Two non-mixing liquids of densities $\mathrm{\rho }$ and $\mathrm{n}\mathrm{\rho }\left(\mathrm{n}>1\right)$ are put in a container. The height of each liquid is $\mathrm{h.}$ A solid cylinder of length $\text{L}$ and density $\mathrm{d}$ is put in this container. The cylinder floats with its axis vertical and length $\mathrm{p}\mathrm{L}\left(\mathrm{p}<1\right)$ in the denser liquid. The density $\mathrm{d}$ is equal to:

• A

  $\left\{1+\left(\mathrm{n}+1\right)\mathrm{p}\right\}\mathrm{\rho }$
• B

  $\left\{2+\left(\mathrm{n}+1\right)\mathrm{p}\right\}\mathrm{\rho }$
• C

  $\left\{2+\left(\mathrm{n}-1\right)\mathrm{p}\right\}\mathrm{\rho }$
• Correct answer
  D

  $\left\{1+\left(\mathrm{n}-1\right)\mathrm{p}\right\}\mathrm{\rho }$

Solution

${\mathrm{f}}_0=\mathrm{m}\mathrm{g}$

$\mathrm{\rho }\mathrm{A}\left(1-\mathrm{p}\right)\mathrm{L}\mathrm{g}+\mathrm{n}\mathrm{\rho }\mathrm{A}\mathrm{p}\mathrm{L}\mathrm{g}=\mathrm{d}\mathrm{A}\mathrm{L}\mathrm{g}$

$\mathrm{\rho }\left(1-\mathrm{p}\right)+\mathrm{n}\mathrm{\rho }\mathrm{p}=\mathrm{d}$

$\left[1-\mathrm{p}+\mathrm{n}\mathrm{p}\right]\mathrm{\rho }=\mathrm{d}$

$\left[1+\left(\mathrm{n}-1\right)\mathrm{p}\right]\mathrm{\rho }=\mathrm{d}$

Hence, the correct answer is option: 'D'.