Fluid Mechanics

A capillary tube is immersed vertically in water and the height of the water column is $x$. When this arrangement is taken into a mine of depth $d$, the height of the water column is $y$. If $R$ is the radius of earth, the ratio $\dfrac xy$ is :

A tank with a small hole at the bottom has been filled with water and kerosene (specific gravity 0.8). The height of water is 3 m and that of kerosene 2 m. When the hole is opened the velocity of fluid coming out from it is nearly : (take $g=10$ ms$^{-2}$ and density of water $=10^3$ kg m$^{-3}$)

An air bubble of radius 0.1 cm is in a liquid having surface tension 0.06 N/m and density $10^3$ kg/m$^3$. The pressure inside the bubble is 1100 Nm$^{-2}$ greater than the atmospheric pressure. At what depth is the bubble below the surface of the liquid ? ($g=9.8$ ms$^{-2}$)

A cylindrical vessel of cross-section A contains water to a height h. There is a hole in the bottom of radius 'a'. The time in which it will be emptied is :

Two soap bubbles coalesce to form a single bubble. If V is the subsequent change in volume of contained air and S the change in total surface area, T is the surface tension and P atmospheric pressure, which of the following relation is correct ?

The velocity of water in a river is 18 km/hr near the surface. If the river is 5 m deep, find the shearing stress between the horizontal layers of water. The co-efficient of viscosity of water $=10^{-2}$ poise.

In the diagram shown, the difference in the two tubes of the manometer is 5 cm, the cross section of the tube at A and B is 6 mm$^2$ and 10 mm$^2$ respectively. The rate at which water flows through the tube is ($g=10$ ms$^{-2}$)

A large number of liquid drops each of radius r coalesce to from a single drop of radius R. The energy released in the process is converted into kinetic energy of the big drop so formed. The speed of the big drop is (given surface tension of liquid T, density ρ)

A solid sphere of radius $R$ and density $\rho$ is attached to one end of a mass-less spring of force constant $k$. The other end of the spring is connected to another solid sphere of radius $R$ and density $3\rho$. The complete arrangement is placed in a liquid of density $2\rho$ and is allowed to reach equilibrium. The correct statement(s) is (are)

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If $\rho_c$ is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

Four point charges, each of $+q$, are rigidly fixed at the four corners of a square planar soap film of side '$a$'. The surface tension of the soap film is $\gamma$. The system of charges and planar film are in equilibrium, and $a=k\left[\dfrac{q^2}\gamma\right]^{1/N}$, where '$k$' is a constant. Then $N$ is

Two solid spheres A and B of equal volumes but of different densities $d_A$ and $d_B$ are connected by a string. They are fully immersed in a fluid of density $d_F$. They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

A tiny spherical oil drop carrying a net charge $q$ is balanced in still air with a vertical uniform electric field of strength $\dfrac{81\pi}{7}\times10^5$ V m⁻¹. When the field is switched off, the drop is observed to fall with terminal velocity $2\times10^{-3}$ m s⁻¹. Given $g=9.8$ m s⁻², viscosity of the air $=1.8\times10^{-5}$ N s m⁻² and the density of oil $=900$ kg m⁻³, the magnitude of $q$ is

If the radius of the opening of the dropper is $r$, the vertical force due to the surface tension on the drop of radius $R$ (assuming $r\ll R$) is

If $r=5\times10^{-4}$ m, $\rho=10^3$ kg m⁻³, $g=10$ m s⁻², $T=0.11$ N m⁻¹, the radius of the drop when it detaches from the dropper is approximately

After the drop detaches, its surface energy is

Column II shows five systems in which two objects are labelled as X and Y. Also in each case a point P is shown. Column I gives some statements about X and/or Y. Match these statements to the appropriate system(s) from Column II.

A cylindrical vessel of height 500 mm has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height H. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 mm. Find the fall in height (in mm) of water level due to opening of the orifice. [Take atmospheric pressure $=1.0\times10^5\ \mathrm{N/m^2}$, density of water $=1000\ \mathrm{kg/m^3}$ and $g=10\ \mathrm{m/s^2}$. Neglect any effect of surface tension.]

Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure 8 N/m². The radii of bubbles A and B are 2 cm and 4 cm, respectively. Surface tension of the soap-water used to make bubbles is 0.04 N/m. Find the ratio $n_B/n_A$, where $n_A$ and $n_B$ are the number of moles of air in bubbles A and B, respectively. [Neglect the effect of gravity.]

STATEMENT-1: The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down.

and

STATEMENT-2: In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.