Fluid Mechanics

A tank contains two immiscible liquids of densities $6\rho$ and $2\rho$. The higher density liquid is filled up to a height $L/2$ from the bottom. A thin rod of density $\rho$ and length $L$ is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is $\dfrac{2\pi}n\sqrt{\dfrac Lg}$, where $g$ is the acceleration due to gravity. The value of $n$ is:

The height (in m) of the liquid in left chamber at $t=500$ s is:

A liquid drop of diameter 2 mm breaks into 512 droplets. The change in surface energy is $\alpha\times10^{-6}$ J. The value of $\alpha$ is ________. (Take surface tension of liquid = 0.08 N/m)

A tub is filled with water and a wooden cube 10 cm × 10 cm × 10 cm is placed in the water. The wooden cube is found to float on the water with a part of it submerged in water. When a metal coin is placed on the wooden cube, the submerged part is increased by 3.87 cm. The mass of the metal coin is ________ gram.
(Take water density as 1 g/cm³ and density of wood as 0.4 g/cm³)

The surface tension of a soap bubble is 0.03 N/m. The work done in increasing the diameter of bubble from 2 cm to 6 cm is $\alpha\pi\times10^{-4}$ J. The value of $\alpha$ is ________. (Take $\pi=3.14$)

If an air bubble of diameter 2 mm rises steadily through a liquid of density 2000 kg/m³ at a rate of 0.5 cm/s, then the coefficient of viscosity of liquid is ________ Poise. (Take $g=10$ m/s²)

The surface tension of a soap solution is $3.5\times10^{-2}$ N/m. The work required to increase the radius of a soap bubble from 1 cm to 2 cm is $\alpha\times10^{-6}$ J. The value of $\alpha$ is ________. ($\pi=22/7$)

A water spray gun is attached to a hose of cross sectional area 30 cm². The gun comprises of 10 perforations each of cross sectional area of 15 mm². If the water flows in the hose with the speed of 50 cm/s, calculate the speed at which the water flows out from each perforation. (Neglect any edge effects)

Eight mercury drops, each of radius $r$, coalesce to form a bigger drop. The surface energy released in this process is ________. ($S$ is the surface tension of mercury).

A cylindrical vessel of 40 cm radius is completely filled with water and its capacity is 528 dm³ (dm : decimeter) The vessel is placed on a solid block of exactly same height as vessel. If a small hole is made at 70 cm below the top of water level, then horizontal range of water falling on the ground in the beginning is ________ cm.

A liquid of density 600 kg/m³ flowing steadily in a tube of varying cross-section. The cross-section at a point $A$ is 1.0 cm² and that at $B$ is 20 mm². Both the points $A$ and $B$ are in same horizontal plane, the speed of the liquid at $A$ is 10 cm/s. The difference in pressures at $A$ and $B$ points is ________ Pa.

A spherical liquid drop of radius $R$ acquires the terminal velocity $v_1$ when falls through a gas of viscosity $\eta$. Now the drop is broken into 64 identical droplets and each droplet acquires terminal velocity $v_2$ falling through the same gas. The ratio of terminal velocities $v_1/v_2$ is ________.

A projectile of mass 200 g is launched in a viscous medium at an angle 60° with the horizontal, with an initial velocity of 270 m/s. It experiences a viscous drag force $\vec F = -c\vec v$ where the drag coefficient $c = 0.1$ kg/s and $\vec v$ is the instantaneous velocity of the projectile. The projectile hits a vertical wall after 2 s. Taking $e = 2.7$, the horizontal distance of the wall from the point of projection (in m) 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 $H = \left(\dfrac{16}{9}\right)h$, then the ratio $t_1/t_2$ is _____.

A table tennis ball has radius $(3/2)\times10^{-2}\,\mathrm{m}$ and mass $(22/7)\times10^{-3}\,\mathrm{kg}$. It is slowly pushed down into a swimming pool to a depth of $d = 0.7\,\mathrm{m}$ below the water surface and then released from rest. It emerges from the water surface at speed $v$, without getting wet, and rises up to a height $H$. Which of the following option(s) is(are) correct?

[Given: $\pi = 22/7$, $g = 10\,\mathrm{m\,s^{-2}}$, density of water $= 1\times10^3\,\mathrm{kg\,m^{-3}}$, viscosity of water $= 1\times10^{-3}$ Pa-s.]

A spherical soap bubble inside an air chamber at pressure $P_0 = 10^5\,\mathrm{Pa}$ has a certain radius so that the excess pressure inside the bubble is $\Delta P = 144\,\mathrm{Pa}$. Now, the chamber pressure is reduced to $8P_0/27$ so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure $\Delta P$ in both the cases to be much smaller than the chamber pressure. The new excess pressure $\Delta P$ in Pa is _____.

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0.1 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 $\dfrac{100}{101}V_0$ 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 cm is _____.

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

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____ $\mathrm{gm\,s^{-1}}$.

When the chimney is closed using a cap at the top, a pressure difference $\Delta P$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $\Delta P$ is _____ $\mathrm{N\,m^{-2}}$.

An ideal gas of density $\rho = 0.2\ \mathrm{kg\,m^{-3}}$ enters a chimney of height $h$ at the rate of $\alpha = 0.8\ \mathrm{kg\,s^{-1}}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_1 = 0.1\ \mathrm m^2$ and the upper end is $A_2 = 0.4\ \mathrm m^2$. The pressure and the temperature of the gas at the lower end are 600 Pa and 300 K, respectively, while its temperature at the upper end is 150 K. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take $g = 10\ \mathrm{m\,s^{-2}}$ and the ratio of specific heats of the gas $\gamma = 2$. Ignore atmospheric pressure.

Which of the following statement(s) is(are) correct?