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Question

A dumbbell of length L having one end of mass M and volume V and another end of mass 2M and volume V is placed in water of density ρ. It is observed that by attaching a mass M of negligible volume at distance d from the 2M end of the rod, the dumbbell floats with the rod horizontal on the surface of water and each sphere exactly half submerged as shown in the figure.


The value of L expressed in terms of distance d is ndρV3MρV2M. Then n = (Answer up to two digits after the decimal points)

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Solution

Drawing the free body diagram of the system,


Balancing torque as the body is in equilibrium
(MgV2ρg) (Ld) =(2Mg V2ρg )d (Mg V2ρg) L = (3Mg Vρg)dL = 3Mg VρgMg V2ρg d = 2d×Vρg3MgVρg2Mg = 2d×(ρV3MρV2M)

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