A dumbbell is made of a rod length L and M, and 2 spheres of radius r and mass m

Question

A dumbbell is made of a rod length L and M, and 2 spheres of radius r and mass m (see figure). The rod is attached at a distance L/3 from the left end to a rotational axis A. The dumbbell is let go under gravity and makes a rotational motion under gravity around the rotational axis A. (Known: g. M. m. L. r) a) Make detailed analysis of the problem Find the distance between the center of mass of the system and the rotational A. Find the moment of inertia (I) of the system about A. What is the maximum angular velocity achieved by the system? What are the maximum speed of each of the spheres? Make the checks of your results.

Explanation / Answer

Assuming the left sphere to be (0,0)
Then the center of mass of system = ml+Ml/2/(2m+M) = l/2
Distance of axis = l/3
b) Hence distance between c.o.m. and axis = l/2 -l/3 = l/6
c) Ia = Ml2/12 + M(l/6)2 + m(l/3)2 + m(2l/3)2 = Ml2/9 + 5ml2/9 = l2/9( M + 5m)

d)

Maximum velocity is achieved when right side sphere reaches the bottom most point

Change in potential energy = 2mgl/3 + Mgl/6 -mgl/3 = mgl/3 + Mgl/6 = (gl/6)*( M + 2m)

Hence increase in kinetic energy = (gl/6)*( M + 2m)

1/2 * (l2/9)*( M + 5m) * (max )2 = (gl/6)*( M + 2m)

max = [3g/l ( (M + 2m)/( M + 5m ) ) ] 1/2

e) Speed maximum of left sphere = l/3 *max= l/3 *[3g/l ( (M + 2m)/( M + 5m ) ) ] 1/2

= [gl/3 ( (M + 2m)/( M + 5m ) ) ] 1/2

Speed maximum of right sphere = 2l/3 *max= [4gl/3 ( (M + 2m)/( M + 5m ) ) ] 1/2

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