Two small drops of mercury, each of radius $R$ coalesce to form a single large drop. The radio of the total surface energies before and after the change is

A solid cylinder of mass $m$ and volume $v$ is suspended from ceiling by a spring of spring constant $k$ . It has cross-section area $A$ . It is submerged in a liquid of density $\rho $ upto half its length. If a small block of mass $M_o$ is kept at the centre of the top, the amplitude of small oscillation will be 

The height of water in a tank is $H$. The range of the liquid emerging out form a hole in the wall of the tank at a depth $\frac{{3H}}{4}$ from the upper surface of water, will be

A pan balance has a container of water with an overflow spout on the right-hand pan as shown. It is full of water right up to the overflow spout. A container on the left-hand pan is positioned to catch any water that overflows. The entire apparatus is adjusted so that it’s balanced. A brass weight on the end of a string is then lowered into the water, but not allowed to rest on the bottom of the container. What happens next?

A large open tank has two holes in the wall. One is a square hole of side $L$ at a depth $y$ from the top and the other is a circular hole of radius $R$ at a depth $4y$ from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same. Then, $R$ is equal to