###### FREE Expert Solution

Moment of inertia,

Solid Sphere,

$\overline{)\begin{array}{rcl}{\mathbf{I}}& {\mathbf{=}}& \frac{\mathbf{2}}{\mathbf{5}}\mathbf{m}{\mathbf{r}}^{\mathbf{2}}\end{array}}$

Solid disk,

$\overline{)\begin{array}{rcl}{\mathbf{I}}& {\mathbf{=}}& \frac{\mathbf{1}}{\mathbf{2}}{\mathbf{mr}}^{\mathbf{2}}\end{array}}$

Based on the coefficients (2/5 and 1/2), the moment of inertia of solid disk is greater than the moment of inertia of a solid sphere.

###### Problem Details

Consider a solid sphere and a solid disk with the same radius and the same mass. Explain why the solid disk has a greater moment of inertia than the solid sphere, even though it has the same overall mass and radius.

Calculate the moment of inertia and the rotational kinetic energy for the following objects spinning about a central axis (in units of Joules):

a) a solid sphere with a mass of 200 grams and a radius of 5.0 cm rotating with an angular speed of 2.5 rad/sec.

b) a thin spherical shell with a mass of 200 grams and a radius of 5.0 cm rotating with an angular speed of 2.5 rad/sec.

c) a solid cylinder with a mass of 200 grams and a radius of 5.0 cm rotating with an angular speed of 2.5 rad/sec.

d) a hoop with a mass of 200 grams and a radius of 5.0 cm rotating with an angular speed of 2.5 rad/sec.