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In rotational kinematics, **angular displacement** refers to the angle through which an object rotates around a particular axis. It is the rotational equivalent of linear displacement and is typically measured in radians. Think of it as the angular "distance" the object covers during its rotation.

Angular displacement is fundamental in calculations involving rotational motion. To better understand, recall that one complete revolution around a circle corresponds to an angular displacement of \(2\pi\) radians. In our exercise, we deal with 6 revolutions, which equate to an angular displacement of \(6 \times 2\pi = 12\pi\) radians.

This transformation from revolutions to radians is common practice in physics, as radians provide a convenient way to relate angular quantities to linear ones. By knowing the angular displacement, we can proceed to use other rotational motion formulas to find more information about the system's state, such as its angular velocity.

The **moment of inertia** is a measure of an object's resistance to change in its rotational motion. It's somewhat analogous to mass in linear motion, but instead of measuring 'how much' matter an object has, it measures 'how far' the mass is distributed from the axis of rotation.

For an object like a hollow sphere, the calculation of the moment of inertia is governed by the formula:

• \(I = \frac{2}{3}MR^2\)

Where \(M\) is the mass, and \(R\) is the radius of the sphere. By substituting the mass (8.20 kg) and radius (0.220 m) into this formula, we determine that the moment of inertia \(I\) is 0.2651 kg m².

Understanding moment of inertia is crucial when dealing with rotational dynamics, as it directly affects the rotational kinetic energy and angular acceleration and plays a critical role when applying Newton's second law in rotational form. It tells us how much torque is required to achieve a desired angular acceleration.

**Rotational kinetic energy** is the energy possessed by a rotating object due to its motion. It's important in understanding the dynamics of systems involving rotation. The formula for rotational kinetic energy \(KE\) shares similarities with linear kinetic energy:

• \(KE = \frac{1}{2}I\omega^2\)

Where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

In our case, after calculating \(I = 0.2651 \text{ kg m}^2\) and the angular velocity \(\omega\), we insert these values into the formula to calculate the rotational kinetic energy. This energy quantifies how effectively the sphere converts its rotational motion into energy.

Rotational kinetic energy is an essential concept, particularly when understanding the conversion of energy forms, such as potential energy into kinetic energy during a rolling motion. This also has practical applications in the analysis of mechanical systems where rotational motion is a factor.