Question

Use the parallel axis theorem (Problem 3.1)

(a) and Example 3, to find the moment of inertia of a solid ball about a line tangent to it;

(b) and Problem 3b to find the moment of inertia of a solid cylinder about a ruling

Short answer

a) The moment of inertia of a solid ball about a line tangent to it is$l_m=\frac{7}{5}M{a}^2$

b) The moment of inertia of a solid cylinder about a ruling is$l_m=\frac{3}{2}M{a}^2$

Step-by-step solution

Given

The solid ball and a solid cylinder.

Concept of Parallel axis theorem

The moment of inertia of a body about an axis parallel to the body passing through its center is equal to the sum of moment of inertia of the body about the axis passing through the center and product of the mass of the body times the square of the distance between the two axes.

Calculate the moment of inertia

a) The moment of inertia of a solid ball about a line tangent to it is calculated as below.

$$\begin{gathered} l_m=\frac{2}{5}M{a}^2 \\ l=l_m+M{a}^2 \\ =\frac{7}{5}M{a}^2 \end{gathered}$$

Hence, the moment of inertia of a solid ball about a line tangent to it is$l_m=\frac{7}{5}M{a}^2$

Calculate the moment of inertia

b) The moment of inertia of a solid cylinder about a ruling is calculated as below.

$$\begin{gathered} l_m=\frac{1}{2}M{a}^2 \\ l=l_m+M{a}^2 \\ =\frac{3}{2}M{a}^2 \end{gathered}$$

Hence, the moment of inertia of a solid cylinder about a ruling is$l_m=\frac{3}{2}M{a}^2$