Question

Two non-mixing liquids of densities $\rho$ and $n\rho (n>1)$ 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 $pL(p<1)$ in the denser liquid. The density $d$ is equal to:

• A. $\left[1+(n+1)p\right]\rho$
• B. $\left[2+(n+1)p\right]\rho$
• C. $\left[2+(n-1)p\right]\rho$
• D. $\left[1+(n-1)p\right]\rho$

Answer 1

The correct option is D $\left[1+(n-1)p\right]\rho$


As we know,
Weight of the wooden block $W=Vdg=ALdg$
where $A$ is the cross section area of the cylinder
Using Archimedes' principle,
Weight of wooden block = Upthrust due to liquid of density $n\rho$ + Upthrust due to liquid of density $\rho$
$ALdg=\left(pL\right)A\left(n\rho \right)g+(1-p)LA\rho g$
$\Rightarrow d=(1-p)\rho +pn\rho =[1+(n-1\left)p\right]\rho$