Answer to Question #218565 in Physics for Yami

Question

A solid sphere, a hollow sphere, a solid disc and a hoop with the same mass and radius are spinning freely about a diameter with the same angular speed on a table. Which of these objects will one do maximum work to stop it?

Answer 1

According to work-kinetic theorem for rotation, the amount of work required to stop a rigid body that rotates under a fixed axis rotation equals to its intial rotational kinetic energy. In turn, the rotational kinetic energy is given as follows:

$$K = \dfrac{I\omega^2}{2}$$

where $I$ is the momet of inertia of the body, and $\omega$ is its angular speed.

Since the angular speed is the same for all bodies, the kinetic energy and, thus, the amount of work are determined only by the momet of inertia.

For the solid sphere:

$$I = \dfrac{2}{5}mr^2$$

For the hollows sphere:

$$I = \dfrac{2}{3}mr^2$$

For the solid disc:

$$I = \dfrac{1}{4}mr^2$$

For the hoop:

$$I = \dfrac{1}{2}mr^2$$

Since the masses $m$ and radii $r$ are also the same, then the maximum moment of inertia belongs to the hollow sphere.

Answer. hollow sphere.