Chapter 8.2, Problem 35SSC

To determine

To Rank: The moments of inertia from least to greatest of sphere, wheel with all its masses at the rim and a solid disk

To explain: The advantages of object with least moment of inertia.

Explanation of Solution

Introduction:

The property of an object that opposes its rotational motion is called moment of inertia of the object.

The mass contained in an object and shape and size of an object depends upon the moment of inertia.

The moment of inertia of sphere, a wheel of all of its mass at the rim and a solid disk are $\frac{2}{5}m{r}^2,m{r}^2$ and $\frac{1}{2}m{r}^2$ respectively.

Based on equal masses and equal radius of each given object, it can be clearly seen from the formulas that the moment of inertia of sphere is less than the moment of inertia of solid disk while the moment of inertia of solid disk is less than the moment of inertia of wheel having mass at its center.

Therefore, the ranking of moment of inertia from least to greatest:

Sphere < solid disk < wheel.

Advantages of less moment of inertia:

Less torque is required to give an object angular acceleration if moment of inertia less.

Conclusion:

Ranking of moment of inertia of a body is:

Sphere < solid disk < wheel.