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Moment of inertia is a physical quantity that reflects how mass is distributed in relation to an axis of rotation. It's a rotational analogue to mass in linear motion, and it influences the resistance of an object to change in its rotational motion.

To calculate the moment of inertia, you use the formula \( I = mr^2 \) for point masses, where \( m \) is the mass and \( r \) is the distance from the axis of rotation. However, for extended bodies, the calculation involves summing up or integrating all the small mass elements \( dm \) multiplied by their respective distance squared from the axis. In the given problem, when the student holds the weights further away, the moment of inertia is greater because the masses are at a larger radius from the axis, making it harder for the student to spin.

Angular speed refers to how quickly an object rotates or revolves relative to another point, expressed in radians per second (rad/s). It represents the angle turned, in radians, through a given amount of time.

In our example, the student's initial angular speed was given as \(0.75 \) rad/s. This means for every second that the student spins, they sweep out an angle of \(0.75 \) radians. Angular speed can change if the moment of inertia changes, assuming angular momentum is conserved, as seen when the student pulls the weights closer to the axis.

In rotational motion, kinetic energy quantifies the energy an object possesses due to its rotation and is given by the formula \( K = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular speed.

For the student, the initial and final kinetic energies are calculated before and after moving the objects closer to the axis. The kinetic energy increases with the square of the angular speed, which explains why even with a smaller moment of inertia, the student's kinetic energy can increase, as seen when they pull the objects nearer.

Conservation of angular momentum is a principle stating that if no external torque acts on a system, the total angular momentum of that system remains constant. Angular momentum is given by \( L = I\omega \), the product of moment of inertia and angular speed.

In the scenario where the student pulls the weights closer, the system's angular momentum is conserved. Initially, with arms extended, the system has a certain angular momentum that is equal to the product of the larger moment of inertia and the slower angular speed. When the student pulls the weights in, decreasing the moment of inertia, the angular speed increases to keep the angular momentum constant.

Rotational motion is the circular movement of an object about a central axis. It includes both spinning and revolving motions — like a planet rotating on its axis or revolving around a star. The study of rotational motion involves quantities such as angular displacement, angular velocity, and angular acceleration, analogous to their linear counterparts.

In the given textbook problem, we analyze the student’s spinning motion on a stool, which is a perfect example of rotational motion. The change in how the weights are positioned affects the rotational motion through changes in moment of inertia, directly influencing the angular velocity and kinetic energy of the system.