Question

Consider the following four objects: a hoop, a flat disk, a solid sphere, and 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 of these objects requires the largest torque to give it the same angular acceleration?

a. the solid sphere.
b. the hollow sphere.
c. the hoop.
d. the flat disk.
e. both the solid and the hollow spheres.

Answer 1

b) The hollow sphere.

Step-by-step explanation:

• The moment of Inertia of a solid rotating object is related with the torque applied and the angular acceleration caused by the torque, by the following relationship, that resembles Newton's 2nd Law for point masses:

`\tau = I * \alpha (1)`

• As it can be seen, for a given angular acceleration α, the larger the moment of inertia, the larger the torque needed to give the object the same angular acceleration.
• From the proposed solids, the one that has the largest moment of Inertia, is the hollow sphere, which moment of Inertia can be written as follows:

`I_(hsph) = (2)/(3) *M*R^(2) (2)`

• So, the hollow sphere requires the largest torque to give the object the same angular acceleration.