Question

A large horizontal circular platform (M=91.7 kg, r=4.16 m) rotates about a frictionless vertical axle. A student (m=76.53 kg) walks slowly from the rim of the platform toward the center. The angular velocity ω of the system is 3.29 rad/s when the student is at the rim.

Find the moment of inertia of platform through the center with respect to the z-axis.

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Find the moment of inertia of the student about the center axis (while standing at the rim) of the platform.

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Find the moment of inertia of the student about the center axis while the student is standing 1.95 m from the center of the platform.

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Find the angular speed when the student is 1.95 m from the center of the platform.

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Answer 1

If we assume that the platform is of the form of the disc , then, the moment of inertia of the platform of mass M and radius R about a vertical axis passing through the center is given by
  
So, putting the given values,
  
we get the moment of inertia of the platform is
  
  

Moment of inertia of the student when it is at the rim is
  
  

Moment of inertia of the student when it is at a distance r = 1.95 m from the center is given by
  
  

From the angular momentum conservation, we get
  
where, is the angular speed initially, and is the angular speed finally. So, the final angular speed is