$A$ stone of relative density $K$ is released from rest on the surface of a lake. If viscous effects are ignored, the stone sinks in water with an acceleration of

$A$ uniform rod of density $\rho$ is placed in a wide tank containing a liquid of density $\rho_0$ $(\rho_0 > \rho)$. The depth of the liquid in the tank is half the length of the rod. The rod is in equilibrium,with its lower end resting on the bottom of the tank. In this position,the rod makes an angle $\theta$ with the horizontal.

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

The weight of a sphere in air is $50 \ g$. Its weight is $40 \ g$ in a liquid at temperature $20^{\circ} C$. When the temperature increases to $70^{\circ} C$,its weight becomes $45 \ g$. The ratio of the densities of the liquid at the two given temperatures is: (in $: 1$)

$A$ boy has a mass of $60\, kg$. He wants to swim in a river with the help of a wooden log. If the relative density of the wood is $0.6$,what is the minimum volume of the wooden log required? (Density of river water is $1000\, kg/m^3$)

$A$ cubical block of wood having a mass of $160 \,g$ has a metal piece fastened underneath as shown in the figure. Find the maximum mass of the metal piece which will allow the block to float in water. The specific gravity of wood is $0.8$, the specific gravity of the metal is $10$, and the density of water is $1 \,g/cm^3$. (in $\,g$)