A playground carousel is free to rotate about its center on frictionless bearing

Question

A playground carousel is free to rotate about its center on frictionless bearings, and air resistance is negligible. The carousel itself (without riders) has a moment of inertia of 125 kg·m2. When one person is standing on the carousel at a distance of 1.50 m from the center, the carousel has an angular velocity of 0.600 rad/s. However, as this person moves inward to a point located 0.939 m from the center, the angular velocity increases to 0.800 rad/s. What is the person’s mass?

2,A thin rod has a length of 0.299 m and rotates in a circle on a frictionless tabletop. The axis is perpendicular to the length of the rod at one of its ends. The rod has an angular velocity of 0.995 rad/s and a moment of inertia of 1.32 x 10-3 kg·m2. A bug standing on the axis decides to crawl out to the other end of the rod. When the bug (whose mass is 5 x 10-3 kg) gets where it's going, what is the change in the angular velocity of the rod?

Explanation / Answer

Q-1) I(tot) = I + mr^2                   ....(1)

From Conservation of angular momentum
I11 = I22
From Eqn(1)

Here

I1 = 125 + m(1.5)^2
1 = 0.6 rad/s

I2 = 125 + m(0.939)^2
2 = 0.8 rad/s

I11 = I22
[125 + m(1.5)^2]*(0.6) = [125 + m(0.939)^2]*(0.8)
m*[0.6(1.5^2) - 0.8(0.939^2) ] = 125(0.8 - 0.6)

m * 0.6446 = 25

m = 38.8 kg

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