A thin uniform tube is bent into a circle of radius $r$ in the virtical plane. Equal volumes of two immiscible liquids, whose densities are ${\rho _1}$ and ${\rho _2}\left( {{\rho _1} > {\rho _2}} \right)$ fill half the circle. The angle $\theta$ between the radius vector passing through the common interface and the vertical is

A barometer kept in an elevator reads $76 \,cm$ when the elevator is accelerating upwards. The most likely pressure inside the elevator (in $cm$ of $Hg$ ) is ........

A given shaped glass tube having uniform cross section is filled with water and is mounted on a rotatable shaft as shown in figure. If the tube is rotated with a constant angular velocity $\omega $then

A fixed thermally conducting cylinder has a radius $\mathrm{R}$ and height $\mathrm{L}_0$. The cylinder is open at its bottom and has a small hole at its top. A piston of mass $M$ is held at a distance $L$ from the top surface, as shown in the figure. The atmospheric pressure is $\mathrm{P}_0$.

$1.$  The piston is now pulled out slowly and held at a distance $2 \mathrm{~L}$ from the top. The pressure in the cylinder between its top and the piston will then be

$(A)$ $\mathrm{P}_0$ $(B)$ $\frac{\mathrm{P}_0}{2}$  $(C)$ $\frac{P_0}{2}+\frac{M g}{\pi R^2}$  $(D)$ $\frac{\mathrm{P}_0}{2}-\frac{\mathrm{Mg}}{\pi \mathrm{R}^2}$

$2.$  While the piston is at a distance $2 \mathrm{~L}$ from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is

$(A)$ $\left(\frac{2 \mathrm{P}_0 \pi \mathrm{R}^2}{\pi \mathrm{R}^2 \mathrm{P}_0+\mathrm{Mg}}\right)(2 \mathrm{~L})$  $(B)$ $\left(\frac{\mathrm{P}_0 \pi R^2-\mathrm{Mg}}{\pi R^2 \mathrm{P}_0}\right)(2 \mathrm{~L})$ 

$(C)$ $\left(\frac{\mathrm{P}_0 \pi \mathrm{R}^2+\mathrm{Mg}}{\pi \mathrm{R}^2 \mathrm{P}_0}\right)(2 \mathrm{~L})$  $(D)$ $\left(\frac{\mathrm{P}_0 \pi \mathrm{R}^2}{\pi \mathrm{R}^2 \mathrm{P}_0-\mathrm{Mg}}\right)(2 \mathrm{~L})$

$3.$  The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is $\rho$. In equilibrium, the height $\mathrm{H}$ of the water column in the cylinder satisfies

$(A)$ $\rho g\left(\mathrm{~L}_0-\mathrm{H}\right)^2+\mathrm{P}_0\left(\mathrm{~L}_0-\mathrm{H}\right)+\mathrm{L}_0 \mathrm{P}_0=0$

$(B)$ $\rho \mathrm{g}\left(\mathrm{L}_0-\mathrm{H}\right)^2-\mathrm{P}_0\left(\mathrm{~L}_0-\mathrm{H}\right)-\mathrm{L}_0 \mathrm{P}_0=0$

$(C)$ $\rho g\left(\mathrm{~L}_0-\mathrm{H}\right)^2+\mathrm{P}_0\left(\mathrm{~L}_0-\mathrm{H}\right)-\mathrm{L}_0 \mathrm{P}_0=0$

$(D)$ $\rho \mathrm{g}\left(\mathrm{L}_0-\mathrm{H}\right)^2-\mathrm{P}_0\left(\mathrm{~L}_0-\mathrm{H}\right)+\mathrm{L}_0 \mathrm{P}_0=0$

Give the answer question $1,2$ and $3.$

Two communicating vessels contain mercury. The diameter of one vessel is $n$ times larger than the diameter of the other. A column of water of height $ h$  is poured into the left vessel. The mercury level will rise in the right-hand vessel ($s =$ relative density of mercury and $\rho  = $ density of water) by

An aeroplane of mass $3 \times 10^4\,kg$ and total wing area of $120\,m^2$ is in a level flight at some height. The difference in pressure between the upper and lower surfaces of its wings in kilopascals is........... $kPa$ $(g=10\,m/s^2)$