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A homogeneous solid cylinder of length $L (L < H/2)$ , cross-sectional area $A$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L/4$ in the denser liquid as shown in the figure. The lower density liquid is open to atmosphere having pressure $P_0$ . Then, density $D$ of solid is given by
$\frac{5}{4}d$
$\frac{4}{5}d$
$4d$
$\frac {d}{5}$
Solution
In floating condition $\mathrm{W}=\mathrm{Th}$
$\mathrm{V} \rho_{\mathrm{B}} \mathrm{g}=\mathrm{V}_{\mathrm{in}} \rho_{\mathrm{L}_{1}} \mathrm{g}+\mathrm{V}_{\mathrm{in}} \rho_{\mathrm{L}_{2}} \mathrm{g}$
$(\mathrm{AL}) \rho_{\mathrm{B}} g=\left(\mathrm{A} \times \frac{3 \mathrm{L}}{4}\right) \mathrm{d} g+\left(\mathrm{A} \times \frac{\mathrm{L}}{4}\right) \times 2 \mathrm{d} \mathrm{g}$
$\Rightarrow \quad \rho_{\mathrm{B}}=\frac{5 \mathrm{d}}{4}$
