Chapter 8.2, Problem 22PP

To determine

To identify: Whether the moment of inertia of solid ball and hollow ball are equal or not. If not, then state the ball that will have greater moment of inertia.

Answer to Problem 22PP

No, the moment of inertia of solid ball and hollow ball are not equal.

Moment of inertia hollow ball is greater than moment inertia of solid ball.

Explanation of Solution

Given :

Dimension and mass of both the balls are equal.

Formula used :

Moment of inertia of solid sphere is ;

  $I=\frac{2}{5}M{R}^2$

Moment of inertia of hollow sphere is ;

  $I=\frac{2}{3}M{R}^2$

Here, M is mass and R is radius.

Calculation :

It is given that both the balls have equal dimension.

Let the radius of both the ball is R and mass is M.

Moment of inertia of solid ball is ;

  $I_1=\frac{2}{5}M{R}^2\mathrm{......1}$

Moment of inertia of hollow ball is ;

  $I_2=\frac{2}{3}M{R}^2\mathrm{......2}$

Dividing equation 1 by equation2.

  $\begin{array}{l}\frac{I_1}{I_2}=\frac{\frac{2}{5}M{R}^2}{\frac{2}{3}M{R}^2}\\ \frac{I_1}{I_2}=\frac{2}{5}\times \frac{3}{2}\\ \frac{I_1}{I_2}=\frac{3}{5}\\ I_1=\frac{3}{5}{I}_2\end{array}$

Conclusion :

Therefore, moment of inertia of hollow ball will be greater than moment of inertia of solid ball.