. The sphere rolls down one ramp without slipping. The small block rolls down the other ramp without friction. Which object reaches the bottom of the ramp first, and why?

A 2.0 kg object with a 0.2 m radius and moment of inertia  $0.0693\ \text{kg}\cdot\text{m}^2\$  is released from rest at a height of 3 m on the curved surface shown. It rolls without slipping down to point  $O$  . It continues to roll up the frictionless surface which starts at point  $O$  . The maximum height that the object will reach on the frictionless side is

A solid disc has rotational inertia that is equal to  $\frac{1}{2}MR^2$ 
 where  $M$   is the disc's mass and  $R$   is the disc's radius. It is rolling along a horizontal surface without slipping with a linear speed of  $v$  . How are the translational kinetic energy and the rotational kinetic energy of the disc related?