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Question

A dumbbell is placed in water of density ρ. It is observed that by attaching a mass m to 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 volume of the mass m is negligible. The value of length l is

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A
d(Vp3m)2(Vp2m)
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B
d(Vp2m)2(Vp3m)
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C
d(Vp+2m)2(Vp3m)
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D
d(Vp2m)2(Vp+2m)
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Solution

The correct option is C d(Vp2m)2(Vp3m)
Since the system is in equilibrium,
Net force acting downward=Net force acting upwards
(m+M+2M)g=V2ρg+V2ρg
m+3M=Vρ..........................(1)
Also net torque about mass 2M must be zero.
Therefore, mgl=(V2ρgMg)d.........................(2)
Solving the equations (1) and (2) gives
l=d(Vρ2M)2(Vρ3M)

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