Question

Why is the moment of inertia of a hoop that has a mass M and a radius R greater than the moment of inertia of a disk that has the same mass and radius? Why is the moment of inertia of a spherical shell that has a mass M and a radius R greater than that of a solid sphere that has the same mass and radius?

Short answer

The hoop has more mass concentrated at the outer radius than the disk. The spherical shell has more mass concentrated at the outer radius than the solid sphere.

Step-by-step solution

Reason for the moment of inertia of the hoop greater than that of a disk.

Moment of inertia is obtained by the addition of the product of mass and the distance squared for all the masses comprising the body.

In the case of a hoop, they have all the masses situated on the outer boundary. Therefore, the presence of mass at the maximum possible distance makes the hoop have more moment of inertia than the disk which has mass distributed equally in its body.

Reason for the moment of inertia of the hoop greater than that of a disk.

Refer to the moment of inertia definition as stated above.

In the case of the spherical shell, they have all the masses situated on the outer boundary. Therefore, the presence of mass at the maximum possible distance makes the spherical shell have more moment of inertia than the solid sphere which has mass distributed equally in its body.