A U-tube of base length $\mathcal{l}$ filled with same volume of two liquids of densities $\rho$ and $2\rho$ is moving with an acceleration $a$ on the horizontal plane. If the height difference between the two surfaces (open to atmosphere) becomes zero, then the height h is given by-

Water stands upto height $\text{'}h\text{'}$ behind the dam as shown in the figure. The front view of the dam gate is also shown in the adjoining figure. Density of water is $\text{'}\rho \text{'}$ and acceleration due to gravity is $\text{'}g\text{'}$. If atmospheric pressure force is also considered, the point of application of total force acting on the dam due to water above $\text{'}o\text{'}$ is_____

An open-ended U-tube of uniform cross-sectional area contains water (density ${10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ ). Initially the water level stands at $0.29 \mathrm{m}$ from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800 \mathrm{kg} {\mathrm{m}}^{-3}$ is added to the left arm until its length is $0.1 \mathrm{m}$, as shown in the schematic figure below. The ratio $\left(\frac{h_1}{h_2}\right)$ of the heights of the liquid in the two arms is

The liquid shown in the figure in the two arms is mercury (specific gravity $13.6$) and water. If the difference of heights of the mercury column is $2 \mathrm{cm}$, the height $h$ of the water column is

Two liquids of densities ${\rho }_1$ and ${\rho }_2\left({\rho }_2=2{\rho }_1\right)$ are filled up behind a square wall of side $10 \mathrm{m}$ as shown in figure. Each liquid has a height of $5 \mathrm{m}.$ The ratio of the forces due to these liquids exerted on upper part $MN$ to that at the lower part $NO$ is (Assume that the liquids are not mixing):

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 $\theta =45°$. $P_1$ and $P_2$ are pressures at point $1$ and $2$, respectively, located at the base of the tube. Let $\beta =\frac{\left(P_1-P_2\right)}{\left(\rho gd\right)}$, where $\rho$ is 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 cylindrical vessel containing a liquid is rotated about its axis so that the liquid rises at its sides as shown in the figure. The radius of vessel is $5 \mathrm{cm}$ and the angular speed of rotation is $\mathrm{\omega } \mathrm{rad} {\mathrm{s}}^{-1}$. The difference in the height, $\mathrm{h} \left(\mathrm{in} \mathrm{cm}\right)$ of liquid at the Centre of vessel and at the sides of the vessel will be :

An empty plastic bottle closed with an airtight stopper is pushed down into a bucket filled with water. As the bottle is pushed down, the force on the bottle increases as shown in the graph. This is because