A person with mass m p = 76 kg stands on a spinning platform disk with a radius

Question

A person with mass mp = 76 kg stands on a spinning platform disk with a radius of R = 1.92 m and mass md = 191 kg. The disk is initially spinning at = 1.6 rad/s. The person then walks 2/3 of the way toward the center of the disk (ending 0.64 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

1)

The total moment of inetia is

Inet = Idisk + Ip

Inet = ( md R2 ) / 2 + mp R2

Inet = ( 191 kg * ( 1.92 m )2 ) / 2 + 76 kg * ( 1.92 m )2

  Inet = 632 Kg.m2   

2)

when the person is at 2/3 of the center of the disk, the total moment of inetia is

I'net = Idisk + I'p

I'net = ( md R2 ) / 2 + mp R'2

I'net = ( 191 kg * ( 1.92 m )2 ) / 2 + 76 kg * ( 0.64 m )2   

  I'net = 383 Kg.m2   

3)

consider the conservation of the angular moment

L = 0 ---> L = L0  

I'net f = Inet 0   

hence

f = ( Inet 0 ) / I'net

f = ( 632 Kg.m2 * 1.6 rad/s ) / ( 383 Kg.m2 )

f = 2.6 rad/s

4)

the change of kinetic energy of the system is

Krot = Krot,f  -  Krot,0

Krot = ( I'net f2 ) / 2 -   ( Inet 02 ) / 2

Krot = ( 383 Kg.m2 * (2.6 rad/s)2 ) / 2 -   ( 632 Kg.m2 * (1.6 rad/s)2 ) / 2

Krot = 485.6 J

5)

the centripetal acceleration is

a = f2 R'

a = ( 2.6 rad/s )2 * 0.64 m

a = 4.3 m/s2   

6)

the angular speed must be 1.6 rad/s

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