Question

A student on a piano stool rotates freely with an angular speed of 2.85 rev/s. The student holds a 1.35 kg mass in each outstretched arm, 0.759 m from the axis of rotation. The combined moment of inertia of the student and the stool, ignoring the two masses, is 5.23 kg ⋅ m², a value that remains constant. As the student pulls his arms inward, his angular speed increases to 3.45 revolutions per second. How far are the masses from the axis of rotation at this time, considering the masses to be points? Calculate the initial kinetic energy of the system. Calculate the final kinetic energy of the system.

Answer 1

To calculate the moment of inertia of the system, sum up the moments of inertia of the disk and annular cylinder. To calculate the rotational kinetic energy of the system, use the formula K = (1/2) * I * ω2.

Step-by-step explanation:

The moment of inertia of the system can be calculated by summing up the moments of inertia of the different components. The moment of inertia of the disk can be calculated using the formula:

Idisk = (1/2) * mdisk * rdisk2

The moment of inertia of the annular cylinder can be calculated using the formula:

Iannular cylinder = (1/2) * mannular cylinder * (router2 + rinner2)

Once the moments of inertia are calculated, the total moment of inertia of the system can be found by summing up the individual moments of inertia.

The rotational kinetic energy of the system can be calculated using the formula:

K = (1/2) * I * ω2

Where K is the rotational kinetic energy, I is the moment of inertia, and ω is the angular speed.