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Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road

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PROPERTIES OF MATTER

C1 All real “rigid” bodies are to some extent elastic, which means that we can change their dimensions slightlyby pulling, pushing, twisting or compressing them.

Hooke’s law states that in elastic deformations, stress (force per unit area) is proportional to strain (relativedeformation) :

modulusElasticStrain

Stress

Three elastic moduli are used to describe the elastic bahaviour (deformations) of objects as they respond toforces that act on them.

1. Longitudinal stress and longitudinal strain : Longitudinal stress is defined as A

F, where F is the force

perpendicular to the plane of cross sectional A. There are two types of longitudinal stress :

(a) Tensile longitudinal stress, and

(b) Compresive longitudinal stress

Tensile stress is tensile force per unit area, A/F . Tensile strain is fractional change in length, l/l0.

Young’s modulus Y is the ratio of tensile stress to tensile strain :

l

l

ll

0

0 A

F

/

A/FY

Compressives stress and strain are defined the same way as tensile stress and strain. For many materials,Young’s modulus has the same value for both tension and compression.

2. Bulk stress or volume stress or hydraulic stress :

The bulk modulus B is the negative of the ratio of pressure change p (bulk stress) a fractional volumechange V/V

0 :

0V/V

pB

Compressibility k is the reciprocal of bulk modulus : k = 1/B.

3. Shear stress is force per unit area F||/A for a force applied parallel to a surface. Shear strain is the angle .

The shear modulus S is the ratio of shear stress to shear strain :

A/F

x

h

A

F

h/x

A/F

strainShear

stressShearS

||||||

The proportional limit is the maximum stress for which stress and strain are proportional. Beyond theproportional limit, Hooke’s law is not valid. The elastic limit is the stress beyond which irreversibledeformation occurs. The breaking stress, or ultimate strength, is the stress at which the material breaks.

Energy stored in a stretched wire per unit volume equals to 2

1 × stress × strain.

Practice Problems :

1. The following four wires are made of the same material. Which of these will have the largestextension when the same tension is applied.

(a) length = 50 cm, diameter = 0.5 mm (b) length = 100 cm, diameter = 1 mm

(c) length = 200 cm, diameter = 2 mm (d) length = 300 cm, diameter = 3 mm

2. The compressibility of water is 4 × 10–5 per unit atmospheric pressure. The decrease in volume of100 cm3 of water under a pressure of 100 atmosphere will be

(a) 0.4 cm3 (b) 4 × 10–5 cm3 (c) 0.025 cm3 (d) 0.004 cm3

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3. Young’s modulus of steel is 2 × 1011 N/m2. A steel wire has a length of 1 m and area of cross section1 mm2. The work required to increase its length by 1 mm is

(a) 0.1 J (b) 1 J (c) 10 J (d) 100 J

4. A substance breaks down by a stress of 106 N/m2. If the density of the material of the wire is3 × 103kg/m3, then the length of the wire of that substance which will break under its own weightwhen suspended vertically is

(a) 3.4 m (b) 34 m (c) 340 m (d) none of these

5. A metal ring of initial radius r and cross-sectional area A is fitted onto a wooden disc of radius R > r.If Young’s modulus of the metal is Y then the tenstion in the ring is

(a)r

AYR(b)

r

)rR(AY (c)

r

rR

A

Y(d)

AR

Yr

6. A massless rod AD consisting of three segments AB, BC and CD joined together is hanging verticallyfrom a fixed support at A. The lengths of the segments are respectively 0.1 m, 0.2 m and 0.15 m. Thecross-section of the rod is uniformly 10–4m2. A weight of 10 kg is hung from D. If Y

AB = 2.5 × 1010

N/m2, YBC

= 4 × 1010 N/m2 and YCD

= 1 × 1010 N/m2 then the ratio of displacement of points B, C andD is

(a) 1 : 2 : 3 (b) 2 : 3 : 7 (c) 3 : 5 : 9 (d) none

7. A steel wire (Young’s modulus = 2 × 1011 N/m2) of diameter 0.8 mm and length 1 m is clamped firmlyat two points A and B which are 1 m apart and in the same plane. A body is hung from the middlepoint of the wire such that the middle point sags 1 cm lower from the original position. The mass ofthe body is

(a) 82 gm (b) 41 gm (c) 22.5 gm (d) 11 gm

8. The bulk modulus of water if its volume changes from 100 litre to 99.5 litre under a pressure of 100atmosphere is

(a) 1.026 × 109 N/m2 (b) 2.026 × 109 N/m2

(c) 3.026 × 109 N/m2 (d) 4.026 × 109 N/m2

9. A rubber cord of length L is suspended vertically. Density of rubber is D and Young’s modulus is Y.If the cord extends by a length l under its own weight, then l is

(a) L2Dg/Y (b) L2Dg/2Y (c) L2Dg/4Y (d)Y

DgL2 2

[Answers : (1) a (2) a (3) a (4) b (5) b (6) d (7) a (8) b (9) b]

C2 Density : Density is mass per unit volume. If a mass m of material has volume V, its density is V

m .

Specific gravity is the ratio of the density of a material to the density of water.

Practice Problems :

1. If equal masses of two liquids of densities d1 and d

2 are mixed together, the density of the mixture is

(a) (d1 + d

2) (b) 2d

1d

2/(d

1 + d

2) (c) d

1d

2/(d

1 + d

2) (d) (d

1 + d

2)/2

2. If equal volume of two liquids of density is d1 and d

2 are mix together then the density of the mixture

is

(a) (d1 + d

2) (b) 2d

1d

2/(d

1 + d

2) (c) d

1d

2/(d

1 + d

2) (d) (d

1 + d

2)/2

3. Due to the change of pressure the density of the liquid will change. If the change in pressure is Pand the bulk modulus of liquid is B then the fractional change in density of the liquid equals to

(a)B

P(b)

B

P2(c)

B2

P(d)

B

P

2

3

[Answers : (1) b (2) d (3) a]

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C3 Pressure : Pressure is normal force per unit area.

Pressure (a scalar quantity) on a surface is defined as

dS

dF

S

Flimp

0s

The units for pressure are Nm–2 or pascal (Pa), or mm of mercury (or any other substance).

C4 Hydrostatic pressure distribution : Pressure in a fluid at rest increases with vertical height ‘h’ according

to the relation gdh

dp .

If the density of the liquid is constant at each point then the pressure at a point A at a depth h below the freesurface is given by p

A = gh + p

0, where p

0 is the pressure at the free surface (atmospheric pressure). Ab-

solute pressure is the total pressure in a fluid; gauge pressure is the difference between absolute pressureand atmospheric pressure.

Hydrostatic Paradox :

Three vessels of equal base area but containing different amounts of liquid upto the same height will havesame force at their bottom.

Practice Problems :

1. The pressure in a water tap at the base of a building is 3 × 106 dynes/cm2 and on its top it is1.6 × 106 dynes/cm2. The height of the building is approximately

(a) 7 m (b) 14 m (c) 70 m (d) 140 m

2. A uniformly tapering vessel is filled with a liquid of density 900 kg/m3. The thrust on the base of thevessel due to the liquid is (g = 10 m/s2)

(a) 3.6 N (b) 7.2 N (c) 10.8 N (d) 14.4 N

3. Consider a liquid of density is placed in a container upto the height h. If the force exerted by theliquid on the side wall is directly proportional to hn, then the value of n is

(a) 0 (b) ½ (c) 1 (d) 2

[Answers : (1) b (2) b (3) d]

C5 Pascal Law : Pascal’s law states that pressure applied to the surface of an enclosed fluid is transmittedundiminished to every portion of the fluid.

Practice Problems :

1. A piston of cross-sectional area 100 cm2 is used in a hydraulic press to exert a force of 107 dynes onthe water. The cross-sectional area of the other piston which supports a truck of mass 2000 kg is

(a) 9.8 × 102cm2 (b) 9.8 × 103cm2 (c) 1.96 × 103cm2 (d) 1.96 × 104cm2

2. A U-tube of uniform cross-section is partially filled with a liquid I. Another liquid II which does notmix with liquid I is poured into one side. It is found that the liquid levels of the two sides of the tubeare the same, while the level of liquid I has risen by 2 cm. If the specific gravity of liquid I is 1.1, thespecific gravity of liquid II must be

(a) 1.12 (b) 1.1 (c) 1.05 (d) 1.0

3. A U-tube is partly filled with a liquid A. Another liquid B, which does not mix with A, is poured intoone side until it stands a height h above the level of A on the other side, which has meanwhile risen aheight l. The density of B relative to that of A is

(a)l

l

h(b)

l

l

2h (c)

l

2l

2h (d)

l

l

h2

[Answers : (1) d (2) b (3) c]

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C6 Archimede’s Principle

When a body is immersed partly or wholly in a fluid, there acts an upward force on it called the buoyancyand its magnitude is equal to the weight of the fluid displaced. The point of the application of buoyancy isat the centre of mass of the displaced fluid and is called the centre of buoyancy. Buoyancy exists because ofpressure gradient. Thus in case of a free fall situation buoyancy is zero.

Principle of floatation

Weight of the object = Buoyancy

sVg =

lV

sg

V : total volume of the object

Vs : submerged volume of the object

s : density of object

l : density of liquid

Practice Problems :

1. A piece of wood of relative density 0.36 floats in oil of relative density 0.90. The fraction of volume ofwood above the surface of oil is

(a) 0.3 (b) 0.4 (c) 0.6 (d) 0.8

2. A large block of ice 10 m thick with a vertical hole drilled through it is floating in a lake. Theminimum length of the rope required to scoop out a bucket full of water through the hole is (densityof ice = 0.9 g/cm3)

(a) 0.5 m (b) 1.0 m (c) 1.2 m (d) 1.8 m

3. A streamlined body of relative density d1 falls from a height h on the surface of a liquid of relative

density d2, where d

2 > d

1. The time for which the body will fall inside the liquid is

(a)g

h2

d

d

2

1 (b)g

h2

d

d

1

2

(c)g

h2

dd

d

12

1

(d)

g

h2

d

dd

2

12

4. A small ball of density is immersed in a liquid of density ( > ) to a depth h and then released.The height above the surface of water up to which the ball will jump is

(a)

h(b) h1

(c) h1

(d)

h

5. A small ball of density is dropped from a height h into a liquid of density ( > ). Neglectingdamping forces, the maximum depth to which the body sinks is

(a)

h(b)

h(c)

)(h(d)

)(h

6. A block (density ) is suspended from a spring and produces an extension ‘x’. If the whole system isdipped in a liquid (density ) then new extention is

(a) x/ (b) x / (c) x (1 – / (d) x (1 – /

[Answers : (1) c (2) b (3) c (4) b (5) b (6) c]

C7 Fluid Dynamics :

An ideal fluid is incompressible and has no viscosity. A flow line is the path of the fluid particle; astreamline is a curve tangent at each point to the velocity vector at that point. A flow tube is a tube boundedat its sides by flow lines. In laminar flow, layers of fluid slide smoothly past each other. In turbulent flowthere is great disorder and a constantly changing flow pattern.

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Principle of Continuity : Conservation of mass in an incompressible fluid is expressed by the equation ofcontinuity; for two cross sections A

1 and A

2 in a flow tube, the flow speed v

1 and v

2 are related by

A1v

1 = A

2v

2.

The product Av is the volume flow rate, dV/dt, the rate at which volume crosses a section of the

tube : Avdt

dV .

Bernoulli’s equation relates the pressure p, flow speed v, and elevation y for steady flow in an ideal fluidwhich is based on conservation of energy principle. For any two points, denoted by subscripts 1 and 2.

2222

2111 v

2

1gypv

2

1gyp

Practice Problems :

1. Two large tanks a and b, open at the top, contains different liquids. A small hole is made in the sideof each tank at the same depth h below the liquid surface, but the hole in a has twice the area of thehole in b. The ratio of the densities of the liquids in a and b so that the mass flux is the same for eachhole should be

(a) 2 (b) 0.5 (c) 4 (d) 0.25

2. In the above problem the ratio of flow rates (volume flux) from the holes in a and b is

(a) 2 (b) 0.5 (c) 4 (d) 0.25

3. Air is streaming past a horizontal aeroplane wing such that its speed is 120 m/s over the uppersurface and 90 m/s at the lower surface. If the density of air is 1.3 kg/m3. If the wing is 10 m long andhas an average width 2 m, the gross lift of the wing is

(a) 5.2 × 104N (b) 6.2 × 104N (c) 7.2 × 104N (d) 8.2 × 104N

4. A horizontal pipe line carries water in a streamline flow. At a point along the pipe where thecross-sectional area is 10 cm2, the water velocity is 1 m/s and the pressure is 2000 Pa. The pressure ofwater at another point where the cross-sectional area is 5 cm2 is

(a) 500 Pa (b) 750 Pa (c) 900 Pa (d) 1100 Pa

5. The rate of flow of glycerine of density 1.25 × 103 kg/m3 through the conical section of a pipe, if theradii of its ends are 0.1 m and 0.04 m and the pressure drop across its length is 10 N/m2 is

(a) 6.28 × 10–3 m3/s (b) 6.28 × 10–4 m3/s

(c) 3.9 × 10–4 m3/s (d) 3.9 × 10–3 m3/s

6. Water flows out of two small holes P and Q in a wall of a tank and the two streams strike the groundat the same point. If the hole P is at a height h above the ground and the level of water stands at aheight H above the ground, then the height of Q is

(a)2

hH (b) H – h (c) H – h/2 (d)

2

hH

[Answers : (1) b (2) a (3) d (4) a (5) b (6) b]

C8 Viscosity : The viscosity of a fluid characterizes its resistance to shear strain. In a Newtonian fluidtheviscous force is propotional to strain rate. The viscous force between two layers of a fluid of area A having

a velocity gradient dv/dx is given by dx

dvAF where is called the coefficient of viscosity. In SI unit

of is poiseuille (1 PI = 1 Ns m–2) and the dimension of is ML–1T–1.

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Practice Problems :

1. The velocity of water (viscosity = 10–3 poiseuille) in a river is 18 km/hr at the surface. If the river is5 m deep, then the shearing stress between the horizontal layers of water is

(a) 0.5 × 10–3 N/m2 (b) 0.8 × 10–3 N/m2

(c) 10–3 N/m2 (d) 1.2 × 10–3 N/m2

[Answers : (1) c]

C9 Stoke’s Law and Terminal Speed : A sphere of radius r moving with speed v through a fluid havingviscosity experiences a viscous resisting force F given by Stoke’s law : F = 6rv.

The following graph shows the variation of velocity v with time t for a small spherical body falling

vertically in a long column of viscous liquid

The terminal speed acheived by a sphere is given by )(gr

9

2v

2

t

where is the density of the

sphere and is the density of the fluid in which sphere is moving.

Practice Problems :

1. The velocity of a small ball of mass m and density d1 when dropped in a container filled with

glycerine becomes constant after some time. The viscous force acting on the ball if density ofglycerine is d

2 is

(a)

1

2

d

d1mg (b) mg (c)

2

1

d

d1mg (d)

1

2

d

dmg

2. The viscosity of glycerine (having density 1.3 gm/cc) if a steel ball of 2 mm radius (density = 8 gm/cc)acquires a terminal velocity of 4 cm/sec in falling freely in the tank of glycerine is

(a) 13.6 poise (b) 14.6 poise (c) 15.6 poise (d) 16.6 poise

3. An air bubble of radius 1 mm is allowed to rise through a long cylindrical column of a viscousliquid of radius 5 cm and travels at a steady rate of 2.1 cm per sec. If the density of the liquid is1.47 gm per cc, then its viscosity is

(a) 1.324 poise (b) 1.424 poise (c) 1.524 poise (d) 1.624 poise

4. ‘n’ equal drops of water are falling through air with a steady velocity v. If the drops coalesced, thenthe new velocity is

(a) (n1/3) v (b) nv (c) (n1/2) v (d) (n2/3) v

5. A spherical ball of radius 1 × 10–4 m and density 104 kg/m3 falls freely under gravity through adistance h before entering a tank of water (viscosity of water is 9.8 × 10–6 N-s/m2). If after enteringthe water the velocity of the ball does not change, the value of h is

(a) 20.4 m (b) 22.4 m (c) 24.4 m (d) 26.4 m

[Answers : (1) a (2) b (3) c (4) d (5) a]

C10 Poiseuille’s Equation :

When such a fluid flows in a cylindrical pipe of inner radius R, and length L is the length if pipe, the totalvolume rate is given by Poiseuille’s equation :

L

ppR

8dt

dV 214

where p1 and p

2 are the pressures at the two ends and is the viscosity.

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Practice Problems :

1. Under a pressure head the rate of orderly volume flow of a liquid through a capillary tube is Q. If thelength of the capillary tube is doubled and the diameter of the bore is halved, the rate of flow wouldbecome

(a) Q/32 (b) Q/8 (c) Q/4 (d) 8 Q

2. Two liquids of coefficients of viscosity 1 and

2 are made to flow through a tube in succession under

the same pressure difference. If V1 and V

2 are, respectively, the volumes of the two liquids flowing

per second, then V1/V

2 is

(a)1

2

(b)

2

1

(c) 2

1

2

2

(d) 2

2

2

1

3. The graph for the variation of capillary rise and radius of the tube for the given liquid is

(a) linear (b) constant (c) hyperbolic (d) exponential

[Answers : (1) a (2) a (3) c]

C11 Reynolds Number : The turbulence flow of a fluid is determined by a dimensionless parameter called the

Reynolds number given by

vdRe where is the density of liquid, v its velocity, its viscosity and d is

the diameter of tube in which liquid will flow. For most cases Re < 1000 signifies laminar flow;

1000 < Re < 2000 is unsteady flow and R

e > 2000 implies turbulent flow.

C12 Surface Tension : The surface of a liquid behaves like a membrane under tension; the force per unit lengthacross a line on the surface is called the surface tension, denoted by T.

C13 Excess Pressure : Excess pressure inside a liquid drop of radius r is given by r

T2. Excess pressure inside

a liquid bubble or air bubble of radius r is given by r

T4.

C14 Capillary Rise or Fall : The rise or fall of a liquid in a capillary tube is given by gr

cosT2h

, where is

the angle of contact, is the density of liquid in the tube and r is the radius of the tube. For a clean glassplate in contact with pure water, = 0.

Practice Problems :

1. A liquid rises to a height h in a capillary tube on the earth. The height to which the same liquid wouldrise in the same tube on the moon is about

(a) 6 h (b) 6 h (c) h/6 (d) h/6

2. n identical spherical drops of a liquid of surface tension T, each of radius r, coalesce to form a singledrop. The surface energy

(a) decreases by 4r2(n – n1/3)T (b) increases by 4r2(n – n1/3)T

(c) decreases by 4r2(n – n2/3)T (d) increases by 4r2(n – n2/3)T

[Answers : (1) a (2) c]

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SINGLE CORRECT CHOIC E TYPE

1. A highly rigid cubical block A of small mass M andside L is fixed rigidly onto another cubical block Bof the same dimenstions and of low modulus ofrigidity such that the lower face of B is rigidlyheld on a horizontal surface. A small force F isapplied perpendicular to one of the side faces of A.After the force is withdrawn, block A executes smalloscillations the time period of which is given by

(a) LM2 (b) L/M2

(c) /ML2 (d) L/M2

2. The normal density of gold is and its bulkmodulus is K. The increase in density of a piece ofgold when a pressure P is applied uniformly fromall sides is

(a)K2

P(b)

K

P

(c)PK

P

(d)

PK

K

3. The length of rubber cord is l1 metres when the

tension 4 N and l2 metres when the tension is 5 N.

The length in metres when the tension is 9 N is

(a) 5l1 – 4l

2(b) 5l

2 – 4l

1

(c) 9l1 – 8l

2(d) 9l

2 – 8l

1

4. A cylindrical vessel of radius r is filled with ahomogenous liquid to a height h. If the forceexerted by the liquid on the side of the vessel is equalto the force exerted by it on the bottom of thevessel, then

(a) h = r (b) h = 2r

(c) h = r/2 (d) h = 3r/2

5. A vertical U-tube contains mercury in both its arms.A glycerine (density 1.3 g/cm3) column of length 10cm is introduced into one of the arms. Oil ofdensity 0.8 g/cm3 is poured into the other arm untilthe upper surfaces of oil and glycerine are at thesame level. The length of the oil column is (densityof mercury = 13.6 g/cm3)

(a) 8.5 cm (b) 9.6 cm

(c) 10.7 cm (d) 11.8 cm

6. A vessel contains oil (density 0.8 g/cm3) overmercury (density 13.6 g/cm3). A homogenous spherefloats with half its volume immersed in mercuryand the other half in oil. The density of thematerial of the sphere in g/cm3 is

(a) 3.3 (b) 6.4

(c) 7.2 (d) 12.8

7. A vessel of cross-sectional area A contains a liquidto a height H

1. If a hole having cross-sectional area

a is made at the bottom of the vessel, then the timetaken by the liquid level to decrease from H

1 to H

2

is

(a) 21 HH2

g

a

A

(b) 21 HHg

2

a

A

(c) 21 HH2

g

A

a

(d) 21 HHg

2

A

a

8. A liquid is kept in a cylindrical vessel which isrotating along its axis. The liquid rises at the sides.If the radius of the vessel is 0.05 m and the speed ofrotation is 2 rev/s, the difference in the height ofthe liquid at the centre of the vessel and at its sidesis

(a) 0.01 m (b) 0.02 m

(c) 0.03 m (d) 0.04 m

9. Two capillary tubes of the same radius and lengthl1 and l

2 are fitted horizontally side by side to the

bottom of a vessel containing water. The length ofa single tube that can replace the two tubes suchthat the rate of steady flow through this tube equalsthe combined rate of flow through the two tubes, is

(a) l1 + l

2(b)

221 ll

(c)21

21

ll

ll

(d)21

21

ll

ll

2

10. Two capillary tubes of the same length and radii r1

and r2 are fitted horizontally side by side to the

bottom of a vessel containing water. The radius ofa single tube that can replace the two tubes suchthat the rate of study flow through this tube equalsthe combined rate of flow through the two tubes, is

(a) r1 + r

2(b) 21rr

(c) 2/122

21 rr (d) 4/14

24

1 rr

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11. Two spherical soap bubbles of radii r1 and r

2 in

vacuum coalesce under isothermal conditions. Theresulting bubble has a radius equal to

(a)2

rr 21 (b)

21

21

rr

rr

(c) 21rr (d) 22

21 rr

12. A long cylindrical glass vessel has a small hole ofradius r at its bottom. The depth to which the ves-sel can be lowered vertically in a deep water bath(surface tenstion T, density d) without any waterentering inside is

(a)rdg

T(b)

rdg

T2

(c)rdg

T3(d)

rdg

T4

13. If a number of little droplets of a liquid of density, surface tenstion T and specific heat c, each ofradius r, coalesce to form a single drop of radius R,the rise in temperature will be

(a)

R

1

r

1

c

T3(b)

R

1

r

1

c

T3

(c)

R

1

r

1

c2

T3(d)

R

1

r

1

c2

T3

14. A copper wire of negligible mass with length 1 mand cross-sectional area 10–6 m2 is kept on a smoothhorizontal table with one end fixed. A ball of mass1 kg is attached to the other end. If the wire and theball are rotating with an angular velocity of20 rad/s then the elongation in the wire is 10–3m.If on increasing the angular velocity to 100 rad/s,the wire breaks down, then the ratio of young’smodulus of the material to the breaking stress ofthe wire is

(a) 20 : 1 (b) 40 : 1

(c) 20 : 3 (d) 40 : 3

15. The depth of a lake at which the density of water is1% greater than at the surface, if thecompressibility of water is 50 × 10–6/atm

(a) 1 km (b) 1.5 km

(c) 2 km (d) 2.5 km

16. A uniform pressure p is exerted on all sides of asolid cube at temperature t0C. The bulk modulusand coefficient of volume expansion of thematerial are b and respectively. Let thetemperature of the cube be raised t in order tobring its volume back to the volume it had beforethe pressure was applied, then t equals to

(a)b

p

(b)

b

p2

(c)b2

p

(d) bp

17. The density of air in atmosphere decreases withheight h and can be expressed by the relation :

= 0e–Ah

where 0 = 1.3 kg/m3 and A = 1.2 × 10–4/m.

If g = 9.8 m/s2 then the atmospheric pressure atsea-level is

(a) 1.06 × 104N/m2 (b) 2.06 × 105N/m2

(c) 3.06 × 105N/m2 (d) 1.06 × 105N/m2

18. A piece of copper of density 8.8 gm/cc having aninternal cavity weighs 264 gm in air and 221 gm inwater. The volume of the cavity is

(a) 11 cc (b) 12 cc

(c) 13 cc (d) 14 cc

19. A piece of brass (alloy of copper and zinc) weighs12.9 gm in air. When completely immersed inwater it weighs 11.3 gm. If the specific gravities ofcopper and zinc are 8.9 and 7.1 respectively thenthe mass of thecopper contained in the alloy is

(a) 7.61 gm (b) 7.25 gm

(c) 6.78 gm (d) 6.25 gm

20. A piece of metal floats on mercury. The coefficientof volume expansion of the metal and mercury are

1 and

2 respectively. If the temperature of both

mercury and metal are increased by an amount T,then the factor of the fraction of the volume of themetal submerged in mercury changes is

(a) 2(2 –

1) T (b) (

2 –

1) T

(c) 2(2 +

1) T (d) (

2 +

1) T

21. A ring is cut from a platinum tube of 8.5 cminternal and 8.7 cm external diameter. It issupported horizontally from a pan of a balance sothat it comes in contact with the water in a glassvessel. It has been found that an extra 3.97 gmweight is required to pull it away from water, thenthe surface tension of water is

(a) 62.18 dyne/cm (b) 68.75 dyne/cm

(c) 72.13 dyne/cm (d) none

22. The lower end of a capallary tube of radius2.00 mm is dipped 8.00 cm below the surface ofwater in a beaker. If surface tension ofwater = 73 × 10–3 N/m, density of water = 103 kg/m3,1 atmosphere = 1.01 × 105 Pa and g = 9.8 m/s2 thenthe pressure required in the tube to blow a bubbleat its end in water is

(a) 1.01 × 105 Pa (b) 1.02 × 105 Pa

(c) 1.03 × 105 Pa (d) 1.04 × 105 Pa

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23. The limbs of a manometer consist of uniformcapillary tubes of radii 1.4 × 10–3 m and7.2 × 10–4 m. The density of the liquid is 103 kg/m3

and surface tension is 72 × 10–3 N/m. It has beenfound that the level of the liquid in narrower tubestands 0.2 m above that in the broader tube, thenthe correct pressure difference is

(a) 1863 Pa (b) 1960 Pa

(c) 1720 Pa (d) 2793 Pa

24. Two separate air bubbles (radii 0.002 m and 0.004m) formed of the same liquid (surface tension 0.07N/m) come together to form a double bubble. Theradius of curvature of the internal film surface

common to both the bubbles is

(a) .002 m (b) .003 m

(c) .004 m (d) .005 m

25. A body of mass 3.14 kg is suspended from one endof a wire of length 10.0 m. The radius of the wire ischanging uniformly from 9.8 × 10–4 m at one end to5.0 × 10–4 m at the other end. The change in lengthof the wire if young’s modulus of the material ofthe wire is 2 × 1011 N/m2

(a) 1 mm (b) 2 mm

(c) 3 mm (d) 4 mm

26. A thin uniform metallic rod of length 0.5 m andradius 0.1 m rotates with an angular velocity 400rad/s in a horizontal plane about a vertical axispassing through one of its ends. The density ofmaterial of the rod is 104 kg/m3 and the Young’smudulus is 2 × 1011 N/m2.The elongation of the rodis

(a) 1 mm (b) 1/2 mm

(c) 1/3 mm (d) 1/4 mm

27. A solid sphere of radius R made of a material ofbulk modulus B is surrounded by a liquid in acylindrical container. A massless piston of area Afloats on the surface of the liquid. The fractionalchange in the radius of the sphere (dR/R) when amass M is placed on the piston to compress theliquid is

(a)AB3

mg(b)

AB2

mg

(c)AB

mg3(d)

AB2

mg3

28.

A cubical block of edge L and density d is floatingin equilibrium in a container of base area 4L2. Asmall hole is made at the lower most right end. Thedensity of the liquid is 2d and the density of thematerial of the block is d. The velocity of efflux att = 0 is

(a)2

)LH8(g (b)

4

)LH8(g

(c)6

)LH8(g (d)

8

)LH8(g

29. Consider an ice cube of edge L kept in a gravityfree hall. Assume that the density of water anddensity of ice is same, the surface area of the waterwhen the ice melts is

(a) (4)1/332/3L2 (b) (4)2/331/3L2

(c) (4)2/332/3L2 (d) (4)1/331/3L2

30. It is found that the movable wire is in equilibriumwhen the upward force 3.45 mN is applied. The wirehas a length of 4.85 cm and linear mass density1.75 × 10–3 kg/m.

The surface tension of the liquid is

(a) 0.027 N/m (b) 0.037 N/m

(c) 0.054 N/m (d) 0.0135 N/m

31. A container of width 2a is filled with a liquid. Athin wire of weight per unit length is gently placedover the liquid surface in the middle of the surfaceas shown in the figure. As a result, the liquidsurface is depressed by a distance y (y << a). Thesurface tension of the liquid is

(a)y2

a(b)

y

a

(c)y

a2(d)

y4

a

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32. Consider a horizontally oriented syringecontaining water located at a height of H above theground. The radius of the plunger is R and thediameter of the nozzle is r. The plunger is pushedwith a constant speed v. The horizontal range ofwater steam on the ground is

(a)

v

r

R

g

H4

2

2

(b)

v

R

r

g

H4

2

2

(c)

v

r

R

g

H24

2

2

(d)

v

R

r

g

H24

2

2

33. A rectangular metal plate has dimensions of10 cm × 20 cm. A thin film of oil separates the platefrom a fixed horizontal surface. The separationbetween the rectangular plate and the horizontalsurface is 0.2 mm. An ideal string is attached to theplate and passes over an ideal pulley to a mass m.When m = 125 gm, the metal plate moves atconstant speed of 5 cm/s across the horizontalsurface. Then the coefficient of viscosity of oil indyne-s/cm2 is (Use g = 1000 cm/s2)

(a) 5 (b) 25

(c) 2.5 (d) 50

34. One thousands water drops of radius of 1mm aremerged to form a bigger drop. The density, surfacetension and specific heat capacity of water is 1g/cc,0.075 N/m and 1 cal/gm0C. Assume that there is noloss of energy which are released then change intemperature of water is

(a) 0.010C (b) 0.0010C

(c) 0.020C (d) none

35. Castor oil, which has a density of 0.96 × 103 kg/m3

at room temperature, is forced through a pipe ofcircular cross section by a pump that maintains agauge pressure of 950 Pa. The pipe has a diameterof 2.6 cm and a length of 65 cm. The castor oilemerging from the free end of the pipe atatmospheric pressure is collected. After 90 s, a

total of 1.23 kg has been collected. The coefficientof viscosity of the castor oil at this temperature is

(a) 1.15 SI unit (b) 2.15 SI unit

(c) 0.15 SI unit (d) 0.25 SI unit

36. A sniper fires a rifle bullet into a gasoline tank,making a hole. The tank was sealed and is under3.10-atm absolute pressure, as shown in the figure.

The stored gasoline has a density of 660 kg/m3. Therange of the liquid comes out immediately aftermaking the hole is

(a) 41 m (b) 82 m

(c) 123 m (d) 144 m

37. Consider a tank of cross-sectional area 1sq.m andfilled with a liquid of density 660 kg/m3. The liquid

is covered by a piston of mass kg103

1.3 4 and a

force of N3

102.6 5 is applied as shown in figure.

A hole of very small area is made. The range of theliquid comes out immediately after making the holeis

(a) 41 m (b) 82 m

(c) 123 m (d) none

38. In Searle’s experiment, which is used to findYoung’s modulus of elasticity, the diameter ofexperimental wire is D = 0.05 cm (measured by ascale of least count 0.001 cm) and length isL = 110 cm (measured by a scale of least count0.1 cm). A weight of 50 N causes an extension ofX = 0.125 cm (measured by a micrometer of leastcount 0.001 cm). Screw gauge and meter scale arefree from error. The maximum possible error in thevalues of Young’s modulus is

(a) 1.09 × 1010 N/m2

(b) 2.09 × 1010 N/m2

(c) 3.09 × 1010 N/m2

(d) 4.09 × 1010 N/m2

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39. A container is filled with a liquid and hole of verysmall area is made at the lower most point. If timetaken to leak out the water for the first half heightis T

1 and time taken to leak out the water for the

next half height is T2 then

2

1

T

T is

(a) 1 (b)2

1

(c) 2 (d) none

40.

Figure shows how the stream of water emergingfrom a faucet “necks down” as it falls. Theindicated cross-sectional areas are A

0 = 1.2 cm2 and

A = 0.35 cm2. The two levels are separated by avertical distance h = 45 mm. The volume flow ratefrom the tap is

(a) 24 cm3/s (b) 29 cm3/s

(c) 34 cm3/s (d) 39 cm3/s

EXCERCISE BASED ON NEW PATTERN

COMPREHENSION TYPE

Comprehension-1

A soap bubble in air has a radius of 3.20 cm. It isthen blown up to a radius of 5.80 cm.The surface tension of the bubble film is26.0 mN/m

1. The pressure diference across the film at the largersize is

(a) 1.79 Pa (b) 0.895 Pa

(c) 3.25 Pa (d) 1.625 Pa

2. The work was done on the atmosphere in blowingup the bubble is

(a) 34.35 J (b) 68.7 J

(c) 108.5 J (d) 125.6 J

3. The work was done in stretching the bubblesurface is

(a) 465 µJ (b) 565 µ J

(c) 656 µJ (d) 765 µJ

Comprehension-2

Two rods of different metals, having the same areaof cross-section A, are placed end to end betweentwo massive walls as shown in figure.

ANSWERS

(SINGLE CORRECT CHOICE TYPE)

1. d

2. b

3. b

4. a

5. b

6. c

7. b

8. b

9. c

10. d

11. d

12. b

13. b

14. b

15. c

16. a

17. d

18. c

19. a

20. b

21. c

22. b

23. a

24. c

25. a

26. c

27. a

28. b

29. a

30. a

31. a

32. c

33. c

34. d

35. a

36. b

37. d

38. a

39. d

40. c

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The first rod has a length L1, coefficient of linear

expansion 1 and Young’s modulus Y

1. The

corresponding quantities for second rod are L2,

2 and Y

2. The temperature of both the rods is now

raised by T degrees. Assume that there is no changein the cross-sectional area of the rods and the rodsdo not bend. There is no deformation of walls.

4. Due to thermal expansion the increase in the lengthof the composite rod is

(a) (L1

1 + L

2

2)T

(b) (L1

1 – L

2

2)T

(c) (L1

1 + L

2

2)T/2

(d) (L1

1 – L

2

2)T/2

5. The force with which the rods act on each other atthe higher temperature is

(a)

2

2

1

1

2211

Y

L

Y

L

T)LL(A

(b)

2

2

1

1

2211

Y

L

Y

L

T)LL(A

(c)

2

2

1

1

2211

Y

L

Y

L2

T)LL(A

(d)

2

2

1

1

2211

Y

L

Y

L2

T)LL(A

6. Let 1 >

2 and Y

1 < Y

2. If the rods have equal

initial length and the lengths of the rods at the

higher temperature is 21 LandL respectively then

(a) 1L must be greater than

2L

(b) 2L must be greater than

1L

(c) 1L must be equal to

2L

(d) can’t be said anything

Comprehension-3

A cubic object of dimensions L = 0.608 m on a sideand weight W = 4450 N in a vacuum is suspendedby a wire in an open tank of liquid of density = 944 kg/m3, as in figure.

7. The totaldownward force exerted by the liquid andthe atmosphere on the top of the object is

(a) 38.4 kN (b) 40.5 kN

(c) 2.3 kN (d) 2.1 kN

8. The total upward force on the bottom of the object.

(a) 38.4 kN (b) 40.5 kN

(c) 2.3 kN (d) 2.1 kN

9. The tension in the wire.

(a) 38.4 kN (b) 40.5 kN

(c) 2.3 kN (d) 2.1 kN

Comprehension-4

Water stands at a depth H behind the vertical faceof a dam and exerts a certain resultant horizontalforce on the dam tending to slide it along its foun-dation and a certain torque tending to overturn thedam about the lower most point O. If the total widthof the dam is L.

10. The total horizontal force is

(a)2gLH

2

1 (b)

2gLH

(c)2gLH

3

1 (d)

2gLH4

1

11. The total torque about O is

(a)3gLH

2

1 (b)

3gLH4

1

(c)3gLH

6

1 (d)

3gLH8

1

12. Moment arm of the resultant horizontal force aboutthe line through O is

(a) H/2 (b) H/3

(c) H/4 (d) H/5

Comprehension-5

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A container of large uniform cross-sectional area Aresting on a horizontal surface, holds twoimmiscible, non-viscous and incompressible liquidsof densities d and 2d each of height (H/2) as shownin figure. The lower density liquid is open to theatmosphere having pressure P

0. A homogeneous

solid cylinder of length L (L < H/2), cross-sectionalarea (A/5) is immersed such that it floats with itsaxis vertical at the liquid-liquid interface withlength (L/4) in the denser liquid.

13. The density of solid is

(a) d4

5(b) d

2

3

(c) d3

5(d) d

3

4

14. The total pressure at the bottom of the container is

(a) dg)LH6(2

1P0

(b) dg)LH6(4

1P0

(c) dg)LH6(6

1P0

(d) dg)LH6(8

1P0

15. The cylinder is slightly depressed verticallydownward and released then

(a) performs oscillatory motion but not SHM

(b) performs SHM

(c) continuously moves downward

(d) none of these

16. The minimum time after which the cylinder willreach it’s original position is

(a)g4

L52

(b)g4

L5

2

(c) never reach it’s original position

(d) none

17. The cylinder is depressed in such a way that its topsurface just below the upper surface of liquid withdensity 2d and is then released. Immediately afterthe release its acceleration is

(a) g5

8 upward (b) g

5

6 upward

(c) g5

3 upward (d) g

5

2 upward

The cylinder is removed and original arrangementis restored. A tiny hole of area s(s << A) is punchedon the vertical side of the container at a height h(h < H/2). This height ‘h’ is such that the horizontaldistance ‘x’ travelled by the liquid initially ismaximum.

18. The initial speed of efflux of the liquid at the hole is

(a)4

gH3(b)

2

gH3

(c) gH3 (d) gH2

19. The maximum value of the distance ‘x’ is

(a)2

H(b) H

4

3

(c) H (d) 2H

Comprehension-6

A large open top container of negligible mass anduniform cross-sectional area A has a small hole ofcross-sectional area A/100 in its side wall near thebottom. The container is kept on a horizontal floorand contains a liquid of density and mass m

0.

Assuming that the liquid starts flowing outhorizontally through the hole at t = 0.

If the surface is frictionless then :

20. The acceleration of the container at t = 0 is

(a)20

g(b)

30

g

(c)40

g(d)

50

g

21. If the height of the liquid above the hole at any timeis h then acceleration depends on h according to

(a) independent of height h

(b) directly proportional to h

(c) directly proportional to h

(d) inversely proportional to h

22. The maximum velocity of the container

(a)A2

gm0(b)

A4

gm0

(c)A3

gm0(d) none

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23. The normal reaction acted by the horizontalsurface on the container will

(a) pass through the center of gravity ofliquid

(b) pass through the left of the center ofgravity

(c) pass through the right of the center ofgravity

(d) none of these

24. If two identical small holes on the opposite side ofthe tank is made at the same height then theacceleration of the container is

(a) zero (b)30

g

(c)40

g(d)

50

g

25. If two identical small holes on the opposite side ofthe tank is made at the different height butseparated by h then the acceleration of the containeris proportional to

(a) h (b) h

(c) h3/2 (d) h2

26. If the surface having some friction then minimumcoefficient of friction such that the container shouldnot move

(a)50

1(b)

75

1

(c)100

1(d)

125

1

Comprehension-7

Consider a large vertical container of cross-sectionalarea A which is filled with a liquid of density =

0e–h where

0 and are constant and h is the

height measured from the bottom.

27. The dimensional formula of

0 is

(a) [ML–3] (b) [ML–4]

(c) [ML–2] (d) dimensionless

28. The total mass of the liquid in the container is

(a)

A0(b)

2

A0

(c)

A2 0(d) infinite

29. If the pressure at the bottom is P0 then the pressure

with height h measured from the bottom will change

(a) linear (b) constant

(c) exponential (d) parabolic

30. A cubical block of length L is floating inequilibrium immersed completely inside the liquid.The bottom of the block is at the height h

0 above

the bottom of the container. If acceleration due togravity g is uniform then the mass of the block is

(a) )Lh(h02

00 eeL

(b) )Lh(h02

00 ee2

L

(c) )Lh(h02

00 ee2

L

(d) )Lh(h02

00 eeL

Comprehension-8

Suppose a spherical body of radius r, density isreleased from rest in a liquid column of largevertical height of viscosity . The density of theliquid is . Assume that acceleration due to gravityg does not change with height.

31. The initial acceleration of the body is

(a) zero (b) g

(c) < g (d) > g

32. The maximum power due to the net force is

(a)

225 g)(r

(b)

225 g)(r2

(c)

2

g)(r 225

(d) none

33. The velocity acheived by the body when the powerof net force will become zero

(a)

9

g)(r2 2

(b)

9

g)(r2

(c)

11

g)(r2 2

(d)

11

g)(r2

34. The total work done by the various forces is

(a) 2

227

243

g)(r4

(b) 2

227

243

g)(r6

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(c) 2

227

243

g)(r8

(d) 2

227

243

g)(r10

35. Due to viscous force, heat is produced. Let therate of production of heat is directly proportionalto rn when the ball acheived constant speed. Thevalue of n is

(a) 4 (b) 5

(c) 6 (d) 7

36. If amount of heat produced is all absorbs by theliquid only then the change in temperature of theliquid depends on time t as tn, once the ball willacheive terminal speed. The value of n is

(a) zero (b) 1

(c) ½ (d) none

37. The total amount of heat produced due to viscousforce until the ball acheived terminal speed is

(a) 2

227

243

g)(r4

(b) 2

227

243

g)(r6

(c) 2

227

243

g)(r8

(d) none

Comprehension-9

In the above container with the dimension as shownin figure is filled with a liquid of density .

38. The force exerted by the liquid on the bottom is

(a) equal to weight of the liquid

(b) greater than the weight of the liquid

(c) less then the weight of the liquid

(d) none

39. The resultant force exerted by the side walls of thecontainer on the liquid is, if atmospheric pressureis p

0

(a) ghr22 + P

0(r

22 – r

12) –

3

h(r

22 – r

12)g

(b) ghr22 + P

0(r

22 + r

12) –

3

h(r

22 – r

12)g

(c) ghr22 + P

0(r

22 – r

12) +

3

h(r

22 – r

12)g

(d) none

40. A hole of very small area is made at the height h/2from the bottom. The horizontal velocity of effluxat t = 0 is

(a) gh

(b)22

12 h)rr(

hgh

(c)22

12

12

h)rr(

)rr(gh

(d)

12 rr

hgh

41. The time after which the water lands on the groundwhich comes out at t = 0 is

(a)g

h(b) >

g

h

(c) < g

h(d) none

Comprehension-10

Consider a tube of very small radius ‘r’ and lengthL which is completely filled with water of surfacetension T and density .

42. The pressure at the bottom of the tube is

(a) P0 + gL (b) P

0 + gL +

r

T2

(c) P0 + gL –

r

T2(d) none

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43. Consider a air bubble at the depth ‘h’ in the tube.The radius of the bubble is r

1. The pressure inside

the bubble is

(a)1

0r

T2gh

r

T2P

(b)1

0r

T2gh

r

T2P

(c)1

0r

T2ghP

(d)1

0r

T2ghP

44. Let the bubble starts rising and the temperature ofthe liquid remains constant. When it just reach thetop most point then the pressure inside the bubbleis

(a)1

0r

T2

r

T2P

(b) > 1

0r

T2

r

T2P

(c) < 1

0r

T2

r

T2P

(d) none

45. If this bubble will burst out then the temperatureof the liquid

(a) remains constant

(b) increases

(c) decreases

(d) can’t be decided

46. Consider the tube without any bubble. Now a verysmall hole is made at the bottom most point of thetube the velocity a efflux of the liquid is

(a)

2

r

T2gL

(b)

2

r

T2gL

(c) gL2 (d) none

Comprehension-11

Consider a fixed container of radius R as shown infigure. The container is half filled with a liquid ofdensity . The atmospheric pressure is P

0

47. The pressure at the bottom most point is

(a) P0 + gR (b) P

0

(c) gR (d) none

48. The resultant force exerted by the liquid on thecontainer is

(a) gR3

2RP4 32

0

(b) gR3

2 3

(c) 20 RP4

(d) none

49. If a hole is made at the height 2

Rabove the

bottom most point then the liquid will land on theground at the distance from the bottom most pointof the container is

(a) R2

3

2

)15(3

(b) R2

3

4

)15(3

(c) R2

3

4

)15(3

(d) none

50. If the hole of very small area ‘a’ is made at thebottom most point such that the liquid will comeout from the hole then the time after which thecontainer will become empty

(a)g2a14

R15 2/5(b)

g2a15

R14 2/5

(c)g2a

R 2/5(d) none

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Comprehension-12

Consider a fixed container of radius R as shown infigure. The container is half filled with a liquid ofdensity . The atmospheric pressure is P

0

A plate is tightly placed on the mouth of thecontainer and the air is completely pumped out.

51. The pressure at the bottom most point

(a) P0 + gR (b) P

0

(c) gR (d) none

52. The force exerted by the liquid on the container is

(a) gR3

2RP4 32

0

(b) gR3

2 3

(c) 20 RP4

(d) none

53. If a very small hole is made at the height 2

Rabove

the bottom most point then the speed with whichliquid will come out is

(a)

0P2

gR (b)

0P2

gR

(c) gR2 (d) gR

54. If the hole of very small area ‘a’ is made at thebottom most point such that the liquid will comeout from the hole then the time after which thecontainer will become empty is

(a) < g2a15

R14 2/5(b)

g2a15

R14 2/5

(c) depends on P0

(d) none

Comprehension-13

Consider a container with the dimension of the base(a × b) and liquid is filled upto the height H. Theliquid has the density . The whole system is placedon the moon and assume that horizonal surface isfrictionless. Mass of the container is negligible. Theacceleration due to gravity on the surface of theearth is g.

55. If this system is on the earth then the total forceexerted on the base is F

1 and in the above case the

force exerted on the base is F2. Then

2

1

F

Fdepends

on

(a) density of the liquid

(b) height of the liquid

(c) atmospheric pressure on the earth

(d) all the above

56. If this system is on the earth then the force exertedby the liquid on the base is F

1 and in the above case

the force exerted by the liquid is F2. Then

2

1

F

F

depends on

(a) density of the liquid

(b) height of the liquid

(c) atmospheric pressure on the earth

(d) none of these

57. The pressure energy per unit volume at the bottomof the container is

(a)6

gH(b) gH

(c)3

gH(d) none

A hole of very small area ‘s’ is made at the bottommost point of the container at the right end.

58. The speed of efflux is

(a) zero (b)6

gH

(c)3

gH(d)

2

gH

59. The acceleration of the container at any time t is

(a)ab3

sg(b)

ab

sg

(c)ab

sg2(d)

ab3

sg2

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60. The time after which the container will becomeempty depends on

(a) area of the hole

(b) density of the liquid

(c) base area of the container

(d) all of the above

Comprehension-14

A cylindrical tank 1 m in radius rests on a platform5 m high. Initially the tank is filled with water up toa height of 5 m. A plug whose area is 10–4 m2 isremoved from an orifice on the side of the tank atthe bottom. Consider the liquid which comes out att = 0.

61. The initial speed with which the water flows fromthe orifice is

(a) 10 m/s (b) 7.5 m/s

(c) 5 m/s (d) 2.5 m/s

62. The horizontal speed of the water that comes out att = 0, at any time during the fall is

(a) 10 m/s (b) 7.5 m/s

(c) 5 m/s (d) 2.5 m/s

63. The speed of the water (that comes at t = 0) strikesthe ground is

(a) 10 m/s (b) 12 m/s

(c) 14 m/s (d) 16 m/s

64. The kinetic energy per unit volume of the waterwhich comes out at t = 0 when it strikes the groundis

(a) 50 kJ/m3 (b) 72 kJ/m3

(c) 98 kJ/m3 (d) 128 kJ/m3

65. Time time taken to empty the tank to half itsoriginal value is

(a) 2.5 h (b) 1.5 h

(c) 1.25 h (d) none

66. Let the time t after which the water completelycomes out from the tank is directly proportional tohn (where h is the height of the container above theground) then the value of n is

(a) zero (b) –½

(c) ½ (d) 1

Comprehension-15

Using the above apparatus, a child can blow a soapbubble of radius ‘r’. The surface tension of the soapsolution is T. Air of density moves with thevelocity v through the tube of radius r

1( << r) and

comes to rest inside the bubble. The circular wirehas the radius R. Assume that the air is fallingnormal to the bubble surface.

67. The force exerted by the circular wire on the thinfilm of soap solution when the child is not blowingis

(a) zero (b) 2RT

(c) 4RT (d) none

68. The energy expended to form the bubble is

(a) 4Tr2 (b) TR2

(c) 4Tr2 – TR2 (d) none

69. The radius ‘r’ of the bubble in terms of T, and vat the time when it will blown out is

(a) 2v

T

(b) 2v2

T

(c)2v4

T

(d)

2v

T4

Comprehension-16

Viscosity of highly viscous liquid can be determinedusing the following appratus.

The apparatus consists of a test tube contains theexperimental liquid (density 1260 kg/m3) and isfitted into a water bath. A tube is fitted in the corkof the test tube through which different metal ballcan be dropped. There are equidistant points P,Qand R marked on the test tube which are separatedby 5 cm. The time taken by the ball to travel thisdistance is measured by a stop watch of least count0.1 s. The radius of the ball is measured by the screwgauge which has the least count of 0.01 cm.

70. The measurement is based on the priciple of

(a) Stoke’s law

(b) Poiseuille’s formula

(c) Searls method

(d) none

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71. If the radius of the sphere is 1cm and its mass is50mg and time to travel the distance of each 5cm is5 second, then the coefficient of viscosity of theexperimental liquid is

(a) 2 poise (b) 4 poise

(c) 8 poise (d) none

72. The maximum possible percentage error in themeasurement of coefficient of viscosity is

(a) 1% (b) 1.5%

(c) 2% (d) none

Comprehension-17

Suppose the beaker is accelerated and it hascomponents of acceleration a

x and a

y in x and y

directions respectively, then the pressure decreasesalong both x and y directions. The equation forpressure gradient is given by

xadx

dP and )ag(

dy

dPy

73. A rectangular container of water undergoesconstant acceleration down an incline as shown.

The slope tan of the free surface using thecoordinate system shown is

(a) 0.23 (b) 0.27

(c) 0.35 (d) 0.39

74. A liquid is kept in a cylindrical vessel which isrotated along its axis. The liquid rises at the sides.If the radius of the vessel is 0.05 m and the speed ofrotation is 2 rev/s, the difference in the height ofthe liquid at the centre of the vessel and its sides is

(a) 1 cm (b) 2 cm

(c) 2.5 cm (d) 3 cm

75. Length of horizontal arm of a U-tube is 20 cm andends of both the vertical arms are open to apressure 1.01 × 103 N/m2. Water is poured into thetube as shown in figure and one of the open ends issealed and the tube is then rotated about a verticalaxis passing through the other vertical arm withangular velocity . Take density ofwater = 103 kg/m3 and g = 10 m/s2. Assumetemperature to be constant.

If length of water in sealed arm is 5 cm then isequal to

(a) 6.15 rad/s (b) 7.15 rad/s

(c) 8.15 rad/s (d) 9.15 rad/s

Comprehension-18

A pitot tube is used to determine the airspeed of anairplane. It consists of an outer tube with anumber of small holes B that allow air into the tube;that tube is connected to one arm of a U-tube. Theother arm of the U-tube is connected to hole A atthe front end of the device, which points in thedirection the plane is headed. At A the air becomestagnant so that v

A = 0. At B, however, the speed of

the air presumably equals the airspeed v of theaircraft. Tube contains alcohol of density810 kg/m3 and the value of h is 26 cm. The densityof air is 1.03 kg/m3. Assume that the accelerationdue to gravity does not change with height. Here his the difference in the fluid level in the tube.

76. Pitot tube works on the principle of

(a) principle of continuity

(b) Bernoulli’s theorem

(c) both (a) and (b)

(d) none of these

77. The speed of plane relative to the air is

(a) 53 m/s (b) 63 m/s

(c) 73 m/s (d) 83 m/s

78. A pitot tube on a high altitude aircraft measures adifferential pressure of 180 Pa. If the density of airis 0.031 kg/m3 then the speed of aircraft is

(a) 132 m/s (b) 142 m/s

(c) 152 m/s (d) 162 m/s

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79. Consider two air crafts moving with the same speedbut a different altitutes. Same types of pitot tube isused to measure the speed of aircraft, then

(a) the value of h in the both pitot tube issame

(b) the value of h in the pitot tube at higherheight is greater

(c) the value of h in the pitot tube at higherheight is less

(d) none

Comprehension-19

A venturi meter is used to measure the flow speedof a fluid. The meter is connected between twosections of the pipe, the cross-sectional area A ofthe entrance and exit of the meter matches the pipe’scross-sectional area. Between the entrance and exit,the fluid flows from the pipe with speed V and thenthrough a narrow “throat” of cross-sectional area‘a’ with speed v. A manometer connects the widerportion of the meter to the narrower portion. Thechange in the fluid’s speed is accompanied by achange p in the fluid’s pressure, which causes aheight difference h of the liquid in the two arms ofthe manometer.

80. Venturi meter works on the principle of

(a) principle of continuity

(b) Bernoulli’s theorem

(c) both (a) and (b)

(d) none of these

81. Suppose that the fluid is fresh water, that thecross-sectional area are 64 cm2 in the pipe and32 cm2 in the throat, and that the pressure is55 kPa in the pipe and 41 kPa in the throat. Therate of water flow in cubic meters per second is

(a) 2.0 × 10–2 (b) 3.0 × 10–2

(c) 4.0 × 10–2 (d) 5.0 × 10–2

82. The value of V in terms of a, A, h, density of fluid

and density of liquid in manometer is

(a))aA(

gha222

2

(b)

)aA(

gha22

2

(c))aA(2

gha22

2

(d) none

MATRIX-MATCH TYPE

Matching-1

Column - A Column - B

(A) Action of paint-gun (p) Bernoulli’stheorem

(B) Velocity of efflux (q) Torricelli’stheorem

(C) Brahma’s press (r) Pascal law

(D) Venturi meter (s) Continuityprinciple

Matching-2

Two identical cylindrical tanks are filled withdifferent liquids of densities

1 and

2 . A small hole

is made in the side of each tank at the same depth hbelow the surface of liquid. The hole in the tankhas area of cross-section twice that of hole in tankB.


(A) The ratio 1/

2 if (p) 1/2

rate of mass flow is same

(B) The ratio of volume flow (q) 2if rate of mass flow is same

(C) The ratio of their speed (r) 1of efflux

(D) The ratio of their (s) 4horizontal distance

Matching-3


(A) Dimension of modulus (p) [M0L0T0]of elasticity

(B) Dimension of (q) [ML–1T–2]coefficient ofviscosity

(C) Dimension of (r) [ML0T–2]surface energy

(D) Dimension of (s) [ML–1T–1]Reynold’s number

Matching-4


(A) Variation of velocity (p) exponentialof falling rain drop increasingon time

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(B) Variation of (q) exponentialacceleration decreasingof falling rain dropon time

(C) Variation of (r) linearaccelerationof falling rain dropon velocity

(D) Variation of power (s) parabolicdue to net force onfalling rain dropwith speed

MULTIPLE CORRECT CHOICE TYPE

1. A bar of cross-section A is subjected to equal andopposite tensile forces F at its ends. Consider a planethrough the bar making an angle with a plane atright angles to the bar.

(a) The tensile stress at this plane ismaximum for equal to zero.

(b) The shearing stress at the plane ismaximum for equals to 450.

(c) The tensile stress at this plane ismaximum for equal to 450.

(d) The shearing stress at the plane ismaximum for equals to zero.

2. A cubical block of iron 5 cm on each side is floatingon mercury in a vessel. The relative density ofmercury is 13.6 and relative density of iron is 7.2.Let the height of the block above mercury level ish

1. Now water is poured into the vessel so that it

just covers the iron block. The height of watercolumn is h

2. Then

(a) h1 = 2.35 cm (b) h

2 = 2.54 cm

(c) h1 = 2.54 cm (d) h

2 = 2.35 cm

3. A beaker containing water is kept on a springbalance B

1. The weight of beaker and water is 5 kg.

A piece of iron (specific gravity 7.5) weighing1.5 kg is hung from a spring balance B

2. If the iron

piece is lowered in water till it is fully immersedbut does not touch the bottom of the beaker, thereadings of B

1 and B

2 are x

1 and x

2 respectively.

(a) x1 = 5.2 kg (b) x

2 = 1.5 kg

(c) x2 = 1.3 kg (d) x

1 = 5.0 kg

4. Figure shows a siphon in action. The liquid flowingthrough the siphon has a density of 1.5 gm/cc. Then

(a) The pressure difference at the pointsA and D is zero

(b) The pressure difference at the pointsB and C is 2.65 × 104N/m2.

(c) The liquid will flow from upper containerto lower container.

(d) none of these

5. A large open top container contains a liquid uptoheight H. A small hole is made in the side of thetank at the height y from the bottom. The liquidemmerges from the hole and lands at a distance xfrom the tank.

(a) If y is increased then x first increases and

then decreases.

(b) x will be maximum when the hole is madeat the middle height.

(c) The maximum possible value of x is H

(d) x doesnot depend upon the density of theliquid.

6. A spring balance reads W1 when a ball is suspended

from it. A weighing machine reads W2 when a tank

of liquid is kept on it. When the ball is immersed inthe liquid, the spring balance reads W

3 and the

weighing machine reads W4. Then

(a) W1 > W

3

(b) W2 < W

4

(c) W1 + W

2 = W

3 + W

4

(d) none

7. A wire forming a loop is dipped into soap solution(surface tension T) and taken out such that a filmof soap solution is formed. A loop of length L of amassless thread is gently put on the film and thefilm is pricked with a needle inside the loop, then

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(a) The threaded loop takes the shape of acircle

(b) The threaded loop remains in the sameshape

(c) The tension in the thread will become

2

TL

(d) There is no tension developed in thethread

Assertion-Reason Type

Each question contains STATEMENT-1 (Assertion)and STATEMENT-2 (Reason). Each question has4 choices (A), (B), (C) and (D) out of which ONLYONE is correct.

(A) Statement-1 is True, Statement-2 is True;Statement-2 is a correct explanationfor Statement-1

(B) Statement-1 is True, Statement-2 is True;Statement-2 is NOT a correctexplanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True

1. STATEMENT-1 : A wire can support a load Wwithout breaking. It is cut into two equal parts. Themaximum load that each part can support is W/2.

STATEMENT-2 : The young’s modulus ofelasticity of the wire does not depend on the lengthof the wire.

2. STATEMENT-1 : The modulus of rigidity (shearmodulus of elasticity) of liquid and gas is zero.

STATEMENT-2 : The bulk modulus of elasticityfor all types of substance is non-zero.

3. STATEMENT-1 : A piece of ice has a stone in it andfloats in a vessel containing water. When the icemelts, the level of water in the vessel would fall.

STATEMENT-2 : The Buoyancy force will arise dueto the vertical pressure gradient.

4. STATEMENT-1 : Hydrolic pump and hydrolicbrake is based on Pascals law.

STATEMENT-2 : Pressure applied to an enclosedfluid is transmitted undiminished to every portionof the fluid and the wall of the containing vessel.

5. STATEMENT-1 : A body floats in a liquid containedin a beaker. The whole system falls freely undergravity. The upthrust on the body due to the liquidis zero.

STATEMENT-2 : In case of free fall, the effectiveacceleration due to gravity is zero.

6. STATEMENT-1 : In order that a floating object bein stable equilibrium, its centre of buoyancy shouldbe vertically above its centre of gravity

STATEMENT-2 : Center of buoyancy will coincidewith the centre of gravity of the displaced liquid.

7. STATEMENT-1 : If enough iron is added to oneend of a uniform wooden stick then it can floatvertically.

STATEMENT-2 : For rotational equilibrium offloating body the meta centre must always be higherthan the center of gravity of body.

8. STATEMENT-1 : The rate of leak from a hole in atank is more if situated near the bottom.

STATEMENT-2 : Bernoulli’s principle is based onconservation of energy.

9. STATEMENT-1 : When a spinning ball is thrown,it deviates from its usual path in flight.

STATEMENT-2 : In accordance with Bernoulli’sprinciple, a pressure difference above and belowthe ball will develop.

10. STATEMENT-1 : Bernoulli’s equation is applicablein the case of stremlined flow of incompressible.

STATEMENT-2 : Principle of continuity is basedon conservation of mass.

11. STATEMENT-1 : Two large tanks a and b, open atthe top, contains different liquids. A small hole ismade in the side of each tank at the same depth hbelow the liquid surface, but the hole in a has twicethe area of the hole in b. The ratio of the densitiesof the liquids in a and b so that the mass flux is thesame for each hole should be 0.5.

STATEMENT-2 : The initial speed of efflux will besame in both cases.

12. STATEMENT-1 : If temperature rises, thecoefficient of viscosity of a liquid decreases whilefor gases increases.

STATEMENT-2 : The viscosity of liquid (exceptwater) increases with increment of pressure whilefor gases it is independent of pressure.

13. STATEMENT-1 : A metal ball immersed in alcoholweighs W

1 at 00C and W

2 at 500C. The coefficient of

cubical expansion of the metal is less than that ofalcohol, assuming that the density of the metal islarge compared to that of alcohol, then W

1 < W

2.

STATEMENT-2 : Density decreases with theincrease of temperature.

14. STATEMENT-1 : An iceberg is floating partiallyimmersed in sea water. If the density of sea water is1.03 g/cc and that of ice is 0.92 g/cc, the fraction ofthe total volume of iceberg above the level of seawater is 0.11.

STATEMENT-2 : It is due to force of buoyancy.

15. STATEMENT-1 : A loaded boat enters the sea fromthe river, it rises.

STATEMENT-2 : Sea water is denser then the riverwater.

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16. STATEMENT-1 : When a piece of ice floating in abeaker of water completely melts, the level of thewater in the beaker will slightly change.

STATEMENT-2 : The density of ice is slightly lowerthan the density of water.

17. STATEMENT-1 : A ball floats on the surface ofwater in a container exposed to atmosphere. If thecontainer is covered and air is compressed, the ballwill sink.

STATEMENT-2 : The force of buoyancy willdecrease.

18. STATEMENT-1 : A wooden piece floats halfsubmerged in a tub of water. If the system istransferred to a lift ascending with acceleration, thepiece will remain half sub-merged.

STATEMENT-2 : The pressure gradient willchange.

19. STATEMENT-1 : People living in houses far remotefrom a municipal water tank often find it difficultto get water on the top floor even if it is situatedlower than the level of water tank.

STATEMENT-2 : There is loss of pressure whenwater is flowing.

20. STATEMENT-1 : Water is flowing through ahorizontal pipe of uniform cross-section underconstant pressure. At some place the pipe becomesnarrow; the pressure of water at this placedecreases.

STATEMENT-2 : The pressure energy will beconverted into kinetic energy.

21. STATEMENT-1 : A thin steel needle floats onwater but when a little soap solution is carefullymixed with the water the needle sinks.

STATEMENT-2 : When a detergent is added towater its surface tension will suddenly decrease.

22. STATEMENT-1 : Hooke’s law is valid at all stress.

STATEMENT-2 : The slope of stress vs strain graphin the proportional limit equals to modulus ofelasticity.

23. STATEMENT-1 : Stress and pressure are thedifferent concept.

STATEMENT-2 : Both are defined as force per unitarea.

24. STATEMENT-1 : Consider a massless rod which issuspended from a ceiling and a block of mass m isattached to a lower end. The change in the lengthof the wire is x. Another uniform wire of the samematerial, same initial length and samecross-section but has mass m is suspended from theceiling and the block is not attached. The change inlength of this wire is x/2.

STATEMENT-2 : In both wire, work done instretching is the same.

25. STATEMENT-1 : Density of a fluid changes withchange in temperature.

STATEMENT-2 : Density of a liquid doesnot changewith change in pressure.

26. STATEMENT-1 : Stoke’s Law is valid only forspherical bodies.

STATEMENT-2 : Viscous force will be experiencedby all types of bodies when they move through theviscous liquid.

27. STATEMENT-1 : The profile of advancing liquidin a tube is a parabola.

STATEMENT-2 : Rate of volume flow of a liquidthrough the tube doesnot depend on the coefficientof viscosity.

28. STATEMENT-1 : N drops of a liquid join to form asingle drop. In this process some energy will bereleased.

STATEMENT-2 : It is due to change of surfaceenergy.

29. STATEMENT-1 : When a capillary tube is dippedin a liquid, the liquid rises to a height h in the tube.The free liquid surface inside the tube ishemispherical in shape. The tube is now pusheddown so that the height of the tube outside theliquid is less than h. The liquid will flow out of thetube slowly.

STATEMENT-2 : The angle of contact at the freeliquid surface inside the tube will change.

30. STATEMENT-1 : There are different types ofvessel with the same base area and they are filledwith the same liquid of different mass but upto sameheight. The force exerted on the base in each casewill be different.

STATEMENT-2 : The force exerted on the base maybe greater than, less than or equal to the weight ofthe liquid.

31. STATEMENT-1 : A container with a liquid is placedin a gravity free surrounding. A hole is made at thebottom of the container then after certain time thecontainer will become empty.

STATEMENT-2 : Bernoulli’s theorem is applicableonly for non-viscous liquid.

32. STATEMENT-1 : Two stream line curve does notintersect each other.

STATEMENT-2 : In a stream line flow the kineticenergy and momentum of all the particles arrivingat a given point are the same.

33. STATEMENT-1 : Surface tension and coefficientof viscosity is a property exist only for the liquid.

STATEMENT-2 : Surface tension and surfaceenergy are numerically equal.

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34. STATEMENT-1 : The viscous force per unit areabetween two layer of liquid is shear stress ortangential stress.

STATEMENT-2 : The origin of viscous force iselectromagnetic.

35. STATEMENT-1 : An ice cube suspended in vacuumin a gravity free space melts. When it melts, its shapechanges to spherical.

STATEMENT-2 : Due to the property of surfacetension, there is a tendency to acquire minimumsurface area. For the given volume the sphere hasthe minimum surface area.

36. STATEMENT-1 : Rain drops are spherical.

STATEMENT-2 : Due to the property of surfacetension, there is a tendency to acquire minimumsurface area. For the given volume the sphere hasthe minimum surface area.

37. STATEMENT-1 : The capillary rise in a tube is less

than gr

T2

where the symbols have their usual

meanings.

STATEMENT-2 : It is due to weight of the liquidcontained in the meniscous.

38. STATEMENT-1 : During a tornado, when a highspeed wind blows over a roof, it blows off the roof.

STATEMENT-2 : According the Beronulli’sprinciple, a low pressure created at the top of theroof.

(Answers) EXCERCISE BASED ON NEW PATTERN

COMPREHENSION TYPE

1. a 2. b 3. d 4. a 5. a 6. d

7. a 8. b 9. c 10. a 11. c 12. b

13. a 14. b 15. b 16. b 17. c 18. a

19. b 20. d 21. a 22. d 23. c 24. a

25. b 26. a 27. c 28. a 29. c 30. a

31. c 32. d 33. a 34. c 35. b 36. b

37. d 38. b 39. d 40. b 41. b 42. c

43. a 44. c 45. b 46. a 47. a 48. a

49. c 50. b 51. c 52. b 53. a 54. c

55. d 56. d 57. a 58. c 59. a 60. d

61. a 62. a 63. c 64. c 65. a 66. a

67. b 68. c 69. d 70. a 71. d 72. d

73. a 74. b 75. d 76. b 77. b 78. c

79. c 80. c 81. a 82. a

MATRIX-MATCH TYPE

1. A-p; B-p, q, s; C-r; D-p, s 2. A-p; B-q; C-r; D-s 3. A-q; B-s; C-r; D-p

4. A-p; B-q; C-r; D-s

MULTIPLE CORRECT CHOICE TYPE

1. [a, b] 2. [a, b] 3. [a, c] 4. [a, b, c] 5. [a, b, c, d]

6. [a, b, c] 7. [a, c]

ASSERTION-REASON TYPE

1. D 2. B 3. B 4. A 5. A 6. B

7. A 8. A 9. A 10. B 11. A 12. B

13. A 14. A 15. A 16. A 17. A 18. B

19. A 20. A 21. A 22. D 23. C 24. C

25. C 26. B 27. C 28. D 29. D 30. D

31. D 32. B 33. D 34. D 35. A 36. A

37. A 38. A

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INITIAL STEP EXERCISE

(SUBJECTIVE)

1. The fresh water behind a reservoir dam is 15 mdeep. A horizontal pipe 4.0 cm in diameter passesthrough the dam 6.0 m below the water surface asshown in figure A plug secures the pipe opening.

(a) Find the friction force between the plug and pipewall. (b) The plug is removed. What volume ofwater flows out of the pipe in 3.0 hours ?

2. A stone of 0.5 kg mass is attached to one end of a0.8 m long aluminium wire of 0.7 mm diameter andsuspended vertically. The stone is now rotated in ahorizontal plane at a rate such that the wire makesan angle of 850 with the vertical. Find the increasein the length of the wire. [Young’s modulus ofaluminium = 7 × 1010 N/m2; sin 850 = 0.9962 andcos 85 = 0.0872].

3. (a) A fluid is rotating at constant angularvelocity about the central vertical axisof a cylindrical container. Show that thevariation of pressure in the radialdirection is given by

rdr

dp 2

(b) Take p = pc at the axis of rotation (r = 0)

and show that the pressure p at any pointr is p = p

c + ½r2r2

(c) Show that the liquid surface is ofparaboloidal form; that is, a vertical crosssection of the surface is the curvey = 2r2/2g.

4. A non-viscous liquid of constant density1000 kg/m3 flows in a stream line motion along thetube of variable cross-section. The tube is keptinclined in the vertical plane as shown in fig. Thearea of cross-section of the tube at two points P andQ at heights of 2 metres and 5 metres arerespectively 4 × 10–3 m2 and 8 × 10–3m2. The velocityof the liquid at point P is 1 m/s.

Find the work done per unit volume by the

pressure and the gravity forces as the fluid flowsfrom point P to Q.

5. A glass capillary sealed at the upper end is of length0.11 m and internal diameter 2 × 10–5 m. The tubeis immersed vertically into a liquid of surfacetension 5.06 × 10–2 N/m. To what length has thecapillary to be immersed so that the liquid levelinside and outside the capillary becomes thesame ? What will happen to the water level insidethe capillary if the seal is now broken ?

6. A cylindrical vessel of radius R is filled with waterto a height of h. It has a capillary tube of length land radius ‘r’ protruding horizontally at itsbottom. If the viscosity of water is , find the timein which the level will fall to a height of h/2.

7. A liquid is flowing through horizontal pipes asshown in figure. Length of different pipes has thefollowing ratio L

AB : L

CD : L

EF : L

GH = 1 : 1 : 2 : 2

Similarly, radii of different pipes has the ratio,

RAB

: RCD

: REF

: RGH

= 1 : 1 : 1 : 2

Pressure at A is 2P0 and pressure at D is P

0. The

volume flow rate through the pipe AB is Q. Find,

(a) Volume flow rates through EF and GH

(b) Pressure at E and F.

8.

A schematic view of a hydraulic jack used to lift anautomobile. The hydraulic fluid is oil(density = 812 kg/m3). A hand pump is used, inwhich a force of magnitude F

i is applied to the

smaller piston (of diameter 2.2 cm) when the handapplies a magnitude F

h to the end of the pump

handle. The combined mass of the car to be liftedand the lifting platform is M = 1980 kg, and thelarge piston has a diameter of 16.4 cm. The lengthL of the pump handle is 36 cm, and the distance xfrom the pivot to the piston is 9.4 cm. (a) What isthe applied force F

h needed to lift the car ? (b) For

each downward stroke of the pump, in which thehand moves a vertical distance of 28 cm, how far isthe car raised ?

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FINAL STEP EXERCISE

(SUBJECTIVE)

1. A rod of length 6 m has a mass 12 kg. It is hinged atone end at a distance of 3 m below water surface.(a) What weight must be attached to the other endof the rod so that 5 m of the rod is submerged ? (b)Find the magnitude and direction of the forceexerted by the hinge on the rod. (Specific gravity ofrod is 0.5).

2. Under isothermal condition two soap bubbles ofradii a and b coalesce to form a single bubble ofradius c. If the external pressure is p

0, show that

surface tension,

)cba(4

)bac(pT

222

3330

3. A conical glass capillary tube of length 0.1 m hasdiameters 10–3 and 5 × 10–4 m at the ends. When itis just immersed in a liquid at 00C with largerdiameter in contact with it, the liquid rises to8 × 10–2 m in the tube. If another cylindrical glasscapillary tube B is immersed in the same liquid at00C, the liquid rises to 6 × 10–2 m height. The rise ofliquid in the tube B is only 5.5 × 10–2 m when theliquid is at 500C. Find the rate at which the surfacetension changes with temperature considering thechange to be linear. The density of the liquid is(1/14) × 104 kg/m3 and angle of contact is zero.Effect of temperature on density of liquid and glassis negligible.

4. A cone made of a material of relative density

64

27s and height 4 m floats with its apex

downward in a big vessel containing water.

(a) Find the submerged height of cone inwater

(b) Find the time period of vertical

oscillation if it is slightly disturbed fromthe equilibrium position

5. A wooden stick of length L, radius R and density has a small metal piece of mass m (of negligiblevolume) attached to its one end. Find the minimumvalue for the mass m (in terms of given parameters)that would make the stick float vertically inequilibrium in a liquid of density .

6. A soap bubble of radius r1 is blown at the end of a

capillary tube of length l and of internal radius a.Calculate the time taken by the bubble to raduceto radius r

2 < r

1. Surface tension of soap is T and

coefficient of viscosity of air is .

7. A hollow cone of radius R and height H is placedon a horizontal surface at its base. If it is filled withwater (density ) to a height h, find the net forceexerted by water on the curved surface of the cone.

8. A liquid of density is filled in a tank of upperradius r

1 and lower radius r

2 as shown in figure.

A capillary tube of length L and inner radius a andouter radius b is attached at the bottom as shownin figure. It has been found that the rate of volumeflow through the tube is if pressure p is appliedat the top of the tank . Now the tube is detachedthen then velocity of the liquid is v

0 coming out of

the hole. Find the coefficient of viscosity of theliquid.

9. A fluid with viscosity fills the space between twolong co-axial cylinders of radii R

1 and R

2, with

R1 < R

2. The inner cylinder is stationary while the

outer one is rotated with a constant angularvelocity

2. The fluid flow is laminar. Taking into

account that the friction force acting on a unit areaof a cylindrical surface of radius r is defined by theformula = r (/r), find :

(a) the angular velocity of the rotating fluidas a function of radius r;

(b) the moment of the friction forces actingon a unit length of the outer cylinder.

10. A tube of length l and radius R carries a steadyflow of fluid whose density is and viscosity . Thefluid flow velocity depends on the distance r fromthe axis of the tube is v = v

0 (1 – r2/R2). Find :

(a) the volume of the fluid flowing across thesection of the tube per unit time;

(b) the kinetic energy of the fluid within thetube’s volume;

(c) the friction force exerted on the tube bythe fluid;

(d) the pressure difference at the ends of thetube.

PPOM – 28


New Delhi – 110 018, Ph. : 9312629035, 8527112111

ANSWERS SUBJECTIVE

(FINAL STEP EXERCISE)

1. (a) 2.33 kg

(b) 56.7 N (downward)

3. –1.4 × 10–2 N/m0C

4. (a) 3m

(b) s98.1

5.

1LR2

6. )rr(Ta

2t 4

24140

l

7.

2

32

H

)hH(Hh3gR

3

1F

9. (a)

221

21

22

22

21

2r

1

R

1

RR

RR

(b)21

22

22

21

2RR

RR4N

10. (a) Q = ½ v0R2

(b) T = 1/6 lR2v02

(c) Ffr = 4lv

0

(d) p = 4lv0/R2

1. (a) 74 N (b) 150 m3

2. 1.668 × 10–3 m

4. 29025 J, –2.64 × 104 J/m3

5. .01 m

6. 2ngr

R84

2

ll

7. Volume rate flow through EF is 17

Q and

through GH is Q17

16

PE = 1.53 P

0, P

F = 1.47 P

0

8. (a) 91 N

(b) 1.3 mm

ANSWERS SUBJECTIVE

(INITIAL STEP EXERCISE)

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