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 = 10 cm and mass M = 0.30 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

Explanation / Answer

mass of the cylider, m1=0.3 kg

radius of the cylider, R=10cm

mass of the cockroach, m2=0.015 kg

distance travelled from the center, r=10cm

speed of the m2 w r to lazy susan, v=0.01 m/sec

by using law of conservation momentum,

I_Ls*w=I_c*w_c

1/2*m1*R^2*w=m2*r^2*(W_c)

but,

W_c=v/r-w

==>

1/2*m1*R^2*w=m2*r^2*(V/r-w)

==>

W=2*m2*r*v/(m2*R^2+2*m1*r^2)

w=2*m2*v/(m2*R+2*m1*r)

W=(2*0.015*0.01)/(2*0.01+2*0.3*0.01)

W=0.0115 rad/sec

angular velocity of susan, W=0.0115 rad/sec

speed of the cockroach w r to floor is,

v'=v-r*w

v'=0.01-0.01*0.0115

v'=9.9*10^-3 m/sec

v'=9.9 mm/sec

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