Question

The figure shows an overhead view of a ring that can rotate about its center like a merry-go-round. Its outer radius R2 is 0.9 m, its inner radius R1 is R2/2, its mass M is 8.0 kg, and the mass of the crossbars at its center is negligible. It initially rotates at an angular speed of 8.5 rad/s with a cat of mass m = M/4 on its outer edge, at radius R2. By how much does the cat increase the kinetic energy of the cat-ring system if the cat crawls to the inner edge, at radius R1?

Answer 1

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when cat is at R2 :

moment of inertia of system, Ii = M * (R2^2 + R1^2)/2 + (M/4)(R2^2)

= M ( R^2/2 + R^2/4 + R1^2 / 2 ) = M (3 R2^2 + 2 R1^2) / 4

when cat is at R1 :

If = M * (R2^2 + R1^2)/2 + (M/4)(R1^2)

= M ( R^2/2 + R^1/4 + R1^2 / 2 ) = M (2 R2^2 + 3 R1^2) / 4

Now Applying Angular momentum conservation,

Ii wi = If wf

[M (3 R2^2 + 2 R1^2) / 4 ] (8.5) = [M (2 R2^2 + 3 R1^2) / 4 ] (wf)

R1 = R2/2

(3.5 R2^2 / 4 ) (8.5) = (2.75 R2^2 / 4) (wf)

wf = 3.5 * 9.6 / 2.75 = 10.8182 rad/s

Ii = 5.67 kg m^2

If = 4.455 kg m^2

Increase in KE = If wf^2 /2 - Ii wi^2 /2

= (4.455 x 10.8182^2 / 2) - (5.67 x 8.5^2 /2 )

= 55.863 J

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