A professor doing a lecture demonstration stands at the center of a frictionless turntable, holding 5.00 -kg masses in each hand with arms extended so that each mass is \(1.20 \mathrm{~m}\) from his centerline. A (carefully selected!) student spins the professor up to a rotational speed of \(1.00 \mathrm{rpm} .\) If he then pulls his arms in by his sides so that each mass is \(0.300 \mathrm{~m}\) from his centerline, what is his new rotation rate? Assume that his rotational inertia without the masses is \(2.80 \mathrm{~kg} \mathrm{~m} / \mathrm{s}\), and neglect the effect on the rotational inertia of the position of his arms, since their mass is small compared to the mass of the body.

To grasp the principle of angular momentum conservation, envision a figure skater spinning. As they pull their arms in, they spin faster, and when they extend their arms, they slow down. This occurs because angular momentum, the rotational equivalent of linear momentum, is conserved in the absence of external torques. In our example, the professor pulling in the weights is similar to the skater drawing in their arms.

Angular momentum, denoted by the symbol \( L \), is the product of an object's moment of inertia \( I \) and its angular velocity \( \omega \). It's given by the equation \( L = I \times \omega \). When the professor changes the position of the weights, the system's angular momentum remains constant because no external forces are acting on it. Therefore, the initial and final angular momenta are equal, and we can write \( L_{initial} = L_{final} \). This allows us to connect the initial and final rotation rates once we've calculated the moments of inertia for each configuration, as shown in the exercise solution.

Understanding Angular Velocity

Rotational kinematics deals with the motion of rotating bodies. In the context of the exercise, when the professor's rotation rate changes, we are observing rotational kinematics in action. Angular velocity, denoted by \( \omega \), is the rate at which an object rotates, analogous to linear velocity in translational motion.

To transition between rotations per minute (rpm) and the standard radians per second (rad/s), we use the fact that there are \( 2\pi \) radians in one rotation and 60 seconds in one minute. This allows us to accurately determine the professor's rotation rates before and after pulling the weights closer to his body.

Rotational Inertia and Point Masses

The moment of inertia, also known as rotational inertia, is the rotational equivalent of mass in linear motion. It measures an object's resistance to changes in its angular velocity. The greater the moment of inertia, the harder it is to change an object's rotational speed.

In the exercise, we calculate the moment of inertia for point masses located at a distance from the axis of rotation using the equation \( I = m \times r^2 \), where \( m \) is the mass and \( r \) is the distance from the rotational axis. By altering the distance \( r \), the professor changes the total moment of inertia of the system. When the weights are closer, the moment of inertia decreases, and according to angular momentum conservation, his rotation rate must increase, which is exactly what the exercise demonstrates.