91Ó°ÊÓ

The moment of inertia is a crucial concept in angular motion. It is the rotational equivalent of mass in linear motion. Moment of inertia indicates how difficult it is to change the rotational speed of an object. It depends on the mass distribution relative to the rotational axis. The greater the distance of mass from the axis, the larger the moment of inertia.
For a solid sphere, the formula for the moment of inertia is \( I_{\text{solid}} = \frac{2}{5} m r^2 \). This means the mass is distributed more towards the center. In contrast, for a thin-walled spherical shell, the mass is concentrated further away from the center \( I_{\text{shell}} = \frac{2}{3} m r^2 \).
Calculating these values:

• Solid sphere: \( 0.024 \text{kg} \cdot \text{m}^2 \)
• Shell: \( 0.040 \text{kg} \cdot \text{m}^2 \)

So, the shell exhibits a larger moment of inertia, indicating it resists changes in rotational speed more than the solid sphere does.

Angular acceleration refers to how quickly an object's rotational velocity changes. It is analogous to linear acceleration. When a torque is applied to an object, the angular acceleration can be calculated by \(\alpha = \frac{\tau}{I}\).
For a given torque, a greater moment of inertia results in a smaller angular acceleration. This means the object will change its angular speed more slowly. For the spheres:

• Solid sphere: \( \alpha_{\text{solid}} = 5\, \text{rad/s}^2 \)
• Shell: \( \alpha_{\text{shell}} = 3\, \text{rad/s}^2 \)

Thus, the spherical shell has a smaller angular acceleration, indicating it slows down at a slower rate when torque is applied.

Torque is the measure of how much a force acting on an object causes it to rotate. In this scenario, a frictional torque of \(0.12\, \text{N} \cdot \text{m}\) acts on both spheres, opposing their motion and leading them to a halt. Torque can be thought of as the rotational equivalent of linear force. The effect of torque relies not only on the force magnitude but also on where and how the force is applied.
The distance from the rotational axis and the angle at which force is applied are key.
Friction plays a vital role here, opposing motion and causing the angular speed to decrease over time. In rotational systems, this torque limits the rotational motion of objects such as these spheres, causing angular deceleration until they eventually stop.

Understanding the differences in how objects with varying mass distributions react to forces is important. In our example, a solid sphere and a thin-walled spherical shell behave differently under the same conditions.
The solid sphere, where mass is distributed closer to the center, has a smaller moment of inertia than the spherical shell. This means it is easier to change its rotational speed, hence, it decelerates faster under a given torque due to friction.
The solid sphere stops in \(4.8\, \text{s}\) while the shell, with mass distributed further out, takes \(8\, \text{s}\) to halt. These principles highlight how mass distribution affects the dynamics of rotational systems, which is essential knowledge in fields like mechanical engineering and physics.