Two solids $P$ and $Q$ float in water. It is observed that $P$ floats with half of its volume immersed and $Q$ floats with $\frac{2}{3}$ of its volume immersed. The ratio of densities of $P$ and $Q$ is

$A$ jar is filled with two non-mixing liquids $1$ and $2$ having densities $\rho_1$ and $\rho_2$ respectively. $A$ solid ball,made of a material of density $\rho_3$,is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which of the following is true for $\rho_1, \rho_2$ and $\rho_3$?

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

$A$ $0.5 \,kg$ block of brass (density $=8 \times 10^3 \,kg \,m^{-3}$) is suspended from a string. What is the tension in the string if the block is completely immersed in water? $(g=10 \,ms^{-2})$

$A$ block of steel of size $5 \text{ cm} \times 5 \text{ cm} \times 5 \text{ cm}$ is weighed in water. If the relative density of steel is $7$,its apparent weight is:

At a given place where acceleration due to gravity is $g \ m/s^2$,a sphere of lead of density $d \ kg/m^3$ is gently released in a column of liquid of density $\rho \ kg/m^3$. If $d > \rho$,the sphere will: