Chapter 12, Problem 45A

A pair of identical 1000-kg space pods in outer space are connected to each other by a 900-m-long cable. They rotate about a common point like a spinning dumbbell as shown in the figure. Calculate the rotational inertia of each pod about the axis of rotation. What is the rotational inertia of the two-pod system about its midpoint? Express your answers in $kg\cdot m^2$.

To determine

The rotational inertia of each pod about the axis of rotation and the rotational inertia of the two-pod system about its midpoint.

Answer to Problem 45A

The rotational inertia of each pod about the axis of rotation is $2\times {10}^8\text{ kg}\cdot {\text{m}}^2$ .

The rotational inertia of the two-pod system about its midpoint is $4\times {10}^8\text{ kg}\cdot {\text{m}}^2$ .

Explanation of Solution

Given info:

The mass of the space pods is $m=1000\text{ kg}$ .

The length of the cable is $l=900\text{ m}$ .

Formula used:

The expression for the rotational inertia as follows:

  $I=m{r}^2$

Here, $m$ is the mass, $v$ is the tangential speed, and $r$ is the distance.

Calculation:

The distance from one end of the cable to its center as follows:

  $\begin{array}{l}r=\frac{l}{2}\\ r=\frac{900\text{ m}}{2}\\ r=450\text{ m}\end{array}$

The rotational inertia of each pod about the axis of rotation as follows:

  $\begin{array}{l}I=m{r}^2\\ I=\left(1000\text{ kg}\right){\left(450\text{ m}\right)}^2\\ I=2\times {10}^8\text{ kg}\cdot {\text{m}}^2\end{array}$

Hence, the rotational inertia of each pod about the axis of rotation is $2\times {10}^8\text{ kg}\cdot {\text{m}}^2$ .

The rotational inertia of the two-pod system about its midpoint as follows:

  $\begin{array}{l}{I}_{\text{system}}=2I\\ I_{\text{system}}=2\times \left(2\times {10}^8\text{ kg}\cdot {\text{m}}^2\right)\\ I_{\text{system}}=4\times {10}^8\text{ kg}\cdot {\text{m}}^2\end{array}$

Hence, the rotational inertia of the two-pod system about its midpoint is $4\times {10}^8\text{ kg}\cdot {\text{m}}^2$ .

Conclusion:

Thus, the rotational inertia of each pod about the axis of rotation is $2\times {10}^8\text{ kg}\cdot {\text{m}}^2$ .

Thus, the rotational inertia of the two-pod system about its midpoint is $4\times {10}^8\text{ kg}\cdot {\text{m}}^2$ .