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?

• 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.

• 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.

• 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.

• 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.

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

• The sphere, because it gains rotational KE, but the block does not

• The sphere, because it gains mechanical energy due to the torque exerted on it, but the block does not

• The block, because it does not lose mechanical energy due to friction, but the sphere does

• The block, because it does not gain rotational kinetic energy, but the sphere does

The figure above shows the forces exerted on a block that is sliding on a horizontal surface: the gravitational force of 40N, the 40 N normal force exerted by the surface, and a frictional force exerted to the left. The coefficient of friction between the block and the surface is 0.2. The acceleration of the block is most nearly

• 1.0 m/s² to the right

• 1.0 m/s² to the left

• 2.0 m/s² to the right

• 2.0 m/s² to the left

The diagram above 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 describes an action by the child that will increase the angular speed of the platform-child system and gives the correct reason why?

• The child moves toward the center of the platform, increasing the total angular momentum of the system

• The child moves toward the center of the platform, decreasing the rotational inertia of the system

• The child moves away from the center of the platform, increasing the total angular momentum of system

• The child moves away from the center of the platform, decreasing the rotational inertia of the system.

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?

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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?

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A wheel with radius 0.33 m and rotational inertia 2.0 kg⋅m²

spins on an axle with an initial angular speed of 3.0 rad/s. Friction in the axle exerts a torque on the wheel, causing the wheel to stop after 6.0 s. The average torque exerted on the wheel as it slows down has magnitude

• 0.5 Nm

• 1 Nm

• 2 Nm

• 3 Nm

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.

• Subtract the greatest value of the angular speed from the smallest value of the angular speed, because α=Δω.

• Determine the slope of the line from 0s to 2s, because the slope represents $\frac{\Delta\omega}{\Delta t}$ .

• Determine the area bounded by the line and the horizontal axis from 0s to 2s, because $\propto=\frac{1}{2}\omega\Delta t$ .

• The angular acceleration cannot be determined without knowing the rotational inertia of the object.

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?

•  $m_2gR=I\alpha$  

•  $T_2R=I\alpha$  

•  $\left(T_2-T_1\right)R=I\alpha$  

•  $\left(m_2-m_1\right)gR=I\alpha$  

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?

• It is the same because there is no external torque acting on the system.

• It is greater because the rotating mass increases, which increases the rotational inertia.

• It is less because the speed of the disk decreases when the clay sticks to it.

• It is less because the angular momentum of the clay opposes that of the disk.