$A$ liquid is kept in a cylindrical vessel which is being rotated about a vertical axis through the centre of the circular base. If the radius of the vessel is $r$ and the angular velocity of rotation is $\omega$,then the difference in the heights of the liquid at the centre of the vessel and the edge is

$A$ motor is used to deliver water at a certain rate through a given horizontal pipe. To deliver $n$-times the water through the same pipe in the same time, the power of the motor must be increased as follows:

State whether the following statements are True or False:
$(i)$ In incompressible flow,the density of a fluid changes with time and position.
$(ii)$ Non-viscous flow can be a streamline flow.
$(iii)$ In a steady flow,the velocity of the fluid at each point remains identical with time.

$A$ thin vertical uniform wooden rod is pivoted at the top and immersed in water as shown. The container is slowly raised. At a certain moment,the equilibrium becomes unstable. If the density of water is $9/5$ times the density of wood,then the ratio of the total length of the rod to the submerged length of the rod at that moment is:

$A$ $U$-tube has a left limb with a cross-sectional area one-quarter that of the right limb. The tube contains mercury (density $13.6 \text{ g/cm}^3$). The level of mercury in the narrow (left) limb is at a distance of $36 \text{ cm}$ from the upper end of the tube. What will be the rise in the level of mercury in the right limb if the left limb is filled to the top with water (in $\text{ cm}$)? (Density of water = $1 \text{ g/cm}^3$)

$A$ tank with a square base of area $1.0 \; m^{2}$ is divided by a vertical partition in the middle. The bottom of the partition has a small hinged door of area $20 \; cm^{2}$. The tank is filled with water in one compartment and an acid (of relative density $1.7$) in the other,both to a height of $4.0 \; m$. Compute the force (in $N$) necessary to keep the door closed.