PPOM – 1
PROPERTIES OF MATTER
C1
All real “rigid” bodies are to some extent elastic, which means that we can change their dimensions slightly
by pulling, pushing, twisting or compressing them.
Hooke’s law states that in elastic deformations, stress (force per unit area) is proportional to strain (relative
deformation) :
Stress
 Elastic modulus
Strain
Three elastic moduli are used to describe the elastic bahaviour (deformations) of objects as they respond to
forces that act on them.
1.
F
, where F is the force
A
perpendicular to the plane of cross sectional A. There are two types of longitudinal stress :
(a)
Tensile longitudinal stress, and
(b)
Compresive longitudinal stress
Longitudinal stress and longitudinal strain : Longitudinal stress is defined as
Tensile stress is tensile force per unit area, F / A . Tensile strain is fractional change in length, l/l0.
Young’s modulus Y is the ratio of tensile stress to tensile strain :
Y
2.
F / A F l 0

l / l 0
A l
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.
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 volume
change V/V0 :
B
3.
p
V / V0
Compressibility k is the reciprocal of bulk modulus : k = 1/B.
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 :
S
Shear stress F|| / A F|| h F|| / A



Shear strain
x/h
A x

The proportional limit is the maximum stress for which stress and strain are proportional. Beyond the
proportional limit, Hooke’s law is not valid. The elastic limit is the stress beyond which irreversible
deformation 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
1.
Practice Problems :
The following four wires are made of the same material. Which of these will have the largest
extension when the same tension is applied.
(a)
length = 50 cm, diameter = 0.5 mm
(b)
length = 100 cm, diameter = 1 mm
(c)
2.
1
× stress × strain.
2
length = 200 cm, diameter = 2 mm
(d)
length = 300 cm, diameter = 3 mm
–5
The compressibility of water is 4 × 10 per unit atmospheric pressure. The decrease in volume of
100 cm3 of water under a pressure of 100 atmosphere will be
(a)
0.4 cm3
Einstein Classes,
(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 section
1 mm2. The work required to increase its length by 1 mm is
(a)
4.
9.
10 J
(d)
100 J
3.4 m
(b)
34 m
(c)
340 m
(d)
none of these
AYR
r
(b)
AY( R  r )
r
(c)
YR r


A r 
(d)
Yr
AR
A massless rod AD consisting of three segments AB, BC and CD joined together is hanging vertically
from a fixed support at A. The lengths of the segments are respectively 0.1 m, 0.2 m and 0.15 m. The
cross-section of the rod is uniformly 10–4m2. A weight of 10 kg is hung from D. If YAB = 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 and
D is
1:2:3
(b)
2:3:7
11
(c)
3:5:9
(d)
none
2
A steel wire (Young’s modulus = 2 × 10 N/m ) of diameter 0.8 mm and length 1 m is clamped firmly
at two points A and B which are 1 m apart and in the same plane. A body is hung from the middle
point of the wire such that the middle point sags 1 cm lower from the original position. The mass of
the body is
(a)
8.
(c)
2
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)
7.
1J
A substance breaks down by a stress of 10 N/m . If the density of the material of the wire is
3 × 103kg/m3, then the length of the wire of that substance which will break under its own weight
when suspended vertically is
(a)
6.
(b)
6
(a)
5.
0.1 J
82 gm
(b)
41 gm
(c)
22.5 gm
(d)
11 gm
The bulk modulus of water if its volume changes from 100 litre to 99.5 litre under a pressure of 100
atmosphere 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
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)
2 L2 Dg
Y
[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  
m
.
V
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 d2 are mixed together, the density of the mixture is
(a)
2.
(b)
2d1d2/(d1 + d2)
(c)
d1d2/(d1 + d2)
(d)
(d1 + d2)/2
If equal volume of two liquids of density is d1 and d2 are mix together then the density of the mixture
is
(a)
3.
(d1 + d2)
(d1 + d2)
(b)
2d1d2/(d1 + d2)
(c)
d1d2/(d1 + d2)
(d)
(d1 + d2)/2
Due to the change of pressure the density of the liquid will change. If the change in pressure is P
and the bulk modulus of liquid is B then the fractional change in density of the liquid equals to
(a)
P
B
(b)
2P
B
(c)
P
2B
(d)
3 P
2 B
[Answers : (1) b (2) d (3) a]
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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C3
Pressure : Pressure is normal force per unit area.
Pressure (a scalar quantity) on a surface is defined as
p  lim
s0
C4
F dF

S
dS
The units for pressure are Nm–2 or pascal (Pa), or mm of mercury (or any other substance).
Hydrostatic pressure distribution : Pressure in a fluid at rest increases with vertical height ‘h’ according
to the relation
dp
 g .
dh
If the density of the liquid is constant at each point then the pressure at a point A at a depth h below the free
surface is given by pA = gh + p0, where p0 is the pressure at the free surface (atmospheric pressure). Absolute pressure is the total pressure in a fluid; gauge pressure is the difference between absolute pressure
and atmospheric pressure.
Hydrostatic Paradox :
1.
Three vessels of equal base area but containing different amounts of liquid upto the same height will have
same force at their bottom.
Practice Problems :
The pressure in a water tap at the base of a building is 3 × 106 dynes/cm2 and on its top it is
1.6 × 106 dynes/cm2. The height of the building is approximately
(a)
2.
(b)
14 m
(c)
70 m
(d)
140 m
3
A uniformly tapering vessel is filled with a liquid of density 900 kg/m . The thrust on the base of the
vessel due to the liquid is (g = 10 m/s2)
(a)
3.
7m
3.6 N
(b)
7.2 N
(c)
10.8 N
(d)
14.4 N
Consider a liquid of density  is placed in a container upto the height h. If the force exerted by the
liquid 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
1.
2.
3.
Pascal Law : Pascal’s law states that pressure applied to the surface of an enclosed fluid is transmitted
undiminished to every portion of the fluid.
Practice Problems :
A piston of cross-sectional area 100 cm2 is used in a hydraulic press to exert a force of 107 dynes on
the 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
A U-tube of uniform cross-section is partially filled with a liquid I. Another liquid II which does not
mix with liquid I is poured into one side. It is found that the liquid levels of the two sides of the tube
are the same, while the level of liquid I has risen by 2 cm. If the specific gravity of liquid I is 1.1, the
specific gravity of liquid II must be
(a)
1.12
(b)
1.1
(c)
1.05
(d)
1.0
A U-tube is partly filled with a liquid A. Another liquid B, which does not mix with A, is poured into
one side until it stands a height h above the level of A on the other side, which has meanwhile risen a
height l. The density of B relative to that of A is
(a)
l
hl
(b)
l
h  2l
(c)
2l
h  2l
(d)
l
2h  l
[Answers : (1) d (2) b (3) c]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 4
C6
1.
Archimede’s Principle
When a body is immersed partly or wholly in a fluid, there acts an upward force on it called the buoyancy
and its magnitude is equal to the weight of the fluid displaced. The point of the application of buoyancy is
at the centre of mass of the displaced fluid and is called the centre of buoyancy. Buoyancy exists because of
pressure gradient. Thus in case of a free fall situation buoyancy is zero.
Principle of floatation
Weight of the object = Buoyancy
sVg = lVsg
V : total volume of the object
Vs : submerged volume of the object
s : density of object
l : density of liquid
Practice Problems :
A piece of wood of relative density 0.36 floats in oil of relative density 0.90. The fraction of volume of
wood above the surface of oil is
(a)
2.
4.
0.5 m
(b)
(a)
d1
d2
(c)
d1
d 2  d1
(c)
0.6
(d)
0.8
1.0 m
(c)
1.2 m
(d)
1.8 m
2h
g
2h
g
(b)
d2
d1
(d)
d 2  d1
d2
2h
g
2h
g
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
h

(b)


  1  h


(c)
 
1   h
 
(d)
h

A small ball of density  is dropped from a height h into a liquid of density  ( > ). Neglecting
damping forces, the maximum depth to which the body sinks is
(a)
6.
0.4
A streamlined body of relative density d1 falls from a height h on the surface of a liquid of relative
density d2, where d2 > d1. The time for which the body will fall inside the liquid is
(a)
5.
(b)
A large block of ice 10 m thick with a vertical hole drilled through it is floating in a lake. The
minimum length of the rope required to scoop out a bucket full of water through the hole is (density
of ice = 0.9 g/cm3)
(a)
3.
0.3
h

(b)
h

(c)
h(    )

(d)
h(    )

A block (density ) is suspended from a spring and produces an extension ‘x’. If the whole system is
dipped 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; a
streamline is a curve tangent at each point to the velocity vector at that point. A flow tube is a tube bounded
at its sides by flow lines. In laminar flow, layers of fluid slide smoothly past each other. In turbulent flow
there is great disorder and a constantly changing flow pattern.
Einstein Classes,
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PPOM – 5
Principle of Continuity : Conservation of mass in an incompressible fluid is expressed by the equation of
continuity; for two cross sections A1 and A2 in a flow tube, the flow speed v1 and v2 are related by
A1v1 = A2v2.
The product Av is the volume flow rate, dV/dt, the rate at which volume crosses a section of the
tube :
dV
 Av .
dt
Bernoulli’s equation relates the pressure p, flow speed v, and elevation y for steady flow in an ideal fluid
which is based on conservation of energy principle. For any two points, denoted by subscripts 1 and 2.
p 1  gy 1 
1.
Practice Problems :
Two large tanks a and b, open at the top, contains different liquids. A small hole is made in the side
of each tank at the same depth h below the liquid surface, but the hole in a has twice the area of the
hole in b. The ratio of the densities of the liquids in a and b so that the mass flux is the same for each
hole should be
(a)
2.
6.
0.5
(c)
4
(d)
0.25
2
(b)
0.5
(c)
4
(d)
0.25
5.2 × 104N
(b)
6.2 × 104N
(c)
7.2 × 104N
(d)
8.2 × 104N
A horizontal pipe line carries water in a streamline flow. At a point along the pipe where the
cross-sectional area is 10 cm2, the water velocity is 1 m/s and the pressure is 2000 Pa. The pressure of
water at another point where the cross-sectional area is 5 cm2 is
(a)
5.
(b)
Air is streaming past a horizontal aeroplane wing such that its speed is 120 m/s over the upper
surface 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 and
has an average width 2 m, the gross lift of the wing is
(a)
4.
2
In the above problem the ratio of flow rates (volume flux) from the holes in a and b is
(a)
3.
1 2
1
v 1  p 2  gy 2  v 22
2
2
500 Pa
(b)
750 Pa
(c)
900 Pa
(d)
1100 Pa
The rate of flow of glycerine of density 1.25 × 103 kg/m3 through the conical section of a pipe, if the
radii 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
Water flows out of two small holes P and Q in a wall of a tank and the two streams strike the ground
at the same point. If the hole P is at a height h above the ground and the level of water stands at a
height H above the ground, then the height of Q is
(a)
Hh
2
(b)
H–h
(c)
H – h/2
(d)
Hh
2
[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 fluidthe
viscous force is propotional to strain rate. The viscous force between two layers of a fluid of area A having
dv
where  is called the coefficient of viscosity. In SI unit
dx
of  is poiseuille (1 PI = 1 Ns m–2) and the dimension of  is ML–1T–1.
a velocity gradient dv/dx is given by F   A
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 6
Practice Problems :
1.
The velocity of water (viscosity = 10–3 poiseuille) in a river is 18 km/hr at the surface. If the river is
5 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 having
viscosity  experiences a viscous resisting force F given by Stoke’s law : F = 6rv.
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 v t 
2 r 2g
(    ) where  is the density of the
9 
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 of
glycerine is d2 is
(a)
2.
mg
(c)

d
mg  1  1
d
2




(d)
d
mg  2
 d1



13.6 poise
(b)
14.6 poise
(c)
15.6 poise
(d)
16.6 poise
1.324 poise
(b)
1.424 poise
(c)
1.524 poise
(d)
1.624 poise
‘n’ equal drops of water are falling through air with a steady velocity v. If the drops coalesced, then
the new velocity is
(a)
5.
(b)
An air bubble of radius 1 mm is allowed to rise through a long cylindrical column of a viscous
liquid of radius 5 cm and travels at a steady rate of 2.1 cm per sec. If the density of the liquid is
1.47 gm per cc, then its viscosity is
(a)
4.



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)
3.

d
mg  1  2
d1

(n1/3) v
(b)
nv
(c)
(n1/2) v
(d)
(n2/3) v
A spherical ball of radius 1 × 10–4 m and density 104 kg/m3 falls freely under gravity through a
distance h before entering a tank of water (viscosity of water is 9.8 × 10–6 N-s/m2). If after entering
the 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 total
volume rate is given by Poiseuille’s equation :
dV   R 4   p 1  p 2 
 


dt
8     L 
where p1 and p2 are the pressures at the two ends and  is the viscosity.
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 7
1.
2.
Practice Problems :
Under a pressure head the rate of orderly volume flow of a liquid through a capillary tube is Q. If the
length of the capillary tube is doubled and the diameter of the bore is halved, the rate of flow would
become
(a)
Q/32
(b)
Q/8
(c)
Q/4
(d)
8Q
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 V2 are, respectively, the volumes of the two liquids flowing
per second, then V1/V2 is
(a)
3.
2
1
(b)
1
2
2
(c)
2
2
1
2
(d)
1
2
2
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 R e 
vd
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 Re > 2000 implies turbulent flow.
C12
Surface Tension : The surface of a liquid behaves like a membrane under tension; the force per unit length
across 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
a liquid bubble or air bubble of radius r is given by
C14
1.
2.
2T
. Excess pressure inside
r
4T
.
r
Capillary Rise or Fall : The rise or fall of a liquid in a capillary tube is given by h 
2T cos 
, where  is
gr
the angle of contact,  is the density of liquid in the tube and r is the radius of the tube. For a clean glass
plate in contact with pure water,  = 0.
Practice Problems :
A liquid rises to a height h in a capillary tube on the earth. The height to which the same liquid would
rise in the same tube on the moon is about
(a)
6h
(b)
6 h
(c)
h/6
(d)
h/6
n identical spherical drops of a liquid of surface tension T, each of radius r, coalesce to form a single
drop. The surface energy
(a)
decreases by 4r2(n – n1/3)T
(b)
increases by 4r2(n – n1/3)T
(c)
decreases by 4r2(n – n2/3)T
(d)
increases by 4r2(n – n2/3)T
[Answers : (1) a (2) c]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 8
SINGLE CORRECT CHOIC E TYPE
1.
A highly rigid cubical block A of small mass M and
side L is fixed rigidly onto another cubical block B
of the same dimenstions and of low modulus of
rigidity  such that the lower face of B is rigidly
held on a horizontal surface. A small force F is
applied perpendicular to one of the side faces of A.
After the force is withdrawn, block A executes small
oscillations the time period of which is given by
(a)
(c)
2.
3.
4.
5.
6.
2 ML
2 ML / 
(b)
(d)
7.
P
2K
(c)
P
KP
(a)
A g
a 2
H
1
 H2

(b)
A 2
a g
H
1
 H2

(c)
a g
A 2
H
1
 H2

(d)
a 2
A g
H
1
 H2

2 M / L
2 M / L
The normal density of gold is  and its bulk
modulus is K. The increase in density of a piece of
gold when a pressure P is applied uniformly from
all sides is
(a)
A vessel of cross-sectional area A contains a liquid
to a height H1. If a hole having cross-sectional area
a is made at the bottom of the vessel, then the time
taken by the liquid level to decrease from H1 to H2
is
(b)
P
K
(d)
K
KP
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 liquid is kept in a cylindrical vessel which is
rotating along its axis. The liquid rises at the sides.
If the radius of the vessel is 0.05 m and the speed of
rotation is 2 rev/s, the difference in the height of
the liquid at the centre of the vessel and at its sides
is
(a)
5l1 – 4l2
(b)
5l2 – 4l1
(a)
0.01 m
(b)
0.02 m
(c)
9l1 – 8l2
(d)
9l2 – 8l1
(c)
0.03 m
(d)
0.04 m
A cylindrical vessel of radius r is filled with a
homogenous liquid to a height h. If the force
exerted by the liquid on the side of the vessel is equal
to the force exerted by it on the bottom of the
vessel, then
(a)
h=r
(b)
h = 2r
(c)
h = r/2
(d)
h = 3r/2
A vertical U-tube contains mercury in both its arms.
A glycerine (density 1.3 g/cm3) column of length 10
cm is introduced into one of the arms. Oil of
density 0.8 g/cm3 is poured into the other arm until
the upper surfaces of oil and glycerine are at the
same level. The length of the oil column is (density
of mercury = 13.6 g/cm3)
(a)
8.5 cm
(b)
9.6 cm
(c)
10.7 cm
(d)
11.8 cm
A vessel contains oil (density 0.8 g/cm3) over
mercury (density 13.6 g/cm3). A homogenous sphere
floats with half its volume immersed in mercury
and the other half in oil. The density of the
material of the sphere in g/cm3 is
(a)
3.3
(b)
6.4
(c)
7.2
(d)
12.8
Einstein Classes,
8.
9.
10.
Two capillary tubes of the same radius and length
l1 and l2 are fitted horizontally side by side to the
bottom of a vessel containing water. The length of
a single tube that can replace the two tubes such
that the rate of steady flow through this tube equals
the combined rate of flow through the two tubes, is
(a)
l1 + l2
(b)
l1  l2
2
(c)
l1l2
l1  l2
(d)
2l1l2
l1  l2
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 of
a single tube that can replace the two tubes such
that the rate of study flow through this tube equals
the combined rate of flow through the two tubes, is
(a)
r1 + r2
(c)
r
1
2
 r2 2
(b)

1/ 2
(d)
r1r2
r
1
4
 r2 4
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1/ 4
PPOM – 9
11.
Two spherical soap bubbles of radii r1 and r2 in
vacuum coalesce under isothermal conditions. The
resulting bubble has a radius equal to
(a)
r1  r2
2
(b)
r1r2
r1  r2
17.
(c)
12.
13.
14.
15.
16.
r1r2
(d)
2
r1  r2
2
A long cylindrical glass vessel has a small hole of
radius r at its bottom. The depth to which the vessel can be lowered vertically in a deep water bath
(surface tenstion T, density d) without any water
entering inside is
(a)
T
rdg
(b)
2T
rdg
(c)
3T
rdg
(d)
4T
rdg
If a number of little droplets of a liquid of density
, surface tenstion T and specific heat c, each of
radius r, coalesce to form a single drop of radius R,
the rise in temperature will be
(a)
3T  1 1 
  
c  r R 
(b)
(c)
3T  1 1 
  
2c  r R 
(d)
A copper wire of negligible mass with length 1 m
and cross-sectional area 10–6 m2 is kept on a smooth
horizontal table with one end fixed. A ball of mass
1 kg is attached to the other end. If the wire and the
ball are rotating with an angular velocity of
20 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’s
modulus of the material to the breaking stress of
the wire is
(a)
20 : 1
(b)
40 : 1
(c)
20 : 3
(d)
40 : 3
The depth of a lake at which the density of water is
1% greater than at the surface, if the
compressibility of water is 50 × 10–6/atm
(a)
1 km
(b)
1.5 km
(c)
2 km
(d)
2.5 km
A uniform pressure p is exerted on all sides of a
solid cube at temperature t0C. The bulk modulus
and coefficient of volume expansion of the
material are b and  respectively. Let the
temperature of the cube be raised t in order to
bring its volume back to the volume it had before
the pressure was applied, then t equals to
Einstein Classes,
p
b
(b)
2p
b
(c)
p
2 b
(d)
pb
The density of air in atmosphere decreases with
height h and can be expressed by the relation :
 = 0e–Ah
where  0 = 1.3 kg/m 3 and A = 1.2 × 10 –4 /m.
If g = 9.8 m/s2 then the atmospheric pressure at
sea-level is
18.
19.
3T  1 1 
  
c  r R 
3T  1 1 
  
2c  r R 
(a)
20.
21.
22.
(a)
1.06 × 104N/m2 (b)
2.06 × 105N/m2
(c)
3.06 × 105N/m2 (d)
1.06 × 105N/m2
A piece of copper of density 8.8 gm/cc having an
internal cavity weighs 264 gm in air and 221 gm in
water. The volume of the cavity is
(a)
11 cc
(b)
12 cc
(c)
13 cc
(d)
14 cc
A piece of brass (alloy of copper and zinc) weighs
12.9 gm in air. When completely immersed in
water it weighs 11.3 gm. If the specific gravities of
copper and zinc are 8.9 and 7.1 respectively then
the mass of thecopper contained in the alloy is
(a)
7.61 gm
(b)
7.25 gm
(c)
6.78 gm
(d)
6.25 gm
A piece of metal floats on mercury. The coefficient
of 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 the
metal submerged in mercury changes is
(a)
2(2 – 1) T
(b)
(2 – 1) T
(c)
2(2 + 1) T
(d)
(2 + 1) T
A ring is cut from a platinum tube of 8.5 cm
internal and 8.7 cm external diameter. It is
supported horizontally from a pan of a balance so
that it comes in contact with the water in a glass
vessel. It has been found that an extra 3.97 gm
weight is required to pull it away from water, then
the surface tension of water is
(a)
62.18 dyne/cm
(b)
68.75 dyne/cm
(c)
72.13 dyne/cm
(d)
none
The lower end of a capallary tube of radius
2.00 mm is dipped 8.00 cm below the surface of
water in a beaker. If surface tension of
water = 73 × 10–3 N/m, density of water = 103 kg/m3,
1 atmosphere = 1.01 × 105 Pa and g = 9.8 m/s2 then
the pressure required in the tube to blow a bubble
at 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
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
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PPOM – 10
23.
24.
25.
26.
27.
The limbs of a manometer consist of uniform
capillary tubes of radii 1.4 × 10 –3 m and
7.2 × 10–4 m. The density of the liquid is 103 kg/m3
and surface tension is 72 × 10–3 N/m. It has been
found that the level of the liquid in narrower tube
stands 0.2 m above that in the broader tube, then
the correct pressure difference is
A cubical block of edge L and density d is floating
in equilibrium in a container of base area 4L2. A
small hole is made at the lower most right end. The
density of the liquid is 2d and the density of the
material of the block is d. The velocity of efflux at
t = 0 is
(a)
1863 Pa
(b)
1960 Pa
(a)
(b)
(c)
1720 Pa
(d)
2793 Pa
g( 8 H  L )
2
g( 8 H  L )
4
(c)
g( 8 H  L )
6
(d)
g( 8 H  L )
8
Two separate air bubbles (radii 0.002 m and 0.004
m) formed of the same liquid (surface tension 0.07
N/m) come together to form a double bubble. The
radius of curvature of the internal film surface
common to both the bubbles is
(a)
.002 m
(b)
.003 m
Consider an ice cube of edge L kept in a gravity
free hall. Assume that the density of water and
density of ice is same, the surface area of the water
when the ice melts is
(c)
.004 m
(d)
.005 m
(a)
(4)1/332/3L2
(b)
(4)2/331/3L2
(c)
(4)2/332/3L2
(d)
(4)1/331/3L2
A body of mass 3.14 kg is suspended from one end
of a wire of length 10.0 m. The radius of the wire is
changing uniformly from 9.8 × 10–4 m at one end to
5.0 × 10–4 m at the other end. The change in length
of the wire if young’s modulus of the material of
the wire is 2 × 1011 N/m2
(a)
1 mm
(b)
2 mm
(c)
3 mm
(d)
4 mm
29.
30.
A thin uniform metallic rod of length 0.5 m and
radius 0.1 m rotates with an angular velocity 400
rad/s in a horizontal plane about a vertical axis
passing through one of its ends. The density of
material of the rod is 104 kg/m3 and the Young’s
mudulus is 2 × 1011 N/m2.The elongation of the rod
is
(a)
1 mm
(b)
1/2 mm
(c)
1/3 mm
(d)
1/4 mm
A solid sphere of radius R made of a material of
bulk modulus B is surrounded by a liquid in a
cylindrical container. A massless piston of area A
floats on the surface of the liquid. The fractional
change in the radius of the sphere (dR/R) when a
mass M is placed on the piston to compress the
liquid is
(a)
mg
3AB
(b)
mg
2AB
(c)
3mg
AB
(d)
3mg
2AB
It is found that the movable wire is in equilibrium
when the upward force 3.45 mN is applied. The wire
has a length of 4.85 cm and linear mass density
1.75 × 10–3 kg/m.
The surface tension of the liquid is
31.
(a)
0.027 N/m
(b)
0.037 N/m
(c)
0.054 N/m
(d)
0.0135 N/m
A container of width 2a is filled with a liquid. A
thin wire of weight per unit length  is gently placed
over the liquid surface in the middle of the surface
as shown in the figure. As a result, the liquid
surface is depressed by a distance y (y << a). The
surface tension of the liquid is
(a)
a
2y
(b)
a
y
(c)
2a
y
(d)
a
4y
28.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
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32.
Consider a horizontally oriented syringe
containing water located at a height of H above the
ground. The radius of the plunger is R and the
diameter of the nozzle is r. The plunger is pushed
with a constant speed v. The horizontal range of
water steam on the ground is
(a)
(c)
33.
34.
35.
4
4
H  R2
g  r 2

v 

2H  R 2
g  r 2

v 

(b)
(d)
4
4
H  r 2
g  R2
5
(b)
25
(c)
2.5
(d)
50
2H  r 2
g  R 2
0.010C
(b)
0.0010C
(c)
0.020C
(d)
none
Castor oil, which has a density of 0.96 × 103 kg/m3
at room temperature, is forced through a pipe of
circular cross section by a pump that maintains a
gauge pressure of 950 Pa. The pipe has a diameter
of 2.6 cm and a length of 65 cm. The castor oil
emerging from the free end of the pipe at
atmospheric pressure is collected. After 90 s, a
Einstein Classes,
(a)
1.15 SI unit
(b)
2.15 SI unit
(c)
0.15 SI unit
(d)
0.25 SI unit
A sniper fires a rifle bullet into a gasoline tank,
making a hole. The tank was sealed and is under
3.10-atm absolute pressure, as shown in the figure.
The stored gasoline has a density of 660 kg/m3. The
range of the liquid comes out immediately after
making the hole is

v


One thousands water drops of radius of 1mm are
merged to form a bigger drop. The density, surface
tension and specific heat capacity of water is 1g/cc,
0.075 N/m and 1 cal/gm0C. Assume that there is no
loss of energy which are released then change in
temperature of water is
(a)
36.

v 

A rectangular metal plate has dimensions of
10 cm × 20 cm. A thin film of oil separates the plate
from a fixed horizontal surface. The separation
between the rectangular plate and the horizontal
surface is 0.2 mm. An ideal string is attached to the
plate and passes over an ideal pulley to a mass m.
When m = 125 gm, the metal plate moves at
constant speed of 5 cm/s across the horizontal
surface. Then the coefficient of viscosity of oil in
dyne-s/cm2 is (Use g = 1000 cm/s2)
(a)
total of 1.23 kg has been collected. The coefficient
of viscosity of the castor oil at this temperature is
37.
(a)
41 m
(b)
82 m
(c)
123 m
(d)
144 m
Consider a tank of cross-sectional area 1sq.m and
filled with a liquid of density 660 kg/m3. The liquid
is covered by a piston of mass
force of
3.1
 10 4 kg and a
3
6.2  10 5
N is applied as shown in figure.
3
A hole of very small area is made. The range of the
liquid comes out immediately after making the hole
is
38.
(a)
41 m
(b)
82 m
(c)
123 m
(d)
none
In Searle’s experiment, which is used to find
Young’s modulus of elasticity, the diameter of
experimental wire is D = 0.05 cm (measured by a
scale of least count 0.001 cm) and length is
L = 110 cm (measured by a scale of least count
0.1 cm). A weight of 50 N causes an extension of
X = 0.125 cm (measured by a micrometer of least
count 0.001 cm). Screw gauge and meter scale are
free from error. The maximum possible error in the
values 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 very
small area is made at the lower most point. If time
taken to leak out the water for the first half height
is T1 and time taken to leak out the water for the
next half height is T2 then
T1
is
T2
(a)
(b)
1
ANSWERS
(SINGLE CORRECT CHOICE TYPE)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1
2
(c)
2
(d)
none
40.
Figure shows how the stream of water emerging
from a faucet “necks down” as it falls. The
indicated cross-sectional areas are A0 = 1.2 cm2 and
A = 0.35 cm2. The two levels are separated by a
vertical distance h = 45 mm. The volume flow rate
from the tap is
(a)
24 cm3/s
(b)
29 cm3/s
(c)
34 cm3/s
(d)
39 cm3/s
d
b
b
a
b
c
b
b
c
d
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
d
b
b
b
c
a
d
c
a
b
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
c
b
a
c
a
c
a
b
a
a
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
a
c
c
d
a
b
d
a
d
c
EXCERCISE BASED ON NEW PATTERN
2.
COMPREHENSION TYPE
Comprehension-1
A soap bubble in air has a radius of 3.20 cm. It is
then blown up to a radius of 5.80 cm.
The surface tension of the bubble film is
26.0 mN/m
1.
3.
The work was done on the atmosphere in blowing
up the bubble is
(a)
34.35 J
(b)
68.7 J
(c)
108.5 J
(d)
125.6 J
The work was done in stretching the bubble
surface is
The pressure diference across the film at the larger
size is
(a)
465 µJ
(b)
565 µ J
(c)
656 µJ
(d)
765 µJ
(a)
1.79 Pa
(b)
0.895 Pa
Comprehension-2
(c)
3.25 Pa
(d)
1.625 Pa
Two rods of different metals, having the same area
of cross-section A, are placed end to end between
two massive walls as shown in figure.
Einstein Classes,
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PPOM – 13
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 Y2. The temperature of both the rods is now
raised by T degrees. Assume that there is no change
in the cross-sectional area of the rods and the rods
do not bend. There is no deformation of walls.
4.
5.
Due to thermal expansion the increase in the length
of the composite rod is
7.
8.
The totaldownward force exerted by the liquid and
the atmosphere on the top of the object is
(a)
38.4 kN
(b)
40.5 kN
(c)
2.3 kN
(d)
2.1 kN
The total upward force on the bottom of the object.
(a)
(L11 + L22)T
(a)
38.4 kN
(b)
40.5 kN
(b)
(L11 – L22)T
(c)
2.3 kN
(d)
2.1 kN
(c)
(L11 + L22)T/2
(d)
(L11 – L22)T/2
9.
The force with which the rods act on each other at
the higher temperature is
The tension in the wire.
(a)
38.4 kN
(b)
40.5 kN
(c)
2.3 kN
(d)
2.1 kN
Comprehension-4
(a)
(b)
(c)
A ( L 1  1  L 2  2 )T
 L1 L 2 



 Y1 Y2 
Water stands at a depth H behind the vertical face
of a dam and exerts a certain resultant horizontal
force on the dam tending to slide it along its foundation and a certain torque tending to overturn the
dam about the lower most point O. If the total width
of the dam is L.
A ( L 1  1  L 2  2 )T
 L1 L 2 



 Y1 Y2 
10.
A ( L 1  1  L 2  2 )T
L
L 
2 1  2 
 Y1 Y2 
11.
(d)
6.
A ( L 1  1  L 2  2 )T
L
L 
2 1  2 
 Y1 Y2 
Let 1 > 2 and Y1 < Y2. If the rods have equal
initial length and the lengths of the rods at the
higher temperature is L  and L  respectively then
1
2
(a)
L 1 must be greater than L 2
(b)
L 2 must be greater than L 1
(c)
L 1 must be equal to L 2
(d)
can’t be said anything
12.
The total horizontal force is
(a)
1
gLH 2
2
(b)
gLH 2
(c)
1
gLH 2
3
(d)
1
gLH 2
4
The total torque about O is
(a)
1
gLH 3
2
(b)
1
gLH 3
4
(c)
1
gLH 3
6
(d)
1
gLH 3
8
Moment arm of the resultant horizontal force about
the line through O is
(a)
H/2
(b)
H/3
(c)
H/4
(d)
H/5
Comprehension-5
Comprehension-3
A cubic object of dimensions L = 0.608 m on a side
and weight W = 4450 N in a vacuum is suspended
by a wire in an open tank of liquid of density
 = 944 kg/m3, as in figure.
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A container of large uniform cross-sectional area A
resting on a horizontal surface, holds two
immiscible, non-viscous and incompressible liquids
of densities d and 2d each of height (H/2) as shown
in figure. The lower density liquid is open to the
atmosphere having pressure P0. A homogeneous
solid cylinder of length L (L < H/2), cross-sectional
area (A/5) is immersed such that it floats with its
axis vertical at the liquid-liquid interface with
length (L/4) in the denser liquid.
13.
14.
15.
16.
The cylinder is removed and original arrangement
is restored. A tiny hole of area s(s << A) is punched
on the vertical side of the container at a height h
(h < H/2). This height ‘h’ is such that the horizontal
distance ‘x’ travelled by the liquid initially is
maximum.
The density of solid is
(a)
5
d
4
(b)
3
d
2
(c)
5
d
3
(d)
4
d
3
18.
(a)
3gH
4
(b)
3gH
2
(c)
3gH
(d)
2gH
The total pressure at the bottom of the container is
(a)
1
P0  (6H  L)dg
2
(b)
1
P0  ( 6H  L )dg
4
(c)
1
P0  ( 6H  L )dg
6
(d)
1
P0  ( 6H  L )dg
8
19.
(a)
performs oscillatory motion but not SHM
(b)
performs SHM
(c)
continuously moves downward
(d)
none of these
20.
5L
4g
 5L
2 4g
(c)
never reach it’s original position
(d)
none
The cylinder is depressed in such a way that its top
surface just below the upper surface of liquid with
density 2d and is then released. Immediately after
the release its acceleration is
(a)
(c)
8
g upward
5
3
g upward
5
Einstein Classes,
H
2
(b)
3
H
4
(c)
H
(d)
2H
If the surface is frictionless then :
21.
(b)
(a)
A large open top container of negligible mass and
uniform cross-sectional area A has a small hole of
cross-sectional area A/100 in its side wall near the
bottom. The container is kept on a horizontal floor
and contains a liquid of density  and mass m0.
Assuming that the liquid starts flowing out
horizontally through the hole at t = 0.
The minimum time after which the cylinder will
reach it’s original position is
2
The maximum value of the distance ‘x’ is
Comprehension-6
The cylinder is slightly depressed vertically
downward and released then
(a)
17.
The initial speed of efflux of the liquid at the hole is
(b)
(d)
22.
The acceleration of the container at t = 0 is
(a)
g
20
(b)
g
30
(c)
g
40
(d)
g
50
If the height of the liquid above the hole at any time
is 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
The maximum velocity of the container
(a)
m 0g
2 A
(b)
(c)
m 0g
3 A
(d)
6
g upward
5
2
g upward
5
m 0g
4 A
none
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23.
24.
The normal reaction acted by the horizontal
surface on the container will
(a)
pass through the center of gravity of
liquid
(b)
pass through the left of the center of
gravity
(c)
pass through the right of the center of
gravity
(d)
none of these
If two identical small holes on the opposite side of
the tank is made at the same height then the
acceleration of the container is
(a)
(c)
25.
26.
30.
zero
g
40
(b)
(d)
A cubical block of length L is floating in
equilibrium immersed completely inside the liquid.
The bottom of the block is at the height h0 above
the bottom of the container. If acceleration due to
gravity g is uniform then the mass of the block is
(a)
L2 0  h 0
e
 e  (h0  L )


(b)
L2 0  h 0
e
 e  (h0  L )
2

(c)
L2  0  h 0
e
 e  (h 0  L )
2

(d)
L2 0  h 0
e
 e  ( h 0  L )


g
30
g
50
h
(b)
h
(c)
h3/2
(d)
h2
If the surface having some friction then minimum
coefficient of friction such that the container should
not move
(a)
1
50
(b)
1
75
(c)
1
100
(d)
1
125



Comprehension-8
Suppose a spherical body of radius r, density  is
released from rest in a liquid column of large
vertical height of viscosity . The density of the
liquid is . Assume that acceleration due to gravity
g does not change with height.
If two identical small holes on the opposite side of
the tank is made at the different height but
separated by h then the acceleration of the container
is proportional to
(a)

31.
32.
The initial acceleration of the body is
(a)
zero
(b)
g
(c)
<g
(d)
>g
The maximum power due to the net force is
(a)
r 5 (    ) 2 g 2

(b)
2r 5 (    ) 2 g 2

(c)
r 5 (    ) 2 g 2
2
(d)
none
Comprehension-7
Consider a large vertical container of cross-sectional
area 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.
28.
The dimensional formula of
(a)
[ML–3]
(b)
[ML–4]
(c)
[ML–2]
(d)
dimensionless
The velocity acheived by the body when the power
of net force will become zero
(a)
2r 2 (  )g
9
(c)
2r 2 (   )g
11
The total mass of the liquid in the container is
(a)
(c)
29.
33.
0
is

0A

(b)
2 0 A

(d)
0A
2
infinite
34.
linear
(b)
constant
(c)
exponential
(d)
parabolic
Einstein Classes,
r 2 (  )g
9
(d)
r 2 (   )g
11
The total work done by the various forces is
(a)
If the pressure at the bottom is P0 then the pressure
with height h measured from the bottom will change
(a)
(b)
(b)
4r 7 (  ) 2 g 2
2432
6r 7 (  ) 2 g 2
2432
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39.
8r 7 (  ) 2 g 2
(c)
2432
10r 7 (  ) 2 g 2
(d)
35.
36.
37.
(a)
4
(b)
5
(c)
6
(d)
7
If amount of heat produced is all absorbs by the
liquid only then the change in temperature of the
liquid depends on time t as tn, once the ball will
acheive terminal speed. The value of n is
(a)
zero
(b)
1
(c)
½
(d)
none
40.
The total amount of heat produced due to viscous
force until the ball acheived terminal speed is
4r 7 (  ) 2 g 2
(a)
(c)
(d)
(a)
ghr22 + P0(r22 – r12) –

h(r22 – r12)g
3
(b)
ghr22 + P0(r22 + r12) –

h(r22 – r12)g
3
(c)
ghr22 + P0(r22 – r12) +

h(r22 – r12)g
3
(d)
none
2432
Due to viscous force, heat is produced. Let the
rate of production of heat is directly proportional
to rn when the ball acheived constant speed. The
value of n is
(b)
The resultant force exerted by the side walls of the
container on the liquid is, if atmospheric pressure
is p0
A hole of very small area is made at the height h/2
from the bottom. The horizontal velocity of efflux
at t = 0 is
(a)
gh
(b)
gh
(c)
gh
2432
6r 7 (  ) 2 g 2
8r 7 (  ) 2 g 2
41.
(r2  r1 ) 2  h 2
(r2  r1 )
(r2  r1 ) 2  h 2
 h 

gh 
 r2  r1 
(d)
2432
h
The time after which the water lands on the ground
which comes out at t = 0 is
2432
h
g
(a)
none
h
g
(b)
>
(d)
none
Comprehension-9
(c)
<
h
g
Comprehension-10
In the above container with the dimension as shown
in figure is filled with a liquid of density .
38.
Consider a tube of very small radius ‘r’ and length
L which is completely filled with water of surface
tension T and density .
42.
The pressure at the bottom of the tube is
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
Einstein Classes,
(a)
P0 + gL
(c)
P0 + gL –
2T
r
(b)
P0 + gL +
(d)
none
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2T
r
PPOM – 17
43.
44.
Consider a air bubble at the depth ‘h’ in the tube.
The radius of the bubble is r1. The pressure inside
the bubble is
(a)
P0 
2T
2T
 gh 
r
r1
(b)
P0 
2T
2T
 gh 
r
r1
(c)
P0  gh 
2T
r1
(d)
P0  gh 
2T
r1
(b)
45.
46.
Consider a fixed container of radius R as shown in
figure. The container is half filled with a liquid of
density . The atmospheric pressure is P0
47.
Let the bubble starts rising and the temperature of
the liquid remains constant. When it just reach the
top most point then the pressure inside the bubble
is
(a)
Comprehension-11
48.
2T 2T
P0 

r
r1
> P0 
(c)
< P0 
(d)
none
2T 2T

r
r1
2T 2T

r
r1
49.
If this bubble will burst out then the temperature
of the liquid
(a)
remains constant
(b)
increases
(c)
decreases
(d)
can’t be decided
Consider the tube without any bubble. Now a very
small hole is made at the bottom most point of the
tube the velocity a efflux of the liquid is
gL 
(a)
2T
r

2
gL 
(b)
2T
r

2
50.
(c)
2gL
(d)
none
The pressure at the bottom most point is
(a)
P0 + gR
(b)
P0
(c)
gR
(d)
none
The resultant force exerted by the liquid on the
container is
(a)
4P0 R 2 
(b)
2
R 3g
3
(c)
4P0 R 2
(d)
none
2
R 3 g
3
R
above the
2
bottom most point then the liquid will land on the
ground at the distance from the bottom most point
of the container is
If a hole is made at the height
(a)
 3 ( 5  1)
3 


R

2
2 

(b)
 3 ( 5  1)
3 


R

4
2 

(c)
 3 ( 5  1)
3 


R

4
2 

(d)
none
If the hole of very small area ‘a’ is made at the
bottom most point such that the liquid will come
out from the hole then the time after which the
container will become empty
(a)
15R 5 / 2
(b)
15a 2g
14a 2g
(c)
R 5 / 2
14R 5 / 2
(d)
none
a 2g
Einstein Classes,
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Comprehension-12
Consider a container with the dimension of the base
(a × b) and liquid is filled upto the height H. The
liquid has the density . The whole system is placed
on the moon and assume that horizonal surface is
frictionless. Mass of the container is negligible. The
acceleration due to gravity on the surface of the
earth is g.
55.
51.
52.
53.
F1
depends
F2
Consider a fixed container of radius R as shown in
figure. The container is half filled with a liquid of
density . The atmospheric pressure is P0
force exerted on the base is F2. Then
A plate is tightly placed on the mouth of the
container and the air is completely pumped out.
(a)
density of the liquid
(b)
height of the liquid
The pressure at the bottom most point
(c)
atmospheric pressure on the earth
(d)
all the above
(a)
P0 + gR
(b)
P0
(c)
gR
(d)
none
on
56.
The force exerted by the liquid on the container is
(a)
4P0 R 2 
(b)
2
R 3g
3
(c)
4P0 R
(d)
none
2
R 3 g
3
(c)
<
2

2P 
 gR  0 
 

(b)
2gR
(d)
14R 5 / 2
(b)
15a 2g
(c)
Einstein Classes,
57.

2P 
 gR  0 
 

(a)
density of the liquid
(b)
height of the liquid
(c)
atmospheric pressure on the earth
(d)
none of these
The pressure energy per unit volume at the bottom
of the container is
(a)
gH
6
(b)
gH
(c)
gH
3
(d)
none
A hole of very small area ‘s’ is made at the bottom
most point of the container at the right end.
gR
58.
The speed of efflux is
(a)
14R 5 / 2
(c)
15a 2g
depends on P0
Comprehension-13
F1
F2
depends on
If the hole of very small area ‘a’ is made at the
bottom most point such that the liquid will come
out from the hole then the time after which the
container will become empty is
(a)
If this system is on the earth then the force exerted
by the liquid on the base is F1 and in the above case
the force exerted by the liquid is F2. Then
R
If a very small hole is made at the height
above
2
the bottom most point then the speed with which
liquid will come out is
(a)
54.
If this system is on the earth then the total force
exerted on the base is F1 and in the above case the
(d)
none
59.
zero
gH
3
(b)
gH
6
(d)
gH
2
The acceleration of the container at any time t is
(a)
sg
3ab
(b)
sg
ab
(c)
2sg
ab
(d)
2sg
3ab
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60.
The time after which the container will become
empty depends on
(a)
area of the hole
(b)
density of the liquid
(c)
base area of the container
(d)
all of the above
67.
Comprehension-14
A cylindrical tank 1 m in radius rests on a platform
5 m high. Initially the tank is filled with water up to
a height of 5 m. A plug whose area is 10–4 m2 is
removed from an orifice on the side of the tank at
the bottom. Consider the liquid which comes out at
t = 0.
61.
62.
63.
64.
65.
66.
Using the above apparatus, a child can blow a soap
bubble of radius ‘r’. The surface tension of the soap
solution is T. Air of density  moves with the
velocity v through the tube of radius r1( << r) and
comes to rest inside the bubble. The circular wire
has the radius R. Assume that the air is falling
normal to the bubble surface.
68.
The initial speed with which the water flows from
the orifice is
(a)
10 m/s
(b)
7.5 m/s
(c)
5 m/s
(d)
2.5 m/s
69.
The horizontal speed of the water that comes out at
t = 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
The force exerted by the circular wire on the thin
film of soap solution when the child is not blowing
is
(a)
zero
(b)
2RT
(c)
4RT
(d)
none
The energy expended to form the bubble is
(a)
4Tr2
(b)
TR2
(c)
4Tr2 – TR2
(d)
none
The radius ‘r’ of the bubble in terms of T,  and v
at the time when it will blown out is
(a)
T
v
2
T
(b)
T
2v 2
4T
The speed of the water (that comes at t = 0) strikes
the ground is
(c)
(a)
10 m/s
(b)
12 m/s
Comprehension-16
(c)
14 m/s
(d)
16 m/s
Viscosity of highly viscous liquid can be determined
using the following appratus.
The kinetic energy per unit volume of the water
which comes out at t = 0 when it strikes the ground
is
(a)
50 kJ/m3
(b)
72 kJ/m3
(c)
98 kJ/m3
(d)
128 kJ/m3
4v 2
(d)
v 2
Time time taken to empty the tank to half its
original value is
(a)
2.5 h
(b)
1.5 h
(c)
1.25 h
(d)
none
The apparatus consists of a test tube contains the
experimental liquid (density 1260 kg/m3) and is
fitted into a water bath. A tube is fitted in the cork
of the test tube through which different metal ball
can be dropped. There are equidistant points P,Q
and R marked on the test tube which are separated
by 5 cm. The time taken by the ball to travel this
distance is measured by a stop watch of least count
0.1 s. The radius of the ball is measured by the screw
gauge which has the least count of 0.01 cm.
Let the time t after which the water completely
comes out from the tank is directly proportional to
hn (where h is the height of the container above the
ground) then the value of n is
(a)
zero
(b)
–½
(c)
½
(d)
1
Comprehension-15
70.
Einstein Classes,
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.
72.
If the radius of the sphere is 1cm and its mass is
50mg and time to travel the distance of each 5cm is
5 second, then the coefficient of viscosity of the
experimental liquid is
(a)
2 poise
(b)
4 poise
(c)
8 poise
(d)
none
The maximum possible percentage error in the
measurement of coefficient of viscosity is
(a)
1%
(b)
1.5%
(c)
2%
(d)
none
If length of water in sealed arm is 5 cm then  is
equal to
Comprehension-17
(a)
6.15 rad/s
(b)
7.15 rad/s
(c)
8.15 rad/s
(d)
9.15 rad/s
Comprehension-18
Suppose the beaker is accelerated and it has
components of acceleration ax and ay in x and y
directions respectively, then the pressure decreases
along both x and y directions. The equation for
pressure gradient is given by
dP
dP
 a x and
 (g  a y )
dx
dy
73.
A rectangular container of water undergoes
constant acceleration down an incline as shown.
A pitot tube is used to determine the airspeed of an
airplane. It consists of an outer tube with a
number of small holes B that allow air into the tube;
that tube is connected to one arm of a U-tube. The
other arm of the U-tube is connected to hole A at
the front end of the device, which points in the
direction the plane is headed. At A the air become
stagnant so that vA = 0. At B, however, the speed of
the air presumably equals the airspeed v of the
aircraft. Tube contains alcohol of density
810 kg/m3 and the value of h is 26 cm. The density
of air is 1.03 kg/m3. Assume that the acceleration
due to gravity does not change with height. Here h
is the difference in the fluid level in the tube.
The slope tan  of the free surface using the
coordinate system shown is
74.
75.
(a)
0.23
(b)
0.27
(c)
0.35
(d)
0.39
A liquid is kept in a cylindrical vessel which is
rotated along its axis. The liquid rises at the sides.
If the radius of the vessel is 0.05 m and the speed of
rotation is 2 rev/s, the difference in the height of
the liquid at the centre of the vessel and its sides is
(a)
1 cm
(b)
2 cm
(c)
2.5 cm
(d)
3 cm
Length of horizontal arm of a U-tube is 20 cm and
ends of both the vertical arms are open to a
pressure 1.01 × 103 N/m2. Water is poured into the
tube as shown in figure and one of the open ends is
sealed and the tube is then rotated about a vertical
axis passing through the other vertical arm with
angular velocity . Take density of
water = 10 3 kg/m 3 and g = 10 m/s 2 . Assume
temperature to be constant.
Einstein Classes,
76.
77.
78.
Pitot tube works on the principle of
(a)
principle of continuity
(b)
Bernoulli’s theorem
(c)
both (a) and (b)
(d)
none of these
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
A pitot tube on a high altitude aircraft measures a
differential pressure of 180 Pa. If the density of air
is 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 speed
but a different altitutes. Same types of pitot tube is
used to measure the speed of aircraft, then
(a)
(a)
2a 2 gh
( A 2  a 2 )
the value of h in the both pitot tube is
same
a 2gh
a 2 gh
(b)
(b)
the value of h in the pitot tube at higher
height is greater
(c)
(c)
the value of h in the pitot tube at higher
height is less
MATRIX-MATCH TYPE
none
Column - A
Column - B
(A)
Action of paint-gun
(p)
Bernoulli’s
theorem
(B)
Velocity of efflux
(q)
Torricelli’s
theorem
(C)
Brahma’s press
(r)
Pascal law
(D)
Venturi meter
(s)
Continuity
principle
(d)
Comprehension-19
2( A 2  a 2 )
(d)
( A 2  a 2 )
none
Matching-1
Matching-2
A venturi meter is used to measure the flow speed
of a fluid. The meter is connected between two
sections of the pipe, the cross-sectional area A of
the entrance and exit of the meter matches the pipe’s
cross-sectional area. Between the entrance and exit,
the fluid flows from the pipe with speed V and then
through a narrow “throat” of cross-sectional area
‘a’ with speed v. A manometer connects the wider
portion of the meter to the narrower portion. The
change in the fluid’s speed is accompanied by a
change p in the fluid’s pressure, which causes a
height difference h of the liquid in the two arms of
the manometer.
80.
81.
Column - A
Column - B
(A)
The ratio 1/2 if
(p)
rate of mass flow is same
1/2
(B)
The ratio of volume flow (q)
if rate of mass flow is same
2
(C)
The ratio of their speed
of efflux
(r)
1
(D)
The ratio of their
horizontal distance
(s)
4
Venturi meter works on the principle of
Matching-3
(a)
principle of continuity
Column - A
Column - B
(b)
Bernoulli’s theorem
(A)
(p)
[M 0L0T0]
(c)
both (a) and (b)
Dimension of modulus
of elasticity
(d)
none of these
(B)
Dimension of
coefficient of
viscosity
(q)
[ML–1T–2]
(C)
Dimension of
surface energy
(r)
[ML0T–2]
(D)
Dimension of
Reynold’s number
(s)
[ML–1T–1]
Suppose that the fluid is fresh water, that the
cross-sectional area are 64 cm2 in the pipe and
32 cm2 in the throat, and that the pressure is
55 kPa in the pipe and 41 kPa in the throat. The
rate of water flow in cubic meters per second is
(a)
(c)
82.
Two identical cylindrical tanks are filled with
different liquids of densities 1 and 2 . A small hole
is made in the side of each tank at the same depth h
below the surface of liquid. The hole in the tank
has area of cross-section twice that of hole in tank
B.
2.0 × 10
–2
–2
(b)
3.0 × 10
4.0 × 10
–2
(d)
5.0 × 10–2
The value of V in terms of a, A, h, density of fluid 
and density of liquid in manometer  is
Einstein Classes,
Matching-4
(A)
Column - A
Column - B
Variation of velocity
of falling rain drop
on time
(p)
exponential
increasing
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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(B)
Variation of
acceleration
of falling rain drop
on time
(q)
exponential
decreasing
(C)
Variation of
acceleration
of falling rain drop
on velocity
(r)
linear
(D)
Variation of power
due to net force on
falling rain drop
with speed
(s)
parabolic
4.
(a)
x1 = 5.2 kg
(b)
x2 = 1.5 kg
(c)
x2 = 1.3 kg
(d)
x1 = 5.0 kg
Figure shows a siphon in action. The liquid flowing
through the siphon has a density of 1.5 gm/cc. Then
MULTIPLE CORRECT CHOICE TYPE
1.
A bar of cross-section A is subjected to equal and
opposite tensile forces F at its ends. Consider a plane
through the bar making an angle  with a plane at
right angles to the bar.
(a)
(b)
(c)
(d)
2.
3.
The tensile stress at this plane is
maximum for  equal to zero.
The shearing stress at the plane is
maximum for  equals to 450.
The tensile stress at this plane is
maximum for  equal to 450.
5.
The shearing stress at the plane is
maximum for  equals to zero.
A cubical block of iron 5 cm on each side is floating
on mercury in a vessel. The relative density of
mercury is 13.6 and relative density of iron is 7.2.
Let the height of the block above mercury level is
h1. Now water is poured into the vessel so that it
just covers the iron block. The height of water
column is h2. Then
(a)
h1 = 2.35 cm
(b)
h2 = 2.54 cm
(c)
h1 = 2.54 cm
(d)
h2 = 2.35 cm
6.
A beaker containing water is kept on a spring
balance B1. The weight of beaker and water is 5 kg.
A piece of iron (specific gravity 7.5) weighing
1.5 kg is hung from a spring balance B2. If the iron
piece is lowered in water till it is fully immersed
but does not touch the bottom of the beaker, the
readings of B1 and B2 are x1 and x2 respectively.
7.
Einstein Classes,
(a)
The pressure difference at the points
A and D is zero
(b)
The pressure difference at the points
B and C is 2.65 × 104N/m2.
(c)
The liquid will flow from upper container
to lower container.
(d)
none of these
A large open top container contains a liquid upto
height H. A small hole is made in the side of the
tank at the height y from the bottom. The liquid
emmerges from the hole and lands at a distance x
from the tank.
(a)
If y is increased then x first increases and
then decreases.
(b)
x will be maximum when the hole is made
at the middle height.
(c)
The maximum possible value of x is H
(d)
x doesnot depend upon the density of the
liquid.
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 in
the liquid, the spring balance reads W3 and the
weighing machine reads W4. Then
(a)
W1 > W3
(b)
W2 < W4
(c)
W1 + W2 = W3 + W4
(d)
none
A wire forming a loop is dipped into soap solution
(surface tension T) and taken out such that a film
of soap solution is formed. A loop of length L of a
massless thread is gently put on the film and the
film is pricked with a needle inside the loop, then
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 23
(a)
The threaded loop takes the shape of a
circle
(b)
The threaded loop remains in the same
shape
(c)
The tension in the thread will become
STATEMENT-1 : If enough iron is added to one
end of a uniform wooden stick then it can float
vertically.
TL
2
STATEMENT-2 : For rotational equilibrium of
floating body the meta centre must always be higher
than the center of gravity of body.
(d)
There is no tension developed in the
thread
STATEMENT-2 : Center of buoyancy will coincide
with the centre of gravity of the displaced liquid.
7.
8.
Assertion-Reason Type
Each question contains STATEMENT-1 (Assertion)
and STATEMENT-2 (Reason). Each question has
4 choices (A), (B), (C) and (D) out of which ONLY
ONE is correct.
1.
(A)
Statement-1 is True, Statement-2 is True;
Statement-2 is a correct explanation
for Statement-1
(B)
Statement-1 is True, Statement-2 is True;
Statement-2 is NOT a correct
explanation for Statement-1
(C)
Statement-1 is True, Statement-2 is False
(D)
Statement-1 is False, Statement-2 is True
STATEMENT-2 : Bernoulli’s principle is based on
conservation of energy.
9.
10.
3.
11.
STATEMENT-1 : A piece of ice has a stone in it and
floats in a vessel containing water. When the ice
melts, the level of water in the vessel would fall.
STATEMENT-2 : The viscosity of liquid (except
water) increases with increment of pressure while
for gases it is independent of pressure.
12.
13.
STATEMENT-1 : Hydrolic pump and hydrolic
brake is based on Pascals law.
STATEMENT-1 : A body floats in a liquid contained
in a beaker. The whole system falls freely under
gravity. The upthrust on the body due to the liquid
is zero.
STATEMENT-2 : In case of free fall, the effective
acceleration due to gravity is zero.
6.
STATEMENT-2 : The initial speed of efflux will be
same in both cases.
STATEMENT-2 : The bulk modulus of elasticity
for all types of substance is non-zero.
STATEMENT-2 : Pressure applied to an enclosed
fluid is transmitted undiminished to every portion
of the fluid and the wall of the containing vessel.
5.
STATEMENT-1 : Two large tanks a and b, open at
the top, contains different liquids. A small hole is
made in the side of each tank at the same depth h
below the liquid surface, but the hole in a has twice
the area of the hole in b. The ratio of the densities
of the liquids in a and b so that the mass flux is the
same for each hole should be 0.5.
STATEMENT-1 : If temperature rises, the
coefficient of viscosity of a liquid decreases while
for gases increases.
STATEMENT-2 : The Buoyancy force will arise due
to the vertical pressure gradient.
4.
STATEMENT-1 : Bernoulli’s equation is applicable
in the case of stremlined flow of incompressible.
STATEMENT-2 : Principle of continuity is based
on conservation of mass.
STATEMENT-1 : A wire can support a load W
without breaking. It is cut into two equal parts. The
maximum load that each part can support is W/2.
STATEMENT-1 : The modulus of rigidity (shear
modulus of elasticity) of liquid and gas is zero.
STATEMENT-1 : When a spinning ball is thrown,
it deviates from its usual path in flight.
STATEMENT-2 : In accordance with Bernoulli’s
principle, a pressure difference above and below
the ball will develop.
STATEMENT-2 : The young’s modulus of
elasticity of the wire does not depend on the length
of the wire.
2.
STATEMENT-1 : The rate of leak from a hole in a
tank is more if situated near the bottom.
STATEMENT-1 : In order that a floating object be
in stable equilibrium, its centre of buoyancy should
be vertically above its centre of gravity
Einstein Classes,
STATEMENT-1 : A metal ball immersed in alcohol
weighs W1 at 00C and W2 at 500C. The coefficient of
cubical expansion of the metal is less than that of
alcohol, assuming that the density of the metal is
large compared to that of alcohol, then W1 < W2.
STATEMENT-2 : Density decreases with the
increase of temperature.
14.
STATEMENT-1 : An iceberg is floating partially
immersed in sea water. If the density of sea water is
1.03 g/cc and that of ice is 0.92 g/cc, the fraction of
the total volume of iceberg above the level of sea
water is 0.11.
STATEMENT-2 : It is due to force of buoyancy.
15.
STATEMENT-1 : A loaded boat enters the sea from
the river, it rises.
STATEMENT-2 : Sea water is denser then the river
water.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 24
16.
STATEMENT-1 : When a piece of ice floating in a
beaker of water completely melts, the level of the
water in the beaker will slightly change.
STATEMENT-2 : The density of ice is slightly lower
than the density of water.
17.
STATEMENT-1 : A wooden piece floats half
submerged in a tub of water. If the system is
transferred to a lift ascending with acceleration, the
piece will remain half sub-merged.
STATEMENT-2 : The pressure gradient will
change.
19.
STATEMENT-1 : People living in houses far remote
from a municipal water tank often find it difficult
to get water on the top floor even if it is situated
lower than the level of water tank.
26.
27.
21.
28.
29.
23.
30.
24.
31.
STATEMENT-1 : Stress and pressure are the
different concept.
STATEMENT-1 : Consider a massless rod which is
suspended from a ceiling and a block of mass m is
attached to a lower end. The change in the length
of the wire is x. Another uniform wire of the same
material, same initial length and same
cross-section but has mass m is suspended from the
ceiling and the block is not attached. The change in
length of this wire is x/2.
STATEMENT-1 : There are different types of
vessel with the same base area and they are filled
with the same liquid of different mass but upto same
height. The force exerted on the base in each case
will be different.
STATEMENT-2 : The force exerted on the base may
be greater than, less than or equal to the weight of
the liquid.
STATEMENT-1 : Hooke’s law is valid at all stress.
STATEMENT-2 : Both are defined as force per unit
area.
STATEMENT-1 : When a capillary tube is dipped
in a liquid, the liquid rises to a height h in the tube.
The free liquid surface inside the tube is
hemispherical in shape. The tube is now pushed
down so that the height of the tube outside the
liquid is less than h. The liquid will flow out of the
tube slowly.
STATEMENT-2 : The angle of contact at the free
liquid surface inside the tube will change.
STATEMENT-1 : A thin steel needle floats on
water but when a little soap solution is carefully
mixed with the water the needle sinks.
STATEMENT-2 : The slope of stress vs strain graph
in the proportional limit equals to modulus of
elasticity.
STATEMENT-1 : N drops of a liquid join to form a
single drop. In this process some energy will be
released.
STATEMENT-2 : It is due to change of surface
energy.
STATEMENT-2 : When a detergent is added to
water its surface tension will suddenly decrease.
22.
STATEMENT-1 : The profile of advancing liquid
in a tube is a parabola.
STATEMENT-2 : Rate of volume flow of a liquid
through the tube doesnot depend on the coefficient
of viscosity.
STATEMENT-1 : Water is flowing through a
horizontal pipe of uniform cross-section under
constant pressure. At some place the pipe becomes
narrow; the pressure of water at this place
decreases.
STATEMENT-2 : The pressure energy will be
converted into kinetic energy.
STATEMENT-1 : Stoke’s Law is valid only for
spherical bodies.
STATEMENT-2 : Viscous force will be experienced
by all types of bodies when they move through the
viscous liquid.
STATEMENT-2 : There is loss of pressure when
water is flowing.
20.
STATEMENT-1 : Density of a fluid changes with
change in temperature.
STATEMENT-2 : Density of a liquid doesnot change
with change in pressure.
STATEMENT-1 : A ball floats on the surface of
water in a container exposed to atmosphere. If the
container is covered and air is compressed, the ball
will sink.
STATEMENT-2 : The force of buoyancy will
decrease.
18.
25.
STATEMENT-1 : A container with a liquid is placed
in a gravity free surrounding. A hole is made at the
bottom of the container then after certain time the
container will become empty.
STATEMENT-2 : Bernoulli’s theorem is applicable
only for non-viscous liquid.
32.
STATEMENT-1 : Two stream line curve does not
intersect each other.
STATEMENT-2 : In a stream line flow the kinetic
energy and momentum of all the particles arriving
at a given point are the same.
33.
STATEMENT-1 : Surface tension and coefficient
of viscosity is a property exist only for the liquid.
STATEMENT-2 : Surface tension and surface
energy are numerically equal.
STATEMENT-2 : In both wire, work done in
stretching is the same.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 25
34.
STATEMENT-1 : The viscous force per unit area
between two layer of liquid is shear stress or
tangential stress.
STATEMENT-2 : The origin of viscous force is
electromagnetic.
35.
36.
STATEMENT-2 : Due to the property of surface
tension, there is a tendency to acquire minimum
surface area. For the given volume the sphere has
the minimum surface area.
37.
STATEMENT-1 : The capillary rise in a tube is less
STATEMENT-1 : An ice cube suspended in vacuum
in a gravity free space melts. When it melts, its shape
changes to spherical.
than
STATEMENT-2 : Due to the property of surface
tension, there is a tendency to acquire minimum
surface area. For the given volume the sphere has
the minimum surface area.
STATEMENT-2 : It is due to weight of the liquid
contained in the meniscous.
2T
where the symbols have their usual
gr
meanings.
38.
STATEMENT-1 : Rain drops are spherical.
STATEMENT-1 : During a tornado, when a high
speed wind blows over a roof, it blows off the roof.
STATEMENT-2 : According the Beronulli’s
principle, a low pressure created at the top of the
roof.
(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
2.
A-p; B-q; C-r; D-s
3.
A-q; B-s; C-r; D-p
3.
[a, c]
4.
[a, b, c]
5.
[a, b, c, d]
MATRIX-MATCH TYPE
1.
A-p; B-p, q, s; C-r; D-p, s
4.
A-p; B-q; C-r; D-s
MULTIPLE CORRECT CHOICE TYPE
1.
[a, b]
2.
[a, b]
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
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 26
INITIAL STEP EXERCISE
(SUBJECTIVE)
1.
2.
3.
The fresh water behind a reservoir dam is 15 m
deep. A horizontal pipe 4.0 cm in diameter passes
through the dam 6.0 m below the water surface as
shown in figure A plug secures the pipe opening.
(a) Find the friction force between the plug and pipe
wall. (b) The plug is removed. What volume of
water flows out of the pipe in 3.0 hours ?
A stone of 0.5 kg mass is attached to one end of a
0.8 m long aluminium wire of 0.7 mm diameter and
suspended vertically. The stone is now rotated in a
horizontal plane at a rate such that the wire makes
an angle of 850 with the vertical. Find the increase
in the length of the wire. [Young’s modulus of
aluminium = 7 × 1010 N/m2; sin 850 = 0.9962 and
cos 85 = 0.0872].
(a)
A fluid is rotating at constant angular
velocity  about the central vertical axis
of a cylindrical container. Show that the
variation of pressure in the radial
direction is given by
5.
6.
7.
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 P0. 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.
dp
 2 r
dr
(b)
4.
Take p = pc at the axis of rotation (r = 0)
and show that the pressure p at any point
r is p = pc + ½r2r2
(c)
Show that the liquid surface is of
paraboloidal form; that is, a vertical cross
section of the surface is the curve
y = 2r2/2g.
A non-viscous liquid of constant density
1000 kg/m3 flows in a stream line motion along the
tube of variable cross-section. The tube is kept
inclined in the vertical plane as shown in fig. The
area of cross-section of the tube at two points P and
Q at heights of 2 metres and 5 metres are
respectively 4 × 10–3 m2 and 8 × 10–3m2. The velocity
of the liquid at point P is 1 m/s.
Find the work done per unit volume by the
Einstein Classes,
pressure and the gravity forces as the fluid flows
from point P to Q.
A glass capillary sealed at the upper end is of length
0.11 m and internal diameter 2 × 10–5 m. The tube
is immersed vertically into a liquid of surface
tension 5.06 × 10–2 N/m. To what length has the
capillary to be immersed so that the liquid level
inside and outside the capillary becomes the
same ? What will happen to the water level inside
the capillary if the seal is now broken ?
A cylindrical vessel of radius R is filled with water
to a height of h. It has a capillary tube of length l
and radius ‘r’ protruding horizontally at its
bottom. If the viscosity of water is , find the time
in which the level will fall to a height of h/2.
A liquid is flowing through horizontal pipes as
shown in figure. Length of different pipes has the
following ratio LAB : LCD : LEF : LGH = 1 : 1 : 2 : 2
8.
A schematic view of a hydraulic jack used to lift an
automobile. The hydraulic fluid is oil
(density = 812 kg/m3). A hand pump is used, in
which a force of magnitude Fi is applied to the
smaller piston (of diameter 2.2 cm) when the hand
applies a magnitude Fh to the end of the pump
handle. The combined mass of the car to be lifted
and the lifting platform is M = 1980 kg, and the
large piston has a diameter of 16.4 cm. The length
L of the pump handle is 36 cm, and the distance x
from the pivot to the piston is 9.4 cm. (a) What is
the applied force Fh needed to lift the car ? (b) For
each downward stroke of the pump, in which the
hand moves a vertical distance of 28 cm, how far is
the car raised ?
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 27
FINAL STEP EXERCISE
(SUBJECTIVE)
1.
2.
A rod of length 6 m has a mass 12 kg. It is hinged at
one end at a distance of 3 m below water surface.
(a) What weight must be attached to the other end
of the rod so that 5 m of the rod is submerged ? (b)
Find the magnitude and direction of the force
exerted by the hinge on the rod. (Specific gravity of
rod is 0.5).
7.
A hollow cone of radius R and height H is placed
on a horizontal surface at its base. If it is filled with
water (density ) to a height h, find the net force
exerted by water on the curved surface of the cone.
8.
A liquid of density  is filled in a tank of upper
radius r1 and lower radius r2 as shown in figure.
Under isothermal condition two soap bubbles of
radii a and b coalesce to form a single bubble of
radius c. If the external pressure is p0, show that
surface tension,
p 0 (c 3  a 3  b 3 )
T
4(a 2  b 2  c 2 )
3.
4.
A conical glass capillary tube of length 0.1 m has
diameters 10–3 and 5 × 10–4 m at the ends. When it
is just immersed in a liquid at 0 0C with larger
diameter in contact with it, the liquid rises to
8 × 10–2 m in the tube. If another cylindrical glass
capillary tube B is immersed in the same liquid at
00C, the liquid rises to 6 × 10–2 m height. The rise of
liquid in the tube B is only 5.5 × 10–2 m when the
liquid is at 500C. Find the rate at which the surface
tension changes with temperature considering the
change 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 glass
is negligible.
6.
9.
A cone made of a material of relative density
27 

s 
 and height 4 m floats with its apex
64 

downward in a big vessel containing water.
5.
A capillary tube of length L and inner radius a and
outer radius b is attached at the bottom as shown
in figure. It has been found that the rate of volume
flow through the tube is if pressure p is applied
at the top of the tank . Now the tube is detached
then then velocity of the liquid is v0 coming out of
the hole. Find the coefficient of viscosity of the
liquid.
(a)
Find the submerged height of cone in
water
(b)
Find the time period of vertical
oscillation if it is slightly disturbed from
the equilibrium position
A wooden stick of length L, radius R and density 
has a small metal piece of mass m (of negligible
volume) attached to its one end. Find the minimum
value for the mass m (in terms of given parameters)
that would make the stick float vertically in
equilibrium in a liquid of density .
10.
A fluid with viscosity  fills the space between two
long co-axial cylinders of radii R1 and R2, with
R1 < R2. The inner cylinder is stationary while the
outer one is rotated with a constant angular
velocity 2. The fluid flow is laminar. Taking into
account that the friction force acting on a unit area
of a cylindrical surface of radius r is defined by the
formula  = r (/r), find :
(a)
the angular velocity of the rotating fluid
as a function of radius r;
(b)
the moment of the friction forces acting
on a unit length of the outer cylinder.
A tube of length l and radius R carries a steady
flow of fluid whose density is  and viscosity . The
fluid flow velocity depends on the distance r from
the axis of the tube is v = v0 (1 – r2/R2). Find :
(a)
the volume of the fluid flowing across the
section of the tube per unit time;
(b)
the kinetic energy of the fluid within the
tube’s volume;
(c)
the friction force exerted on the tube by
the fluid;
(d)
the pressure difference at the ends of the
tube.
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 raduce
to radius r2 < r1. Surface tension of soap is T and
coefficient of viscosity of air is .
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PPOM – 28
ANSWERS SUBJECTIVE
(INITIAL STEP EXERCISE)
1.
(a)
2.33 kg
(b)
56.7 N (downward)
(a)
2.
1.668 × 10–3 m
4.
29025 J, –2.64 × 104 J/m3
3.
–1.4 × 10–2 N/m0C
5.
.01 m
4.
(a)
3m
(b)
1.98 s
8  lR 2
7.
gr 4
(b)
150 m3
1.
6.
74 N
ANSWERS SUBJECTIVE
(FINAL STEP EXERCISE)
ln 2
Volume rate flow through EF is
through GH is
16
Q
17
Q
and
17
5.
 

R 2 L 
 1
 

6.
t0 
7.
F
9.
(a)
  2
(b)
N  4 2
(a)
Q = ½ v0R2
(b)
T = 1/6 lR2v02
(c)
Ffr = 4lv0
(d)
p = 4lv0/R2
PE = 1.53 P0, PF = 1.47 P0
8.
(a)
91 N
(b)
1.3 mm
10.
Einstein Classes,
2l
4
a T
(r14  r24 )

1
(H  h)3 
R 2 g  3h  H 

3
H 2 

R 12 R 22  1
1 
 2
2
2 
2
R 2  R 1  R 1 r 
R 12 R 22
R 22  R 12
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111