A container of height 2 m , length 2 m and breadth 1 m is made of insulating vertical walls and two large area horizontal metal plates ( $\mathrm{M}_1$ and $\mathrm{M}_2$ ) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area $\sqrt{10} \mathrm{~cm}^2$ near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant $\epsilon_r=15$ and the right chamber is empty ( $\epsilon_r=1$ ). At time $t=0$, the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has $\epsilon_r=1$ and is maintained at atmospheric pressure. The schematic of the container at a time $t>0$ is shown in the figure.

[Given : acceleration due to gravity is $10 \mathrm{~ms}^{-2}$.]

The height (in m ) of the liquid in left chamber at $t=500 \mathrm{~s}$ is :

Two large, identical water tanks, 1 and 2 , kept on the top of a building of height $H$, are filled with water up to height $h$ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2 , and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are $t_1$ and $t_2$, respectively. If $H=\left(\frac{16}{9}\right) h$, then the ratio $t_1 / t_2$ is ___________.

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius $0.1 \mathrm{~mm}$ is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$ to $\frac{100}{101} V_0$ with the movable piston. Considering air as an ideal gas, the height $(h)$ of the liquid column in the capillary above the liquid level in $\mathrm{cm}$ is _______.

[Given: Surface tension of the liquid is $0.075 \mathrm{~N} \mathrm{~m}^{-1}$, atmospheric pressure is $10^5 \mathrm{~N} \mathrm{~m}^{-2}$, acceleration due to gravity $(\mathrm{g})$ is $10 \mathrm{~m} \mathrm{~s}^{-2}$, density of the liquid is $10^3 \mathrm{~kg} \mathrm{~m}^{-3}$ and contact angle of capillary surface with the liquid is zero]

A train with cross-sectional area S_t is moving with speed vt inside a long tunnel of cross-sectional area S₀ (S₀ = 4S_t). Assume that almost all the air (density $$\rho $$) in front of the train flows back between its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be p₀. If the pressure in the region between the sides of the train and the tunnel walls is p, then
p₀ - p = $${7 \over {2N}}\rho v_t^2$$. The value of 𝑁 is ________.