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  $p_i$  perpendicular to the rod. The sphere rebounds along its original line of motion with momentum of magnitude  $p_f$  . What is the magnitude of the angular momentum of the rod immediately after the collision?

The diagram provided shows a top view of a child of mass  $M$  on a circular platform of mass  $2M$  that is rotating counterclockwise. Assume the platform rotates without friction. Which of the following statements describes an action by the child that will increase the angular speed of the platform-child system, and gives the correct reason why?

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 has friction so that the sphere rolls without slipping, 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 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?

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 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?