A lazy Susan consists of a heavy plastic disk mounted on a frictionless bearing

Question

A lazy Susan consists of a heavy plastic disk mounted on a frictionless bearing resting on a vertical shaft through its center. The cylinder has a radius R = 20 cm and mass M = 0.33 kg. A cockroach (mass m = 0.015 kg) is on the lazy Susan, at a distance of 10 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 10 cm from the axis of the shaft. If the speed of the cockroach with respect to the lazy Susan is 0.01 m/s, what is the speed of the cockroach with respect to the room?

Explanation / Answer

Note that

w = v/r

Thus, the cockroach is actually moving with

w1 = 0.1 rad/s

The moment of inertia of the cockroach is

I1 = m1 r^2 = 1.5E-4 kg*m^2

Thus, as they are both at rest, the lazy susan must counter this angular momentum.

For the lazy susan,

I2 = 1/2 m2 R^2 = 5.2E-3 kg*m^2

Thus, by conservation of momentum,

w2 = -2.88E-3 rad/s

Thus, the relative angular velocity is

wrel = w1 - w2 = 0.1029 rad/s

Thus,

vrel = wrel * r

= 0.0103 m/s

Racalibrating this as the vrel = 0.01 m/s, the actual vrel is

vrel = 0.00972 m/s

OR

vrel = 9.72 mm/s [ANSWER]

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