A person with mass m p = 72 kg stands on a spinning platform disk with a radius

Question

A person with mass mp = 72 kg stands on a spinning platform disk with a radius of R = 1.83 m and mass md = 196 kg. The disk is initially spinning at ? = 2 rad/s. The person then walks 2/3 of the way toward the center of the disk (ending 0.61 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

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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 = 189.0 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?

answer

mp = 75 kg; R = 1.77 m; md = 189 kg; ?1 = 1.5 rad/s; r1 = R = 1.77 m; r2 = R/3 = 0.59 m

The moment of inertia of a disk is Id = (1/2)mdR2; the moment of inertia of a point mass (the person) is Ip = mpr2

(1) The total moment of inertia is simply the algebraic sum of inertias:

I1 = Id + Ip = (1/2)mdR2 + mpr12

(2) I2 = Id + Ip = (1/2)mdR2 + mpr22

(3) Conservation of angular momentum, L = I ?:

I1?1 = I2?2

?2 = (I1/I2)?1

(4) Rotational kinetic energy is K = (1/2)I ?2

?K = K2 - K1 = (1/2)I2?22 - (1/2)I1?12

(5) ac = r ?2 = (R/3)(?22)

(6) Because of the conservation of angular momentum, the angular speed is the same as the initial,

? = ?1

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