Find the depth of a lake at which the density of water is $1\%$ greater than at the surface. Given the compressibility of water $K = 50 \times 10^{-6} \text{ atm}^{-1}$. (Take $1 \text{ atm} = 10^5 \text{ Pa}$) (in $\text{ km}$)

$A$ litre of alcohol weighs

$A$ cylindrical vessel of base radius $R$ and height $H$ has a narrow neck of height $h$ and radius $r$ at one end (see figure). The vessel is filled with water (density $\rho_w$) and its neck is filled with immiscible oil (density $\rho_0$). Then,the pressure at

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

Which of the following quantities is expressed as force per unit area?

In Guericke's experiment to show the effect of atmospheric pressure,two copper hemispheres were tightly fitted to each other to form a hollow sphere and the air from the sphere was pumped out to create a vacuum inside. If the radius of each hemisphere is $R$ and the atmospheric pressure is $p$,then the minimum force required (when the two hemispheres are pulled apart by the same force) to separate the hemispheres is