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 placed in this container. The cylinder floats with its axis vertical and a length $pL$ $(p < 1)$ submerged in the denser liquid. The density $d$ is equal to:

$A$ machine is blowing spherical soap bubbles of different radii filled with helium gas. It is found that,if the bubbles have a radius smaller than $1 \, cm$,then they sink to the floor in still air. Larger bubbles float in the air. Assume that the thickness of the soap film in all bubbles is uniform and equal. Assume that the density of soap solution is same as that of water $\left(= 1000 \, kg \, m^{-3}\right)$. The density of helium inside the bubbles and air are $0.18 \, kg \, m^{-3}$ and $1.23 \, kg \, m^{-3}$,respectively. Then,the thickness of the soap film of the bubbles is .......... $\mu m$ (Note: $1 \, \mu m = 10^{-6} \, m$)

$A$ 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 (density $2d$) and $3L/4$ in the lighter liquid (density $d$). The lower density liquid is open to the atmosphere having pressure $P_0$. Then, the density $D$ of the solid is given by:

$A$ body of density $\rho$ is dropped from rest at a height $h$ into a lake of density $\sigma$ $(\sigma > \rho)$. The maximum depth to which the body sinks before returning to float on the surface is (neglect air dissipative forces).

$A$ rectangular block of mass $M$ and cross-sectional area $A$ floats on a liquid of density $\rho$. It is given a small vertical displacement from equilibrium; it starts oscillating with frequency $n$. Then:

$A$ solid ball is completely immersed in a liquid. The coefficients of volume expansion of the ball and liquid are $3 \times 10^{-6} / ^\circ C$ and $8 \times 10^{-6} / ^\circ C$ respectively. The percentage change in upthrust when the temperature is increased by $100 ^\circ C$ is ....... $\%$