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 hot air balloon is carrying some passengers, and a few sandbags of mass 1 kg each so that its total mass is 480 kg. Its effective volume giving the balloon its buoyancy is V. The balloon is floating at an equilibrium height of 100 m. When N number of sandbags are thrown out, the balloon rises to a new equilibrium height close to 150 m with its volume V remaining unchanged. If the variation of the density of air with height h from the ground is
$$\rho \left( h \right) = {\rho _0}{e^{ - {h \over {{h_0}}}}}$$, where $$\rho $$₀ = 1.25 kg m⁻³ and h₀ = 6000 m, the value of N is _________.

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 ________.

A cubical solid aluminium (bulk modulus = $$ - V{{dP} \over {dV}} = 70GPa$$) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be 10³ kg m^-3 and 10 ms^-2, respectively, the change in the edge length of the block in mm is _______.