A person with mass mp = 75.0 kg stands on a spinning platform disk with a radius

Question

A person with mass mp = 75.0 kg stands on a spinning platform disk with a radius of R = 1.77 m and mass md = 188.0 kg. The disk is initially spinning at ? = 1.5 rad/s. The person then walks 2/3 of the way toward the center of the disk (ending 0.59 m from the center). 1) What is the total moment of inertia of the system about the center of the disk when the person stands on the rim of the disk? 2) What is the total moment of inertia of the system about the center of the disk when the person stands at the final location 2/3 of the way toward the center of the disk? 3) What is the final angular velocity of the disk? 4) What is the change in the total kinetic energy of the person and disk? (A positive value means the energy increased.) 5) What is the centripetal acceleration of the person when she is at R/3? 6) If the person now walks back to the rim of the disk, what is the final angular speed of the disk?

Explanation / Answer

The total rotational kinetic energy must remain the same, so we will calculate that first. (We will assume for the sake of this problem that the person is extremely thin, so we can treat him/her as a point. Otherwise we'd have to take into account his width.) Rotational Kinetic Energy = 1/2 * I * w^2, where w = angular velocity. Rotational Kinetic Energy = 1/2 * (500 kg m^2) * (2.9 rad/s)^2 Rotational Kinetic Energy = 2,102.5 Joules This will be the rotational kinetic energy of the system both before and after "the walk" (and thus the answer to part b of your question). The moment of inertia of the person once they are on the edge of the merry go round can be calculated as: I = mr^2 I = 75kg * (2.9m)^2 I = 630.75 kg m^2 The total moment of inertia, then, is that of the MGR plus that of the person. Total I = (630.75 + 500) kg m^2 Total moment of Inertia = 1130.75 kg m^2 Now since our kinetic energy (rotational) stays the same: KE = 1/2 * I * w^2 Solve for w: w = sqrt (2*KE / I) w = sqrt (2*2102.5/1130.75) w = 1.93 rad/s

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