A lazy Susan consists of a heavy plastic cylinder mounted on a frictionless bear

Question

A lazy Susan consists of a heavy plastic cylinder mounted on a frictionless bearing resting on a vertical shaft through its center. The cylinder has a radius R = 15 cm and mass M = 0.25 kg. A cockroach (mass m = 0.015 kg) is on the lazy Susan, at a distance of 8.0 cm from the center. Both the cockroach and the lazy Susan are initially at rest. The cockroach then walks along a circular path concentric with the axis of the lazy Susan at a constant distance of 8.0 cm from the axis of the shaft. If the speed of the cockroach with respect to the lazy Susan is 0.010 m/s, what is the speed of the cockroach with respect to the room?

Explanation / Answer

Angular velocity of cockrocah (w1) = v/r = 0.01 /(0.08) = 0.125 rad/s
Moment of inertia of cockroach I1 = mr2 = 0.015*(0.08)2 = 9.6*10-5 kgm2
Angular momentum of cockroach = I1w1 = 1.2*10-5
This angular momentum must be balanced by the lazy susan
= I2W2 = mr2W2
where m is mass of lazy susan and r is her radius from the shaft.
= 0.25*(0.15)2W2 = 5.625*10-3 W2
equating the two angular momentum
W2= 1.2*10-5 /5.625*10-3 = -2.133*10-3 rad/s
-ve sign shows the opposite direction.
Thus relative angular velocity
W1 - W2 = 0.123 rad/s
Hence the linear relative velocity (Vrel) = 0.123*(0.08) = 9.84*10-3 m/s
Or = 9.84 mm/s

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