A potter's wheel with rotational inertia \(6.40 \mathrm{kg} \cdot \mathrm{m}^{2}\) is spinning freely at 19.0 rpm. The potter drops a 2.70 -kg lump of clay onto the wheel, where it sticks \(46.0 \mathrm{cm}\) from the rotation axis. What's the wheel's subsequent angular speed?

Rotational inertia, also known as 'moment of inertia,' refers to an object's resistance to change in its rotation. It depends not only on the mass of an object but also on how this mass is distributed relative to the axis of rotation. Imagine a spinning ice skater pulling in their arms; by doing so, they are decreasing their rotational inertia, leading to an increase in angular velocity to conserve angular momentum.

Every object that rotates, like a potter's wheel, has a certain amount of rotational inertia. When the potter drops a lump of clay onto the wheel, he adds mass at a certain distance from the central axis, effectively increasing the wheel's rotational inertia. According to the law of conservation of angular momentum, if no external torque acts on the system, the product of the rotational inertia and angular velocity before the clay is added will equal the product of those after the clay has been added.

Angular velocity is a measure of how fast an object rotates or spins around an axis. It is usually denoted by the Greek letter omega \(\omega\) and can be expressed in units such as revolutions per minute (rpm) or radians per second (rad/s), with the latter being the standard unit in physics.

For example, the potter's wheel in the exercise is initially spinning at 19 rpm. To use this value in calculations dealing with angular momentum, we need to convert it into rad/s. The significance of angular velocity becomes apparent when we talk about how an object’s speed of rotation changes, such as when the potter's wheel slows down due to the added mass of clay. Understanding how angular velocity is affected by changes in mass distribution is crucial for solving problems involving rotational dynamics.

The moment of inertia is key to understanding how an object's mass distribution affects its rotational motion. It is basically the rotational equivalent of mass for linear motion and is used to predict the angular acceleration that will result when a torque is applied. The greater the moment of inertia, the harder it is to change the object's rotational speed.

In our exercise, the moment of inertia changes when the lump of clay is added because the mass is no longer all concentrated at the wheel's axis; some of the mass is now a distance away, at \(46 \text{cm}\). This increased distance results in a larger moment of inertia, \(I_2 = I_1 + mr^2\), which ultimately decreases the angular velocity because of the conservation of angular momentum, as seen by the final calculation in the step-by-step solution.