Chapter 8.2, Problem 24PP

(a)

To determine

To identify:Whether the moment of inertia of the system is same or different for the given situation. If different, then state the case when moment of inertia is greater.

Answer to Problem 24PP

The moment of inertia of the system rotated about sphere A and rotated about sphere C is different.

The moment of inertia is greater when the system is rotated about sphere A.

Explanation of Solution

Given :

The given figure is shown below.

The mass of three spheres connected by a rod is same. First system is rotated about sphere A and then rotated about sphere C.

Formula used:

The expression for the moment of inertia of a point mass is,

$I=m{r}^2$

Here, $m$ is the mass of the object and $r$ is the distance of the object from the axis of rotation.

Calculation:

Consider the mass of each sphere as m and the spacing between each sphere as r .

The moment of inertia of the system when it is rotated about sphere A is,

$\begin{array}{l}I=m{r}^2+m{\left(2r\right)}^2\\ I=5m{r}^2\end{array}$

The moment of inertia of the system when it is rotated about sphere C is,

$\begin{array}{l}I=m{r}^2+m{r}^2\\ I=2m{r}^2\end{array}$

Hence, the moment of inertia of the system rotated about sphere A and rotated about sphere C is different. Also, the moment of inertia is greater when the system is rotated about sphere A.

Conclusion:

Thus, the moment of inertia of the system rotated about sphere A and rotated about sphere C is different. Also, the moment of inertia is greater when the system is rotated about sphere A.

(b)

To determine

The moment of inertia of the system rotated about sphere A and rotated about sphere C.

Answer to Problem 24PP

The moment of inertia of the system rotated about sphere A is $0.020\text{ kg}\cdot {\text{m}}^2$ .

The moment of inertia of the system rotated about sphere C is $0.008\text{ kg}\cdot {\text{m}}^2$ .

Explanation of Solution

Given :

The given figure is shown below.

The mass of three spheres connected by a rod is same. First system is rotated about sphere A and then rotated about sphere C.

The mass of each sphere is $m=0.10\text{ kg}$ .

The distance between spheres A and C is $r=0.20\text{ m}$ .

Formula used:

The expression for the moment of inertia of a point mass is

$I=m{r}^2$

Here, $m$ is the mass of the object and $r$ is the distance of the object from the axis of rotation.

Calculation:

The moment of inertia of the system when it is rotated about sphere A is,

$\begin{array}{l}I=m{r}^2+m{\left(2r\right)}^2\\ I=5\times \left(0.10\text{ kg}\right)\times {\left(0.20\text{ m}\right)}^2\\ I=0.020\text{ kg}\cdot {\text{m}}^2\end{array}$

The moment of inertia of the system when it is rotated about sphere C is,

$\begin{array}{l}I=m{r}^2+m{r}^2\\ I=2m{r}^2\\ I=2\times \left(0.10\text{ kg}\right)\times {\left(0.20\text{ m}\right)}^2\\ I=0.008\text{ kg}\cdot {\text{m}}^2\end{array}$

Conclusion:

Thus, the moment of inertia of the system rotated about sphere A is $0.020\text{ kg}\cdot {\text{m}}^2$ and the moment of inertia of the system rotated about sphere C is $0.008\text{ kg}\cdot {\text{m}}^2$ .