Is the center of mass of the dumbbell in FIGURE Q12.1 at point 1, 2, or 3? The solid spheres are made of the same material. Explain.

FIGURE Q12.1

If the angular velocity $\omega$ is held constant, by what factor must R change to double the rotational kinetic energy of the dumbbell in FIGURE Q12.2?

FIGURE Q12.2

FIGURE Q12.3 shows three rotating disks, all of equal mass. Rank in order, from largest to smallest, their rotational kinetic energies $K_{\text{A}}$ to $K_{\text{C}}$.

FIGURE Q12.3

Must an object be rotating to have a moment of inertia? Explain.

The moment of inertia of a uniform rod about an axis through its center is $\frac{1}{12}m{L}^2$. The moment of inertia about an axis at one end is $\frac{1}{3}m{L}^2$. Explain why the moment of inertia is larger about the end than about the center.

You have two solid steel spheres. Sphere 2 has twice the radius of sphere 1. By what factor does the moment of inertia $I_2$ of sphere 2 exceed the moment of inertia $I_1$ of sphere 1?

The professor hands you two spheres. They have the same mass, the same radius, and the same exterior surface. The professor claims that one is a solid sphere and the other is hollow. Can you determine which is which without cutting them open? If so, how? If not, why not?

FIGURE Q12.8 shows force $\vec{F}$ being applied at the center of a rod. At which numbered point should a downward force $2\vec{F}$ be applied so that the net torque about the hinge is zero?

FIGURE Q12.8

A student gives a quick push to a ball at the end of a massless, rigid rod, as shown in FIGURE Q12.9, causing the ball to rotate clockwise in a horizontal circle. The rod’s pivot is frictionless. As the student is pushing, does the torque vector point (a) toward the ball, (b) away from the ball, (c) in the direction of the push, (d) out of the page, or (e) into the page?

FIGURE Q12.9

Rank in order, from largest to smallest, the angular accelerations ${\alpha }_{\text{A}}$ to ${\alpha }_{\text{D}}$ of the disks in FIGURE Q12.10. Explain.

FIGURE Q12.10

The solid cylinder and cylindrical shell in FIGURE Q12.11 have the same mass, same radius, and turn on frictionless, horizontal axles. (The cylindrical shell has lightweight spokes connecting the shell to the axle.) A rope is wrapped around each cylinder and tied to a block. The blocks have the same mass and are held the same height above the ground. Both blocks are released simultaneously. Which hits the ground first? Or is it a tie? Explain.

FIGURE Q12.11

A diver in the pike position (legs straight, hands on ankles) usually makes only one or one-and-a-half rotations. To make two or three rotations, the diver goes into a tuck position (knees bent, body curled up tight). Why?

Is the angular momentum of disk A in FIGURE Q12.13 larger than, smaller than, or equal to the angular momentum of disk B? Explain.

FIGURE Q12.13