$A$ homogeneous solid cylinder of length $L$ $(L < H/2)$ and cross-sectional area $A/5$ 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 the atmosphere having pressure $P_0$. Then,the density $D$ of the solid is given by:

$A$ wooden block,with a coin placed on its top,floats in water as shown in the figure. The distances $l$ and $h$ are shown. After some time,the coin falls into the water. Then:

$A$ hemispherical portion of radius $R$ is removed from the bottom of a cylinder of radius $R$. The volume of the remaining cylinder is $V$ and its mass is $M$. It is suspended by a string in a liquid of density $\rho$ where it stays vertical. The upper surface of the cylinder is at a depth $h$ below the liquid surface. The force on the bottom of the liquid is

$A$ solid cube and a solid sphere,both made of the same material,are completely submerged in water but at different depths. The sphere and the cube have the same surface area. The buoyant force is

$A$ 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: $1 \, \mu m = 10^{-6} \, m$)

An object with uniform density $\rho$ is attached to a spring that is known to stretch linearly with applied force. When the spring-object system is immersed in a liquid of density $\rho_1$,the spring stretches by an amount $x_1$ (where $\rho > \rho_1$). When the experiment is repeated in a liquid of density $\rho_2$ (where $\rho_2 < \rho_1$),the spring stretches by an amount $x_2$. Neglecting any buoyant force on the spring,the density of the object is: