$A$ drum of radius $R$ full of liquid of density $d$ is rotated at angular velocity $\omega \ rad/s$. The increase in pressure at the centre of the drum will be

$A$ cylindrical tube,with its base as shown in the figure,is filled with water. It is moving down with a constant acceleration $a$ along a fixed inclined plane with angle $\theta=45^{\circ}$. $P_1$ and $P_2$ are pressures at points $1$ and $2$,respectively,located at the base of the tube. Let $\beta=(P_1-P_2) / (\rho g d)$,where $\rho$ is the density of water,$d$ is the inner diameter of the tube,and $g$ is the acceleration due to gravity. Which of the following statement$(s)$ is(are) correct?
$(A)$ $\beta=0$ when $a=g / \sqrt{2}$
$(B)$ $\beta>0$ when $a=g / \sqrt{2}$
$(C)$ $\beta=\frac{\sqrt{2}-1}{\sqrt{2}}$ when $a=g / 2$
$(D)$ $\beta=\frac{1}{\sqrt{2}}$ when $a=g / 2$

The diagram shows a cup of tea seen from above. The tea has been stirred and is now rotating without turbulence. $A$ graph showing the speed $v$ with which the liquid is crossing points at a distance $X$ from $O$ along a radius $XO$ would look like:

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density $\rho$. The height of the liquid in one vessel is ${h_1}$ and that in the other vessel is ${h_2}$. The area of either base is $A$. The work done by gravity in equalizing the levels when the two vessels are connected is:

Different processes are given in Column-$I$ and their reasons are given in Column-$II$. Match them appropriately.

Column-$I$	Column-$II$
$(a)$ Rain drops move downwards with constant velocity.	$(i)$ Viscous liquids
$(b)$ Floating clouds at a height in air.	$(ii)$ Viscosity
	$(iii)$ Less density

Explain why bullets of a gun are cylindrical in shape.