Fluid Mechanics

Consider a cylindrical tank of radius 1 m is filled with water. The top surface of water is at 15 m from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of 5 m from the bottom. A force of $5\times10^5$ N is applied an the top surface of water using a piston. The speed of efflux from the hole will be :

(given atmospheric pressure $P_A=1.01\times10^5$ Pa, density of water $\rho_w$ = 1000 kg/m$^3$ and gravitational acceleration g = 10 m/s$^2$)

Given below are two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).

Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension (S), if density of the liquid is ρ and radius of the drop is r, then $T=K\sqrt{\dfrac{\rho r^3}{S^{3/2}}}$ is dimensionally correct, where K is dimensionless.

Reason (R) : Using dimensional analysis we get R.H.S. having different dimension than that of time period.

In the light of above statements, choose the correct answer from the options given below.

Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : Clothes containing oil or grease stains cannot be cleaned by water wash.

Reason (R) : Because the angle of contact between the oil/grease and water is obtuse.

In the light of the above statements, choose the correct answer from the option given below.

The velocity of a small ball of mass 0.3 g and density 8 g/cc when dropped in a container filled with glycerine becomes constant after some time. If the density of glycerine is 1.3 g/cc, then the value of viscous force acting on the ball will be $x\times10^{-4}$ N. The value of $x$ is _______. [use g = 10 m/s$^2$]

The metallic bob of simple pendulum has the relative density 5. The time period of this pendulum is 10 s. If the metallic bob is immersed in water, then the new time period becomes $5\sqrt x$ s. The value of $x$ will be _______.

A water drop of radius 1 cm is broken into eight equal droplets. Surface tension of water is 0.075 N m$^{-1}$. The gain in surface energy is _______ $\times10^{-7}$ J. (Take π = 3.14)

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration $a$ along a fixed inclined plane with angle $\theta = 45^\circ$. $P_1$ and $P_2$ are pressures at points 1 and 2, respectively, located at the base of the tube. Let $\beta = (P_1 - P_2)/(\rho g d)$, where $\rho$ is density of water, $d$ is the inner diameter of the tube and $g$ is the acceleration due to gravity. Which of the following statement(s) is(are) correct?

The value of $X$ is ____.

The value of $Y$ is ____.

A hydraulic press can lift 100 kg when a mass 'm' is placed on the smaller piston. It can lift _______ kg when the diameter of the larger piston is increased by 4 times and that of the smaller piston is decreased by 4 times keeping the same mass 'm' on the smaller piston.

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

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$) of potassium permanganate ($\ce{KMnO4}$) of different concentrations $n_1$ and $n_2\ (n_1 > n_2)$ molecules per unit volume with $\Delta n = (n_1 - n_2) \ll n_1$. When they are connected by a tube of small length $l$ and cross-sectional area $S$, $\ce{KMnO4}$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order $(\Delta n)^2$, which of the following is/are correct? ($k_B$ is the Boltzmann constant)

When water is filled carefully in a glass, one can fill it to a height $h$ above the rim of the glass due to the surface tension of water. To calculate $h$ just before water starts flowing, model the shape of the water above the rim as a disc of thickness $h$ having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are $10^3\ \mathrm{kg\,m^{-3}}$, $0.07\ \mathrm{N\,m^{-1}}$ and $10\ \mathrm{m\,s^{-2}}$, respectively, the value of $h$ (in mm) is ______.

A train with cross-sectional area $S_t$ is moving with speed $v_t$ inside a long tunnel of cross-sectional area $S_0\ (S_0 = 4S_t)$. 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 $p_0$. If the pressure in the region between the sides of the train and the tunnel walls is $p$, then $p_0 - p = \dfrac{7}{2N}\rho v_t^2$. The value of $N$ is ______.

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 $\rho(h) = \rho_0 e^{-\frac{h}{h_0}}$, where $\rho_0 = 1.25\ \mathrm{kg\,m^{-3}}$ and $h_0 = 6000$ m, the value of $N$ is ______.

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800\ \mathrm{kg\,m^{-3}}$ and dielectric constant 21. If the angle between the strings remains the same after the immersion, then

A cubical solid aluminium (bulk modulus $= -V\dfrac{dP}{dV} = 70$ GPa) 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 $10^3\ \mathrm{kg\,m^{-3}}$ and $10\ \mathrm{m\,s^{-2}}$, respectively, the change in the edge length of the block in mm is ______.

A cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries T1 and T2 of different materials having water contact angles of 0° and 60°, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct?

[Surface tension of water = 0.075 N/m, density of water = 1000 kg/m³, take $g = 10\ \mathrm{m/s^2}$]

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$. The angle of contact between water and the wall of the capillary tube is $\theta$. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_0$. Which of the following statements is (are) true?