A particularly large playground merry-go-round is essentially a uniform solid di

Question

A particularly large playground merry-go-round is essentially a uniform solid disk of mass 3M and radius R that can rotate with no friction about a central axis. You, with a mass M, are a distance of R/2 from the center of the merry-go-round, rotating together with it at an angular velocity of 3.20 rad/s clockwise (when viewed from above). You then move to the outside of the merry-go-round so you are a distance R from the center, still rotating with the merry-go-round. Consider you and the merry-go-round to be one system.

(a) When you reach the outer edge of the merry-go-round, what is the angular velocity of the you and merry-go-round system?

(b) You then start running around the outer edge of the merry-go-round. At what angular speed would you have to run to make the merry-go-round alone come to a complete stop?

Explanation / Answer

a) initial moment of inertia = 3M R^2 /2 + (M ( R/2)^2) = 7MR^2 / 4

final momentum of inertia = (3M R^2 /2 ) + ( MR^2) = 5MR^2 /2

Using momentum conservation,

Iwi = I ( wf)

( 7MR^2 / 4 ) x 3.20 = ( 5MR^2 /2 ) wf

wf = 2.24 rad/s

b) Applying angular momentum conservation,

( 7MR^2 / 4 ) x 3.20   = 0 + ( M R^2) w

w = 5.6 rad /s

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