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  • A dumbbell is placed in water of density . It is observed that by attaching a mass m to t

A dumbbell is placed in water of density \mathrm{\rho}. 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

 

Option: 1

\mathrm{\frac{d\left(V_\rho-3 M\right)}{2\left(V_\rho-2 M\right)}}


Option: 2

\mathrm{\frac{d\left(V_\rho-2 M\right)}{2\left(V_\rho-3 M\right)}}


Option: 3

\mathrm{\frac{d\left(V_\rho+2 M\right)}{2\left(V_\rho-3 M\right)}}


Option: 4

\mathrm{\frac{d\left(V_\rho-2 M\right)}{2\left(V_\rho+3 M\right)}}


Answers (1)

best_answer

\mathrm{ (3 M+m) g=\rho v g }
Torque balance about \mathrm{B}
\mathrm{ \begin{aligned} & \mathrm{mg}(\mathrm{d}-\ell)+\rho v g \frac{\ell}{2} \\ & =2 \mathrm{mg} \ell+\frac{\rho v g}{2}(\mathrm{~d}-\ell) \quad \ell=\frac{\mathrm{d}(\mathrm{v} \rho-2 \mathrm{M})}{2(\rho v-3 M)} \end{aligned} }

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