A spherical ball falls through viscous medium with terminal velocity V. If this ball is replaced by another ball of the same mass but half the radius, then the terminal velocity will be (neglect the effect of buoyancy.)

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 ? remaining unchanged. If the variation of the density of air with height h from the ground is ρ ( h ) = ρ 0 e − h h 0 , where ρ 0 = 1.25 kgm − 3 and h 0 =6000 m, the value of N is .

The velocity of the liquid coming out of a small hole of a large vessel containing two different liquids of densities 2 ρ and ρ as shown in Figure is

One end of a long iron chain of linear mass density λ is fixed to a sphere of mass m and specific density 1/3 while the other end is free. The sphere along with the chain is immersed in a deep lake. If specific density of iron is 7, the height h above the bed of the lake at which the sphere will float in equilibrium is x m λ . F i n d x . (Assume that the part of the chain lying on the bottom of the lake exerts negligible forces on the upper part of the chain)

When liquid medicine of density ρ is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop. We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension T when the radius of the drop is R. When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.

A ball is thrown upwards into air with a speed greater than its terminal speed. It lands at the same place from where it was thrown. Mark the correct statement(s).

A cubical block of wood of side a and density ρ floats in water of density 2 ρ . The lower surface of the cube just touches the free end of a massless spring of force constant K fixed at the bottom of the vessel. The weight W put over the block so that it is completely immersed in water without wetting the weight is

Length o f a horizontal arm o f a U -tube is L = 1 m and ends of both the vertical arms are open to atmospheric pressure P 0 = 10 5 N / m 2 . A liquid of density ρ = 10 3 kg / m 3 is poured in the tube such that liquid just fills the horizontal part of the tube as shown in figure. Now one end of the open ends is sealed and the tube is then rotated about a vertical axis passing through the other vertical arm with angular speed ω 0 = 40 / 3 rad/sec . If length of each vertical arm is a = 1 m and in the sealed end liquid rises to a height y = 1/2 m , find pressure in the sealed tube during rotation in 10 5 N / m 2

Equal volume of two immiscible liquids of densities ρ and 2 ρ are filled in a vessel as shown in the figure. Two small holes are punched at depths h/2 and 3h/2 from the surface of lighter liquid. If v 1 and v 2 are the velocities of efflux at these two holes, then v 1 /v 2 is

A cube of mass m = 800 g floats on the surface of water. Water wets it completely. The cube is 10 cm on each edge. By what additional distance (in mm) is it buoyed up or down by surface tension? Surface of water = 0.07 Nm -1 .

A homogeneous solid cylinder of length L < H 2 , cross-sectional area A 5 is immersed such that it floats with its axis vertical at the liquid interface with length L 4 in the denser liquid as shown in Figure. The lower density liquid is open to atmosphere having pressure P 0 . Then, density D of solid is given by

A fully filled hemispherical tank of radius R has an orifice of small area a at its bottom. Time required to completely empty the tank, will be (assume that the top surface area of liquid is always much greater than the orifice area)

A container is containing mercury and water. A uniform solid object stays at equilibrium at the interface between mercury and water such that 20% of the volume of object is in mercury. Find the density ( g c c − 1 ) of the object. [Density of mercury is 13.6 g c c − 1 , density of water is 1 g c c − 1 ]

A hollow sphere of relative density 2 floats completely immersed on the surface of water. If the exterior(total) volume of the sphere be V, then the volume of the cavity will be

A light cylindrical vessel which is kept on a frictional horizontal surface, its base area A and height H. A hole of cross –sectional area ‘a’ is made at a height H 4 from it’s base (bottom).Then choose correct option/s.

A U tube containing a liquid of density ρ has parameters as shown in figure. The tube is given a horizontal acceleration such that the pressure at point A is equal to atmospheric pressure. If a minimum = n × g . Find n.

A container is containing mercury and water. A uniform solid object stays at equilibrium at the interface between mercury and water such that 20 % of the volume of object is in mercury. Find the density ( g c c − 1 ) of the object. [Density of mercury is 13.6 g c c − 1 , density of water is 1 g c c − 1 ]

A cube of mass m and side length ‘a’ wettable by water (contact angle ), floats on the surface of water. Distance between the lower face of cube and the surface of water is a N . Value of N is (Take m = ρ a 3 3 ; surface tension of water = ρ a 2 g 24 where ρ is density of water)

Two spherical soap bubbles in vacuum are connected through a narrow tube. Radius of left bubble is R 0 and that of right bubble is 2 R 0 . Air flows from one side to another very slowly maintaining spherical shape of bubbles . At any instant before steady state is reached, r 1 , A 1 , V 1 , n 1 are radius, surface area, volume and number of moles of gas in the left bubble and r 2 , A 2 , V 2 , n 2 are radius, surface area, volume and number of moles in the right bubble. Assume that temperature T remains constant throughout the process. Let n 1 = 2 initially and S be the surface tension of soap solution. Choose the correct option(s). (R is universal gas constant)

A conical capillary tube as shown in figure is submerged in a liquid. Contact angle between the liquid and capillary is 0 ° and the weight of liquid inside the meniscus is to be neglected. T is surface tension of the liquid, r m is radius of the meniscus, g is acceleration due to gravity and ρ is density of the liquid. Semi-vertex angle of conical tube is θ

A cylindrical shaft of radius R rotating with angular speed ω is supported on a horizontal surface such that there exists a thin film of viscous fluid of thickness h. Coefficient of viscosity of liquid is η . The power supplied to the shaft must be proportional to the nth power of angular speed of the shaft. Then n is

Two capillary tubes of same diameter are put vertically in each of two liquids. Whose relative densities are 0.8 and 0.6 and surface tensions are 60 and 50 dyne/cm respectively. Ratio of heights of liquids in two tubes are k : 10 . Find k. (Contact angles for both capillary is same)

A rectangular gate 3 × 1 m 2 stands vertical with water on one side of it hinged at middle. The force F required to be applied at the bottom to keep the gate in equilibrium is K × 10 3 N . ρ w a t e r = 1 × 10 3 k g / m 3 a n d g = 10 m / s 2 . Find K

An open-ended U-tube of uniform cross-sectional area contains water (density 10 3 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 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 h 1 h 2 of the heights of the liquid in the two arms is-

As shown schematically in the figure, two vessels contain water solutions (at temperature ?) of potassium permanganate (KMnO 4 ) of different concentrations ? 1 and ? 2 (? 1 > ? 2 ) molecules per unit volume with Δ n = n 1 − n 2 ≪ n 1 . . When they are connected by a tube of small length ℓ and cross-sectional area S, KMnO 4 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 − β v on each molecule, where β is a constant. Neglecting all terms of the order ( Δ n ) 2 , which of the following is/are correct? (? ? 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 kg m −3 , 0.07 Nm −1 and 10 ms −2 , respectively, the value of h (in mm) is

A train with cross-sectional area S t is moving with speed ? ? inside a long tunnel of cross-sectional area S 0 (S 0 = 4S t ). Assume that almost all the air (density ρ ) 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 = 7 2 N ρ v t 2 . The value of N is

Three liquids of densities ρ 1 , ρ 2 and ρ 3 and heights as shown are in equilibrium in a U-tube. Match the columns. Column-I Column-II (P) ρ 3 = 2 ρ 2 − ρ 1 (1) K = 2 h (Q) ρ 3 = ρ 2 − ρ 1 2 (2) K = h / 2 (R) ρ 1 = ρ 2 (3) K = 0 (S) ρ 2 = 3 ρ 1 + ρ 3 3 (4) None of the above

In two figures,

A liquid flows steadily through a series combination of three capillary tubes of radii r, 2r and 3r, all of the same length L. If the pressure difference across the combination is 28 cm of mercury, the pressure difference (in cm of Hg) across the tube of radius 2r is very nearly equal to cm of Hg

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 θ = 45 ∘ . P 1 and P 2 are pressure at points 1 and 2, respectively, located at the base of the tube Let β = P 1 − P 2 / ( ρgd ) , where ρ 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 ?

A long capillary tube of radius 0.1 cm open at both ends is filled with water and placed vertically. What will be the height of the column of water left in the capillary (in cm)?Given, surface tension of water = 75 dyne cm and density of water = 1 g cm -3 . (Take g = 1000 cm/s 2 )

In the figure, the cross-sectional area of the smaller tube is a and that of the larger tube is 2a. A block of mass m is kept in the smaller tube having the same base area a, as that of the tube. The difference between water levels of the two tubes is

A sphere of brass released in a long liquid column attains a terminal speed v 0 . If the terminal speed is attained by a sphere of marble of the same radius and released in the same liquid is nv 0 then the value of n will be (Given: The specific gravities of brass, marble and liquid are 8.5, 2.5 and 0.8 respectively)

When at rest, a liquid stands at the same level in the tubes as shown in the figure. But as indicated, a height difference h occurs when the system is given an acceleration a towards the right. Then h is equal to

A container partially filled with water is moved horizontally with acceleration a = g 3 . A small wooden ball of mass m is tied to the bottom of the container using a string. The ball remains inside water with the string inclined at an angle θ to the horizontal. Assuming that the density of ball is half the density of water, if, then m = 3 10 kg find the tension in the string (in N). (Take g = 10 m/s 2 )

A drop of liquid of density ρ is floating half-immersed in a liquid of density d. If σ is the surface tension, the diameter of the drop of the liquid is

A small metal ball of diameter 4 mm and density 10.5 g/cm 3 in dropped in glycerine of density 1.5 g/cm 3 . The ball attains a terminal velocity of 8/cm s -1 . The coefficient of viscosity of glycerine is

The area of two holes A and B are 2a and a, respectively. The holes are at height (H/3) and (2H/3) from the surface of water. Find the correct option(s):

A solid cylinder of height h and mass m is floating in a liquid of density ρ as shown in the figure. Find the acceleration of the vessel (in m/s 2 ) containing liquid for which the relative downward acceleration of the completely immersed cylinder w.r.t. vessel becomes equal to one-third of that of the vessel. (Take g = 10 m/s 2 )

A wooden plank of length 1 m and uniform cross section is hinged at one end to the bottom of a tank as shown in the figure. The tank is filled with water up to a height of 0.5 m. The specific gravity of the plank is 0.5. If the angle θ by the inclination of that the plank makes with the vertical in the equilibrium position (exclude the case θ = 0). Find the value of 1 cos 2 θ .

An open vessel containing liquid is moving with constant acceleration a on a levelled horizontal surface. For this situation mark out the correct statement(s).

A ball rises to the surface of a liquid with constant velocity. The density of the liquid is four time the density of the material of the ball. The frictional force of the liquid on the rising ball is greater than the weight of the ball by a factor of

In the Figure shown, the heavy cylinder (radius R) resting on a smooth surface separates two liquids of densities 2 ρ and 3 ρ . The height h for the equilibrium of cylinder must be