Fluid Mechanics

A drop of liquid of radius $R = 10^{-2}$ m having surface tension $S = \frac{0.1}{4\pi}\ \mathrm{Nm^{-1}}$ divides itself into $K$ identical drops. In this process the total change in the surface energy $\Delta U = 10^{-3}$ J. If $K = 10^\alpha$ then the value of $\alpha$ is

The following observations were taken for determining surface tension T of water by capillary method :

diameter of capillary, $D=1.25\times10^{-2}$ m

rise of water, $h=1.45\times10^{-2}$ m.

Using $g=9.80$ m/s$^2$ and the simplified relation $T=\dfrac{rhg}2\times10^3$ N/m, the possible error in surface tension is closest to :

Two tubes of radii $r_1$ and $r_2$, and lengths $l_1$ and $l_2$, respectively, are connected in series and a liquid flows through each of them in stream line conditions. $P_1$ and $P_2$ are pressure differences across the two tubes. If $P_2$ is $4P_1$ and $l_2$ is $\dfrac{l_1}4$, then the radius $r_2$ will be equal to :

Consider two solid spheres P and Q each of density $8\ \mathrm{gm\,cm^{-3}}$ and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density $0.8\ \mathrm{gm\,cm^{-3}}$ and viscosity $\eta=3$ poiseulles. Sphere Q is dropped into a liquid of density $1.6\ \mathrm{gm\,cm^{-3}}$ and viscosity $\eta=2$ poiseulles. The ratio of the terminal velocities of P and Q is

Which of the following option correctly describes the variation of the speed $v$ and acceleration '$a$' of a point mass falling vertically in a viscous medium that applies a force F = −kv, where 'k' is a constant, on the body ? (Graphs are schematic and not drawn to scale)

Consider a water jar of radius R that has water filled up to height H and is kept on a stand of height h (see figure). Through a hole of radius r (r << R) at its bottom, the water leaks out and the stream of water coming down towards the ground has a shape like a funnel as shown in the figure. If the radius of the cross-section of water stream when it hits the ground is $x$. Then :

Two spheres $P$ and $Q$ of equal radii have densities $\rho_1$ and $\rho_2$, respectively. The spheres are connected by a massless string and placed in liquids $L_1$ and $L_2$ of densities $\sigma_1$ and $\sigma_2$ and viscosities $\eta_1$ and $\eta_2$, respectively. They float in equilibrium with the sphere $P$ in $L_1$ and sphere $Q$ in $L_2$ and the string being taut (see figure). If sphere $P$ alone in $L_2$ has terminal velocity $\vec V_P$ and $Q$ alone in $L_1$ has terminal velocity $\vec V_Q$, then

A spherical body of radius $R$ consists of a fluid of constant density and is in equilibrium under its own gravity. If $P(r)$ is the pressure at $r\,(r<R)$, then the correct option(s) is(are)

If it takes 5 minutes to fill a 15 litre bucket from a water tap of diameter $\dfrac2{\sqrt\pi}$ cm then the Reynolds number for the flow is (density of water = $10^3$ kg/m$^3$ and viscosity of water = $10^{-3}$ Pa.s) close to :

If two glass plates have water between them and are separated by very small distance (see figure), it is very difficult to pull them apart. It is because the water in between forms cylindrical surface on the side that gives rise to lower pressure in the water in comparison to atmosphere. If the radius of the cylindrical surface is R and surface tension of water is T then the pressure in water between the plates is lower by :

A cylindrical block of wood (density = 650 kg m$^{-3}$), of base area 30 cm$^2$ and height 54 cm, floats in a liquid of density 900 kg m$^{-3}$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly) :

A glass capillary tube is of the shape of a truncated cone with an apex angle $\alpha$ so that its two ends have cross sections of different radii. When dipped in water vertically, water rises in it to a height $h$, where the radius of its cross section is $b$. If the surface tension of water is $S$, its density is $\rho$, and its contact angle with glass is $\theta$, the value of $h$ will be ($g$ is the acceleration due to gravity)

If the piston is pushed at a speed of $5\ \mathrm{mm\,s^{-1}}$, the air comes out of the nozzle with a speed of

If the density of air is $\rho_a$ and that of the liquid $\rho_\ell$, then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to

A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance of 1.2 m from the person. In the following, state of the lift's motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the list.

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends 90° angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical. Ratio $\dfrac{d_1}{d_2}$ is :

On heating water, bubbles being formed at the bottom of the vessel detatch and rise. Take the bubbles to be spheres of radius $R$ and making a circular contact of radius $r$ with the bottom of the vessel. If $r\ll R$, and the surface tension of water is $T$, value of $r$ just before bubbles detatch is :

(density of water is $\rho_w$)

An open glass tube is immersed in mercury in such a way that a length of 8 cm extends above the mercury level. The open end of the tube is then closed and sealed and the tube is raised vertically up by additional 46 cm. What will be length of the air column above mercury in the tube now ?

(Atmospheric pressure $=76$ cm of Hg)

Water is flowing at a speed of 1.5 ms$^{-1}$ through a horizontal tube of cross-sectional area $10^{-2}$ m$^2$ and you are trying to stop the flow by your palm. Assuming that the water stops immediately after hitting the palm, the minimum force that you must exert should be (density of water $=10^3$ kgm$^{-3}$).

The amplitude of a simple pendulum, oscillating in air with a small spherical bob, decreases from 10 cm to 8 cm in 40 seconds. Assuming that Stokes law is valid, and ratio of the coefficient of viscosity of air to that of carbon dioxide is 1.3, the time in which amplitude of this pendulum will reduce from 10 cm to 5 cm in carbondioxide will be close to ($\ln5=1.601$, $\ln2=0.693$).