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JEE Mains | PYQ

A small ball of mass \( m \) and density \( \rho \) is dropped in a viscous liquid of density \( \rho_{0} \). After sometime, the ball falls with constant velocity. The viscous force on the ball is:

The velocity of 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 \( x \times 10^{-4} \, N \), The value of \( x \) is __________. [use \( g = 10 \, m/s^{2} \) ]

A soap bubble of surface tension \( 0.04 \, \text{N/m} \) is blown to a diameter of \( 7 \, \text{cm} \). If \( (15000 - x) \, \mu \text{J} \) of work is done in blowing it further to make its diameter \( 14 \, \text{cm} \), then the value of \( x \) is ____________ . \( \left( \pi = \dfrac{22}{7} \right) \).

A pressure-pump has a horizontal tube of cross sectional area \( 10\,\text{cm}^{2} \) for the outflow of water at a speed of \( 20\,\text{m/s} \). The force exerted on the vertical wall just in front of the tube which stops water horizontally flowing out of the tube, is:
[given : density of water \( =1000\,\text{kg}/\text{m}^{3} \) ]

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: When you squeeze one end of a tube to get toothpaste out from the other end, Pascal’s principle is observed.
Reason R: A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.
In the light of the above statements, choose the most appropriate answer from the options given below

A cubical block of density \( \rho_b = 600 \, \text{kg/m}^3 \) floats in a liquid of density \( \rho_e = 900 \, \text{kg/m}^3 \). If the height of block is \( H = 8.0 \, \text{cm} \) then height of the submerged part is ________________ \( \text{cm} \).

A liquid of density \( 600 \, \text{kg} \, \text{m}^{-3} \) flowing steadily in a tube of varying cross-section. The cross-section at a point A is \( 1.0 \, \text{cm}^2 \) and that at B is \( 20 \, \text{mm}^2 \). Both the points A and B are in same horizontal plane, the speed of the liquid at A is \( 10 \, \text{cm} \, \text{s}^{-1} \). The difference in pressures at A and B points is ______ \( \text{Pa} \).

Small water droplets of radius \( 0.01 \, \text{mm} \) are formed in the upper atmosphere and falling with a terminal velocity of \( 10 \, \text{cm/s} \). Due to condensation, if \( 8 \) such droplets are coalesced and formed a large drop, the new terminal velocity will be _______ \( \text{cm/s} \).

The increase in pressure required to decrease the volume of a water sample by \( 0.2\% \) is \( P \times 10^{5} \, \text{N m}^{-2} \). Bulk modulus of water is \( 2.15 \times 10^{9} \, \text{N m}^{-2} \). The value of \( P \) is __________.

An air bubble of diameter \( 6 \, \text{mm} \) rises steadily through a solution of density \( 1750 \, \text{kg} / \text{m}^3 \) at the rate of \( 0.35 \, \text{cm}/\text{s} \). The co-efficient of viscosity of the solution (neglect density of air) is ________ \( \text{Pas} \). 
(Given, \( g = 10 \, \text{m}\,\text{s}^{-2} \) )

A water drop of radius \( 1 \, \mu \text{m} \) falls in a situation where the effect of buoyant force is negligible. Co-efficient of viscosity of air is \( 1.8 \times 10^{-5} \, \text{N s m}^{-2} \) and its density is negligible as compared to that of water \( 10^{6} \, \text{g m}^{-3} \). Terminal velocity of the water drop is: (Take acceleration due to gravity \( = 10 \, \text{m s}^{-2} \))

Two narrow bores of diameter \( 5.0 \, \text{mm} \) and \( 8.0 \, \text{mm} \) are joined together to form a U-shaped tube open at both ends. If this U-tube contains water, what is the difference in the level of two limbs of the tube.
[Take surface tension of water \( T = 7.3 \times 10^{-2} \, \text{N m}^{-1} \), angle of contact \( = 0 \), \( g = 10 \, \text{ms}^{-2} \) and density of water \( = 1.0 \times 10^{3} \, \text{kg m}^{-3} \).]

Surface tension of two liquids (having same densities), \( T_1 \) and \( T_2 \), are measured using capillary rise method utilizing two tubes with inner radii of \( r_1 \) and \( r_2 \) where \( r_1 > r_2 \). The measured liquid heights in these tubes are \( h_1 \) and \( h_2 \) respectively. [Ignore the weight of the liquid about the lowest point of meniscus]. The heights \( h_1 \) and \( h_2 \) and surfaces tensions \( T_1 \) and \( T_2 \) satisfy the relation:

Given below are two statements:
Statement I: The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well.
Statement II: The rise of a liquid in a capillary tube does not depend on the inner radius of the tube.
In the light of the above statements, choose the correct answer from the options given below:

The amount of work done to break a big water drop of radius \( 'R' \) into \( 27 \) small drops of equal radius is \( 10 \, \text{J} \). The work required to break the same big drop into \( 64 \) small drops of equal radius will be:

Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure gauge attached to the pipe is \( P_1 \). The reading of the pressure gauge falls to \( P_2 \) when the valve is opened. The speed of water flowing in the pipe is proportional to:

A small ball of mass \( M \) and density \( \rho \) is dropped in a viscous liquid of density \( \rho_{0} \). After some time, the ball falls with a constant velocity. What is the viscous force on the ball?

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 \( \dfrac{v_1}{v_2} \) is ______.

Two cylindrical vessels of equal cross-sectional area \( 16 \, \text{cm}^{2} \) contain water upto heights \( 100 \, \text{cm} \) and \( 150 \, \text{cm} \) respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take, density of water \( = 10^{3} \, \text{kg/m}^{3} \) and \( g = 10 \, \text{m s}^{-2} \) ]:

Given below are two statements:
Statement I: When speed of liquid is zero everywhere, pressure difference at any two points depends on equation \( P_1 - P_2 = \rho g (h_2 - h_1) \)
Statement II: In ventury tube shown \( 2gh = v_1^2 - v_2^2 \).

In the light of the above statements, choose the most appropriate answer from the options given below:

A plane is in level flight at constant speed and each of its two wings has an area of \( 40 \, \text{m}^2 \). If the speed of the air is \( 180 \, \text{km/h} \) over the lower wing surface and \( 252 \, \text{km/h} \) over the upper wing surface, the mass of the plane is ______ \( \text{kg} \). [Take air density to be \( 1 \, \text{kg m}^{-3} \) and \( g = 10 \, \text{m s}^{-2} \)]

Two liquids \( A \) and \( B \) have \( \theta_A \) and \( \theta_B \) as contact angles in a capillary tube. If \( K = \cos \theta_A / \cos \theta_B \), then identify the correct statement:

There is an air bubble of radius \( 1.0 \, \text{mm} \) in a liquid of surface tension \( 0.075 \, \text{N m}^{-1} \) and density \( 1000 \, \text{kg m}^{-3} \) at a depth of \( 10 \, \text{cm} \) below the free surface. The amount by which the pressure inside the bubble is greater than the atmospheric pressure is ________ \( \text{Pa} \) \( \left( g = 10 \, \text{m s}^{-2} \right) \)

A small spherical ball of radius \( r \), falling through a viscous medium of negligible density has terminal velocity \(' v '\). Another ball of the same mass but of radius \( 2r \), falling through the same viscous medium will have terminal velocity:

The height of liquid column raised in a capillary two of certain radius when dipped in liquid \( A \) vertically is, \( 5 \, \text{cm} \). If the tube is dipped in a similar manner in another liquid \( B \) of surface tension and density double the values of liquid \( A \), the height of liquid column raised in liquid \( B \) would be _______ \( \text{m} \).

Consider a completely full cylindrical water tank of height \( 1.6 \, \text{m} \) and of cross-sectional area \( 0.5 \, \text{m}^{2} \). It has a small hole in its side at a height \( 90 \, \text{cm} \) from the bottom. Assume, the cross-sectional area of the hole to be negligibly small as compared to that of the water tank. If a load \( 50 \, \text{kg} \) is applied at the top surface of the water in the tank then the velocity of the water coming out at the instant when the hole is opened is: \( (g = 10 \, \text{m}/\text{s}^{2}) \)

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

Consider a cylindrical tank of radius \( 1 \, \text{m} \) is filled with water. The top surface of water is at \( 15 \, \text{m} \) from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of \( 5 \, \text{m} \) from the bottom. A force of \( 5 \times 10^{5} \, \text{N} \) is applied on the top surface of water using a piston. The speed of ifflux from the hole will be:
(given atmospheric pressure \( P_A = 1.01 \times 10^{5} \, \text{Pa} \), density of water \( \rho_w = 1000 \, \text{kg/m}^{3} \) and gravitational acceleration \( g = 10 \, \text{m/s}^{2} \))

A water drop of radius \( 1 \, \text{cm} \) is broken into \( 729 \) equal droplets. If surface tension of water is \( 75 \, \text{dyne}/\text{cm} \), then the gain in surface energy upto first decimal place will be:
(Given \( \pi = 3.14 \))

Sixty four rain drops of radius \( 1 \, \text{mm} \) each falling down with a terminal velocity of \( 10 \, \text{cm/s} \) coalesce to form a bigger drop. The terminal velocity of bigger drop is ______ \( \text{cm/s} \)

If \( \rho \) is the density and \( \eta \) is coefficient of viscosity of fluid which flows with a speed \( v \) in the pipe of diameter \( d \), the correct formula for Reynolds number \( R_{e} \) is:

A hydraulic press containing water has two arms with diameters as mentioned in the figure. A force of \( 10 \, \text{N} \) is applied on the surface of water in the thinner arm. The force required to be applied on the surface of water in the thicker arm to maintain equilibrium of water is _____ \( \text{N} \).

Given below are two statements :
Statement (I) : Viscosity of gases is greater than that of liquids.
Statement (II) : Surface tension of a liquid decreases due to the presence of insoluble impurities.
In the light of the above statements, choose the most appropriate answer from the options given below :

A liquid column of height \( 0.04 \, \text{cm} \) balances excess pressure of a soap bubble of certain radius. If density of liquid is \( 8 \times 10^{3} \, \text{kg m}^{-3} \) and surface tension of soap solution is \( 0.28 \, \text{N m}^{-1} \), then diameter of the soap bubble is ______ \( \text{cm} \).
[If \( g = 10 \, \text{m s}^{-2} \)]

The area of cross-section of a large tank is \( 0.5 \, \text{m}^{2} \). It has a narrow opening near the bottom having area of cross-section \( 1 \, \text{cm}^{2} \). A load of \( 25 \, \text{kg} \) is applied on the water at the top in the tank. Neglecting the speed of water in the tank, the velocity of the water, coming out of the opening at the time when the height of water level in the tank is \( 40 \, \text{cm} \) above the bottom, will be ________ \( \text{cm s}^{-1} \). [Take \( g = 10 \, \text{m s}^{-2} \)]

A cylindrical block of mass \( M \) and area of cross section \( A \) is floating in a liquid of density \( \rho \) and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is _______.

The radius of pressure metre attached with a closed pipe is \( 4.5 \times 10^{4} \, \text{N}/\text{m}^{2} \). On opening the value, water starts flowing and the reading of pressure metre falls to \( 2.0 \times 10^{4} \, \text{N}/\text{m}^{2} \). The velocity of water is found to be \( \sqrt{V} \, \text{m}/\text{s} \). The value of V is ____________ .

A big drop is formed by coalescing \( 1000 \) small droplets of water. The surface energy will become:

Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is \( 5 \, \text{cm} \) and the area of cross-sections at A and B are \( 6 \, \text{cm}^{2} \) and \( 3 \, \text{cm}^{2} \) respectively. The rate of flow will be ______________ \( \text{cm}^{3} / \text{s} \). \( \text{(take } g = 10 \, \text{m} / \text{s}^{2} \text{)} \)

In the experiment for measurement of viscosity '\( \eta \)' of given liquid with a ball having radius \( R \), consider following statements.
A. Graph between terminal velocity \( V \) and \( R \) will be a parabola.
B. The terminal velocities of different diameter balls are constant for a given liquid.
C. Measurement of terminal velocity is dependent on the temperature.
D. This experiment can be utilized to assess the density of a given liquid.
E. If balls are dropped with some initial speed, the value of \( \eta \) will change.
Choose the correct answer from the options given below:

A spherical ball of radius \( 1 \, \text{mm} \) and density \( 10.5 \, \text{g/cc} \) is dropped in glycerin of coefficient of viscosity \( 9.8 \) poise and density \( 1.5 \, \text{g/cc} \). Viscous force on the ball when it attains constant velocity is: \( 3696 \times 10^{-x} \, \text{N} \). The value of \( x \) is: (Given, \( g = 9.8 \, \text{m/s}^2 \) and \( \pi = \frac{22}{7} \))

A water spray gun is attached to a hose of cross sectional area \( 30 \, \text{cm}^2 \). The gun comprises of \( 10 \) perforations each of cross sectional area of \( 15 \, \text{mm}^2 \). If the water flows in the hose with the speed of \( 50 \, \text{cm/s} \), calculate the speed at which the water flows out from each perforation. (Neglect any edge effects)

When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height \( h \). If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by \(1\%\) each, then the height of the liquid in the tube will change by ________ \(\%\).

A light cylindrical vessel is kept on a horizontal surface. Area of base is \( A \). A hole of cross-sectional area \( a \) is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impart force of the emerging liquid is \( (a \ll A) \).

A small rigid spherical ball of mass \( M \) is dropped in a long vertical tube containing glycerine. The velocity of the ball becomes constant after some time. If the density of glycerine is half of the density of the ball, then the viscous force acting on the ball will be (consider \( g \) as acceleration due to gravity)

Given below are two statements:
Statement I: The hot water flows faster than cold water.
Statement II: Soap water has higher surface tension as compared to fresh water.
In the light above statements, choose the correct answer from the options given below:

A big drop is formed by coalescing \( 1000 \) small droplets of water. The ratio of surface energy of \( 1000 \) droplets to that of energy of big drop is \( \dfrac{10}{x} \). The value of \( x \) is __________.

Given below are two statements:
Statement I : When a capillary tube is dipped into a liquid, the liquid neither rises nor falls in the capillary. The contact angle may be \( 0^\circ \).
Statement II : The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well.
In the light of the above statement, choose the correct answer from the options given below:

The velocity of a small ball of mass ‘\( m \)’ and density \( d_1 \), when dropped in container filled with glycerine, becomes constant after some time. If the density of glycerine is \( d_2 \), then the viscous force acting on the ball, will be:

A cube having a side of \( 10 \, \text{cm} \) with unknown mass and \( 200 \, \text{gm} \) mass were hung at two ends of an uniform rigid rod of \( 27 \, \text{cm} \) long. The rod along with masses was placed on a wedge keeping the distance between wedge point and \( 200 \, \text{gm} \) weight as \( 25 \, \text{cm} \). Initially the masses were not at balance. A beaker is placed beneath the unknown mass and water is added slowly to it. At given point the masses were in balance and half volume of the unknown mass was inside the water. (Take the density of unknown mass is more than that of the water, the mass did not absorb water and water density is \( 1 \, \text{gm}/\text{cm}^3 \)). The unknown mass is ______ kg.

A tube of length \( 50 \, \text{cm} \) is filled completely with an incompressible liquid of mass \( 250 \, \text{g} \) and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity \( x\sqrt{F} \) \( \text{rad s}^{-1} \). If \( F \) be the force exerted by the liquid at the other end then the value of \( x \) will be _______.

An air bubble of radius \( 0.1 \, \text{cm} \) lies at a depth of \( 20 \, \text{cm} \) below the free surface of a liquid of density \( 1000 \, \text{kg/m}^3 \). If the pressure inside the bubble is \( 2100 \, \text{N/m}^2 \) greater than the atmospheric pressure, then the surface tension of the liquid in SI unit is (use \( g = 10 \, \text{m/s}^2 \)).

Two small drops of mercury each of radius \( R \) coalesce to form a single large drop. The ratio of total surface energy before and after the change is:

A tub is filled with water and a wooden cube \( 10 \, \text{cm} \times 10 \, \text{cm} \times 10 \, \text{cm} \) 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 \( 3.87 \, \text{cm} \). The mass of the metal coin is ______ gram. (Take water density as \( 1 \, \text{g}/\text{cm}^3 \) and density of wood as \( 0.4 \, \text{g}/\text{cm}^3 \))

A liquid of density \( 750\,\text{kg m}^{-3} \) flows smoothly through a horizontal pipe that tapers in cross-sectional area from \( A_1 = 1.2 \times 10^{-2}\,\text{m}^2 \) to \( A_2 = \frac{A_1}{2} \). The pressure difference between the wide and narrow sections of the pipe is \( 4500\,\text{Pa} \). The rate of flow of liquid is ____________ \( \times 10^{-3}\,\text{m}^3\text{s}^{-1} \).

A vessel with square cross-section and height of \( 6 \, \text{m} \) is vertically partitioned. A small window of \( 100 \, \text{cm}^2 \) with hinged door is fitted at a depth of \( 3 \, \text{m} \) in the partition wall. One part of the vessel is filled completely with water and the other side is filled with the liquid having density \( 1.5 \times 10^3 \, \text{kg} / \text{m}^3 \). What force one needs to apply on the hinged door so that it does not get opened?
(Acceleration due to gravity \( = 10 \, \text{m} / \text{s}^2 \))

The surface tension of a soap solution is \( 3.5\times 10^{-2} \, \text{N} / \text{m} \). The work required to increase the radius of a soap bubble from \( 1 \, \text{cm} \) to \( 2 \, \text{cm} \) is \( \alpha \times 10^{-6} \, \text{J} \). The value of \( \alpha \) is ______. \( \left( \pi=\dfrac{22}{7} \right) \)

A small liquid drop of radius \( R \) is divided into \( 27 \) identical liquid drops. If the surface tension is \( T \), then the work done in the process will be:

If \( 1000 \) droplets of water of surface tension \( 0.07 \, \text{N/m} \), having same radius \( 1 \, \text{mm} \) each, combine to from a single drop. In the process the released surface energy is: (Take \( \pi = \frac{22}{7} \))

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:

Two soap bubbles of radius \( 2 \, \text{cm} \) and \( 4 \, \text{cm} \), respectively, are in contact with each other. The radius of curvature of the common surface, in \( \text{cm} \), is _________.

Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes \( 0.01 \, \mathrm{cm^{3}} \) of oleic acid per \( \mathrm{cm^{3}} \) of the solution. Then you make a thin film of this solution (monomolecular thickness) of area \( 4 \, \mathrm{cm^{2}} \) by considering \( 100 \) spherical drops of radius \( \left( \dfrac{3}{40\pi} \right)^{1/3} \times 10^{-3} \, \mathrm{cm} \). Then the thickness of oleic acid layer will be \( x \times 10^{-14} \, \mathrm{m} \). Where \( x \) is _______.

The diameter of an air bubble which was initially \( 2 \, \text{mm} \), rises steadily through a solution of density \( 1750 \, \text{kg m}^{-3} \) at the rate of \( 0.35 \, \text{cms}^{-1} \). The coefficient of viscosity of the solution is ______ poise (in nearest integer). (the density of air is negligible).

In a hydraulic lift, the surface area of the input piston is \( 6 \, \text{cm}^2 \) and that of the output piston is \( 1500 \, \text{cm}^2 \). If \( 100 \, \text{N} \) force is applied to the input piston to raise the output piston by \( 20 \, \text{cm} \), then the work done is __________ kJ.

Given below are two statements:
Statement I : Pressure in a reservoir of water is same at all points at the same level of water.
Statement II: The pressure applied to enclosed water is transmitted in all directions equally.
In the light of the above statements, choose the correct answer from the options given below:

What will be the nature of flow of water from a circular tap, when its flow rate increased from \( 0.18 \, \text{L/min} \) to \( 0.48 \, \text{L/min} \)? The radius of the tap and viscosity of water are \( 0.5 \, \text{cm} \) and \( 10^{-3} \, \text{Pa s} \), respectively.
(Density of water : \( 10^{3} \, \text{kg/m}^{3} \))

An object is located at \( 2 \, \text{km} \) beneath the surface of the water. If the fractional compression \( \dfrac{\Delta V}{V} \) is \( 1.36\% \), the ratio of hydraulic stress to the corresponding hydraulic strain will be ________.
[Given : density of water is \( 1000 \, \text{kg m}^{-3} \) and \( g = 9.8 \, \text{m s}^{-2} \)]

A drop of liquid of density \( \rho \) is floating half immersed in a liquid of density \( \sigma \) and surface tension \( 7.5 \times 10^{-4} \, \text{N cm}^{-1} \). The radius of drop in cm will be: \( ( g = 10 \, \text{ms}^{-2} ) \)

The terminal velocity of a metallic ball of radius \( 6 \, \text{mm} \) in a viscous fluid is \( 20 \, \text{cm/s} \). The terminal velocity of another ball of same material and having radius \( 3 \, \text{mm} \) in the same fluid will be ______ \( \text{cm/s} \).

A tube of length \( 1 \, \text{m} \) is filled completely with an ideal liquid of mass \( 2M \), and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is \( F \) then angular velocity of the tube is \( \sqrt{\frac{F}{\alpha M}} \) in SI unit. The value of \( \alpha \) is __________.

In a test experiment on a model aeroplane in wind tunnel, the flow speeds on the upper and lower surfaces of the wings are \( 70 \, \text{m s}^{-1} \) and \( 65 \, \text{m s}^{-1} \) respectively. If the wing area is \( 2 \, \text{m}^2 \), the lift of the wing is __________ N.
(Given density of air \( = 1.2 \, \text{kg m}^{-3} \))

A sphere of relative density \( \sigma \) and diameter \( D \) has concentric cavity of diameter \( d \). The ratio of \( \dfrac{D}{d} \), if it just floats on water in a tank is:

The excess pressure inside a soap bubble \( A \) in air is half the excess pressure inside another soap bubble \( B \) in air. If the volume of the bubble \( A \) is \( n \) times the volume of the bubble \( B \), then, the value of \( n \) is __________.

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : A spherical body of radius \( (5 \pm 0.1) \, \text{mm} \) having a particular density is falling through a liquid of constant density. The percentage error in the calculation of its terminal velocity is \( 4\% \).
Reason R : The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius.
In the light of the above statements, choose the correct answer from the options given below:

A mercury drop of radius \( 10^{-3} \, \text{m} \) is broken into \( 125 \) equal size droplets. Surface tension of mercury is \( 0.45 \, \text{N m}^{-1} \). The gain in surface energy is:

An ideal fluid of density \( 800 \,\mathrm{kg\, m^{-3}} \), flows smoothly through a bent pipe (as shown in figure) that tapers in cross-sectional area from \( a \) to \( \frac{a}{2} \). The pressure difference between the wide and narrow sections of pipe is \( 4100 \,\mathrm{Pa} \). At wider section, the velocity of fluid is \( \frac{\sqrt{x}}{6} \,\mathrm{m\, s^{-1}} \) for \( x = \) ____________. (Given \( g = 10 \,\mathrm{m\, s^{-2}} \) )

A soap bubble of radius \( 3 \, \text{cm} \) is formed inside the another soap bubble of radius \( 6 \, \text{cm} \). The radius of an equivalent soap bubble which has the same excess pressure as inside the smaller bubble with respect to the atmospheric pressure is ________ \( \text{cm} \).

Consider a water tank as shown in the figure. Its cross-sectional area is \( 0.4 \, \text{m}^{2} \). The tank has an opening B near the bottom whose cross-section area is \( 1 \, \text{cm}^{2} \). A load of \( 24 \, \text{kg} \) is applied on the water at the top when the height of the water level is \( 40 \, \text{cm} \) above the bottom, the velocity of water coming out the opening B is \( v \, \text{m s}^{-1} \). The value of \( v \), to the nearest integer, is: [Take value of \( g \) to be \( 10 \, \text{m s}^{-2} \)]

Mercury is filled in a tube of radius \( 2 \, \text{cm} \) up to a height of \( 30 \, \text{cm} \). The force exerted by mercury on the bottom of the tube is _______________ \( \text{N} \).
(Given, atmospheric pressure \( = 10^{5} \, \text{N m}^{-2} \), density of mercury \( = 1.36 \times 10^{4} \, \text{kg m}^{-3} \), \( g = 10 \, \text{m s}^{-2} \), \( \pi = \dfrac{22}{7} \))

A soap bubble is blown to a diameter of \( 7 \) \( \text{cm} \). \( 36960 \) \( \text{erg} \) of work is done in blowing it further. If surface tension of soap solution is \( 40 \) \( \text{dyne/cm} \) then the new radius is _________ \( \text{cm} \). Take \( \left( \pi = \frac{22}{7} \right) \).

A small metallic sphere of diameter \( 2 \, \text{mm} \) and density \( 10.5 \, \text{g/cm}^3 \) is dropped in glycerine having viscosity \( 10 \, \text{Poise} \) and density \( 1.5 \, \text{g/cm}^3 \) respectively. The terminal velocity by the sphere is ______ \( \text{cm/s} \).
\( (\pi = \frac{22}{7} \text{ and } g = 10 \, \text{m/s}^2) \)

Correct Bernoulli’s equation is: [Symbols have their usual meaning]

The terminal velocity \( (v_t) \) of the spherical rain drop depends on the radius \( (r) \) of the spherical rain drop as:

Consider following statements:
A. Surface tension arises due to extra energy of the molecules at the interior as compared to the molecules at the surface, of a liquid.
B. As the temperature of liquid rises, the coefficient of viscosity increases.
C. As the temperature of gas increases, the coefficient of viscosity increases.
D. The onset of turbulence is determined by Reynold’s number.
E. In a steady flow two stream lines never intersect.

A spherical drop of liquid splits into \( 1000 \) identical spherical drops. If \( u_i \) is the surface energy of the original drop and \( u_f \) is the total surface energy of the resulting drops, the (ignoring evaporation), \( \dfrac{u_f}{u_i} = \left( \dfrac{10}{x} \right) \). Then value of \( x \) is ________.

A cube of ice floats partly in water and partly in kerosene oil. The ratio of volume of ice immersed in water to that in kerosene oil (specific gravity of Kerosene oil \( = 0.8 \), specific gravity of ice \( = 0.9 \)):

A cylindrical vessel of \( 40 \, \text{cm} \) radius is completely filled with water and its capacity is \( 528 \, \text{dm}^3 \) \( (\text{dm} : \) decimeter\() \). The vessel is placed on a solid block of exactly same height as vessel. If a small hole is made at \( 70 \, \text{cm} \) below the top of water level, the horizontal range of water falling on the ground in the beginning is ______ \( \text{cm} \).

Two water drops each of radius ‘\( r \)’ coalesce to form a bigger drop. If ‘\( T \)’ is the surface tension, the surface energy released in this process is: