Question
Two non-mixing liquids of densities $$\rho $$ and $$n\rho \left( {n > 1} \right)$$ are put in a container. The height of each liquid is $$h.$$ A solid cylinder of length $$L$$ and density $$d$$ is put in this container. The cylinder floats with its axis vertical and length $$pL\left( {p < 1} \right)$$ in the denser liquid. The density $$d$$ is equal to
A.
$$\left\{ {2 + \left( {n + 1} \right)p} \right\}\rho $$
B.
$$\left\{ {2 + \left( {n - 1} \right)p} \right\}\rho $$
C.
$$\left\{ {1 + \left( {n - 1} \right)p} \right\}\rho $$
D.
$$\left\{ {1 + \left( {n + 1} \right)p} \right\}\rho $$
Answer :
$$\left\{ {1 + \left( {n - 1} \right)p} \right\}\rho $$
Solution :
According to question, the situation can be drawn as following.
Applying Archemedies principle
Weight of cylinder $$ = {\left( {{\text{upthrust}}} \right)_1} + {\left( {{\text{upthrust}}} \right)_2}$$
i.e. $$ALdg = \left( {1 - P} \right)LA\rho g + \left( {PLA} \right)n\rho g$$
$$\eqalign{ & \Rightarrow d = \left( {1 - P} \right)\rho + Pn\rho = \rho - P\rho + n\,P\rho \cr & = \rho + \left( {n - 1} \right)P\rho = \rho \left[ {1 + \left( {n - 1} \right)\rho } \right] \cr} $$
According to question, the situation can be drawn as following.
Applying Archemedies principle
Weight of cylinder $$ = {\left( {{\text{upthrust}}} \right)_1} + {\left( {{\text{upthrust}}} \right)_2}$$
i.e. $$ALdg = \left( {1 - P} \right)LA\rho g + \left( {PLA} \right)n\rho g$$
$$\eqalign{ & \Rightarrow d = \left( {1 - P} \right)\rho + Pn\rho = \rho - P\rho + n\,P\rho \cr & = \rho + \left( {n - 1} \right)P\rho = \rho \left[ {1 + \left( {n - 1} \right)\rho } \right] \cr} $$
