Question

A thin spherical shell has a radius of 1.90 m.An applied torque of 960 N.m gives the shell an angular acceleration of 6.20 rad/s² about an axis through the center of the shell. What are (a) the rotational inertia of the shell about that axis and (b) the mass of the shell?

Short answer

1. Rotational inertia of the shell about axis is $154.8 \text{kg}.{\text{m}}^2.$
2. Mass of the shell is data-custom-editor="chemistry" $64.3\text{kg.}$

Step-by-step solution

Step 1: Given

1. Radius of the thin shell is $1.90m$
2. Torque is$960\text{N}$
3. Angular acceleration is $6.20\text{rad}/{\text{s}}^2$

Determining the concept

Use the basic formula for torque in terms of inertia and angular acceleration to find the rotational inertia. Mass can be found from the value of rotational inertia using the formula for the rotational inertia in terms of mass and radius.

Formulae are as follow:

$\tau =I\times \alpha$

$I=\frac{2}{3}M{R}^2$

Where,

$\tau$ is torque, M is mass, R is radius, I is moment of inertia and $\alpha$ is angular acceleration.

(a) Determining the rotational inertia of the shell about axis

By using formula for torque as follows,

$$\begin{gathered} \tau =I\times \alpha \\ 960=I\times 6.20 \\ I=154.8\text{kg}-{\text{m}}^2 \end{gathered}$$

Hence, rotational inertia of the shell about axis is $154.8\text{kg}-{\text{m}}^2$

(b) Determining the mass of the shell

Now, using the following formula, mass can be found,

$I=\frac{2}{3}M{R}^2$

This isfor a sphericalshell,

$154.8=\frac{2}{3}M\times 1.{90}^2$

$M=64.3\text{kg}$

Hence, mass of the shell is $64.3\text{kg.}$

Therefore, from the given torque and angular acceleration, the rotational inertia can be found. Using the formula for rotational inertia of the sphere, the mass of the sphere can be found.