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 = 10 cm and mass M = 0.31 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?

mm/s

Answer 1

Angular momentum of lady Susan = Iw = (1/2)MR²*w
We know that (w) = v/R = 0.01/(0.1) = 0.1 rad/s
Angular momentum of lady Susan = (1/2)*(0.31)*(0.1)²*0.1 = 1.55*10^-4
Relative angular speed of cockroach with floor
v_f = v -wr
Angular momentum of the cockroach = I₂*[(v/R)-w]= mR²*[(v/R)-w]
Now using conservation of momentum
mR²*[(v/R)-w] = (1/2)MR²*w
0.015*[(0.01/0.1) - w₂] = (1/2)*0.31*(0.1)

w = 0.009 rad/s
Now the relative veloctiy
v_f = v - wR
= 0.01 - [0.009*0.1] = 9.1*10^-3 m/s
= 9.1 mm/s