Question

Consider the following four objects: a hoop, a solid sphere, a flat disk, a hollow sphere. Each of the objects has mass M and radius R. The axis of rotation passes through the center of each object, and is perpendicular to the plane of the hoop and the plane of the flat disk. Which object requires the largest torque to give it the same angular acceleration ?

Answer 1

Answer:

Hoop.

Explanation:

The angular acceleration performed at a given torque:

$\alpha = \frac{\tau}{I}$

The moments of inertia of each element are described below:

Hoop

$I = M\cdot R^{2}$

Solid sphere

$I = \frac{2}{5}\cdot M \cdot R^{2}$

Flat disk

$I = \frac{1}{2}\cdot M \cdot R^{2}$

Hollow sphere

$I = \frac{2}{3}\cdot M \cdot R^{2}$

The greater the moment of inertia, the greater the torque to obtain the same angular acceleration. Therefore, the hoop requires the largest torque to receive the same angular acceleration.