Physics

5.7.4 Escape Velocity: 5.8 Earth Satellites:
The objects which revolve around the
When any object is thrown vertically up, it
Earth are called Earth satellites. moon is the
falls back to the Earth after reaching a certain only natural satellite of the Earth. It revolves
in almost a circular orbit around the Earth
height. Higher the speed with which the object is with period of revolution of nearly 27.3 days.
Artificial satellites have been launched by
thrown up, greater will be the height. If we keep several countries including  India. These
satellites have different periods of revolution
on increasing the velocity, a stage will come according to their practical use like navigation,
surveillance, communication, looking into
when the object will reach heights so large that space and monitoring the weather.
Communication Satellites: These are
it will escape the gravitational field of the Earth geostationary satellites.They revolve around the
Earth in equatorial plane. They have same sense
and will not fall back on the Earth. This initial of rotation as that of the Earth and the same period
of rotation as that of the Earth, i. e., one day
velocity is called the escape velocity. or 24 hours. Due to this, they appear stationary
from the Earth’s surface. Hence they are called
Thus, the minimum velocity with which a geostationary satellites or geosynchronous
satellites. These are used for communication,
body should be thrown vertically upwards from television transmission, telephones and
radiowave signal transmission, e.g., INSAT
the surface of the Earth so that it escapes the group of satellites launched by India.
Polar Satellites: These satellites are placed
Earth’s gravitational field, is called the escape in lower polar orbits. They are at low altitude
500 km to 800 km. Polar satellites are used
velocity (ve) of the body. Obviously, as the for weather forecasting and meteorological
purpose. They are also used for astronomical
gravitational force due to Earth becomes zero observations and study of Solar radiations.

only at infinite distance, the object has to reach Period of revolution of polar satellite is
nearly 85 minutes, so it can orbit the Earth16
infinite distance in order to escape. time per day. They go around the poles of the
Earth in a north-south direction while the Earth
Let us consider the kinetic and potential rotates in an east-west direction about its own
axis. The polar satellites have cameras fixed
energies of an object thrown vertically upwards on them. The camera can view small stripes of
the Earth in one orbit. In entire day the whole
with escape velocity ve, when it is at the surface Earth can be viewed strip by strip. Polar and
of the Earth and when it reaches infinite distance. equatorial regions at close distances can be
viewed by these satellites.
On the surface of the Earth,
5.8.1 Projection of Satellite:
K.E. = 1 mv 2
2 e For the projection of an artificial satellite,

PTo.Eta.l=en-eGrgMRy m= P.E. + K.E. it is necessary for the satellite to have a certain

= 1 mv 2 - GMm --- (5.33) velocity and a minimum two stage rocket. A
2 e R
single stage rocket can not achieve this. When
The kinetic energy of the object will go
the fuel in first stage of rocket is ignited on
on decreasing with time as it is pulled back by
the surface of the Earth, it raises the satellite
Earth’s gravitational force. It will become zero

when it reaches infinity. Thus at infinite distance

from the Earth

K.E. = 0

Also, P.E. = GMm = 0
f

∴ Total energy = P.E. + K.E. = 0

As energy is conserved

1 mv 2 - GMm =0
2 e R

or, v = 2GM --- (5.34)
e R

Using the numerical values of G, M and R.

the escape velocity is 11.2 km/s.

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vertically. The velocity of projection of satellite During this elliptical path, if the satellite
normal to the surface of the Earth is the vertical
velocity. If this vertical velocity is less that the passes through the Earth’s atmosphere, it
escape velocity (v ), the satellite returns to the
experiences a nonconservative force of air
e
resistance. As a result it loses energy and spirals
Earth’s surface. While, if the vertical velocity is
greater than or equal to the escape velocity, the down to the Earth.
satellite will escape from Earth’s gravitational
influence and go to infinity. Hence launching Case (II) vh=vc
of a satellite in an orbit round the Earth can If the horizontal velocity is exactly equal
not take place by use of single stage rocket. It
requires minimum two stage  rocket. to the critical velocity, the satellite moves in a

With the help of first stage of rocket, stable circular orbit round the Earth.
satellite can be taken to a desired height above
the surface of the Earth. Then the launcher is Case (III) vc<vh<ve
rotated in horizontal direction i.e. through 900 If horizontal velocity is greater than
using remote control and the first stage of the
rocket is detached. Then with the help of second the critical velocity and less than the escape
stage of rocket, a specific horizontal velocity
(v ) is given to satellite so that it can revolve velocity at that height, the satellite again moves

h in an elliptical orbit round the Earth with the

in a circular path round the Earth. The exact point of projection as perigee (point closest to
horizontal velocity of projection that must be
given to a satellite at a certain height so that it the Earth).
can revolve in a circular orbit round the Earth is
called the critical velocity or orbital velocity Case (IV) vh = ve
(vc) If horizontal speed of projection is equal

Asatellite follows different paths depending to the escape speed at that height, the satellite
upon the horizontal velocity provided to it. Four
different possible cases are shown in Fig. 5.10. travels along parabolic path and never returns
Case (I) vh<vc:
to the point of projection. Its speed will be zero
If tangential velocity of projection vh is less
than the critical velocity, the orbit of satellite is at infinity.
an ellipse with point of projection as apogee
(farthest from the Earth) and Earth at one of the Case (V) vh > ve
foci. If horizontal velocity is greater than the

Fig. 5.10: Various possible orbits depending escape velocity, the satellite escapes from
on the value of vh.
gravitational influence of Earth transversing a

hyperbolic path.

Expression for critical speed

Consider a satellite of mass m revolving

round the Earth at height h above its surface.

Let M be the mass of the Earth and R be

its radius. If the satellite is moving in a circular

orbit of radius (R+h) = r, its speed must be the

magnitude of critical velocity vc.
The centripetal force necessary for circular

motion of satellite is provided by gravitational

force exerted by the Earth on the satellite.

∴ Centripetal force = Gravitational force

mv 2 = GMm
c r2

r

? vc2 = GM
r

?vc = GM
r

?vc = GM gh (R +h) --- (5.35)
(R +h)

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This is the expression for critical speed in ?vc = 2R GSU
the orbit of radius (R + h) 3

It is clear that the critical speed of a When a satellite revolves very close to
satellite is independent of the mass of the
satellite. It depends upon the mass of the Earth the surface of the Earth, motion of satellite
and the height at which the satellite is revolving
or gravitational acceleration at that altitude. gets affected by the friction produced due to
The critical speed of a satellite decreases with
increase in height of satellite. resistance of air. In deriving the above expression
Special case
the resistance of air is not considered.
When the satellite is revolving close to the
surface of the Earth, the height is very small as 5.8.2 Weightlessness in a Satellite:
compared to the radius of the Earth. Hence the
height can be neglected and radius of the orbit is According to Newton’s second law of
nearly equal to R (i.e R>>h, R+h ≈ R) motion, F = ma , where F is the net force acting
on an object having acceleration a.

Let us consider the example of a lift or
elevator from an inertial frame of reference.

... Critical speed vc = GM Whether the lift is at rest or in motion, a
R passenger in it experiences only two forces:
As G is related to acceleration due to (i) Gravitational force mg directed vertically
downwards (towards centre of the earth) and
gravity by the relation, (ii) normal reaction force N directed vertically
upwards, exerted by the floor of the lift. As these
g = GM forces are oppositely directed, the net force in
R2 the downward direction will be F = ma - N .

∴GM = gR2 Though the weight of a body (passenger, in
... Critical speed in terms of acceleration this case) is the gravitational force acting upon
due to gravity can be obtained as it, we experience or feel our weight only due
to the normal reaction force N exerted by the
vc = gR2 = gR floor. This, in turn, is equal and opposite to the
R relative force between the body and the lift. If
you are standing on a weighing machine in a
= 7.92 km/s lift, the force recorded by the weighing machine
Obviously, this is the maximum possible critical is nothing but the normal reaction N.
speed. This is at least 25 times the speed of the
fastest passenger aeroplanes. Case I: Lift having zero acceleration
Example 5.9: Show that the critical velocity of
a body revolving in a circular orbit very close This happens when the lift is at rest or is
to the surface of a planet of radius R and mean moving upwards or downwards with constant
velocity:
density ρ is 2R GSU .
3
Solution : Since the body is revolving very

close to the planet, h = 0 The net force F = 0 = mg - N ∴ mg =N

density U M = M Hence in this case we feel our normal
V weight mg .
4 S R3
3
Case II: Lift having net upward acceleration a
?M = 4 S R3 U u
3
This happens when the lift just starts
Critical Velocity moving upwards or is about to stop at a lower
floor during its downward motion (remember,
GM G 4 S R3 U while stopping during downward motion, the
R 3 acceleration must be upwards).
vc = =
R

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As the net acceleration is upwards, the not falling on the earth? The reason is that the
upward force must be greater. revolving satellite is having a tangential velocity
which manages to keep it moving in a circular
m∴gF, h=enmcea,u = N - mg ∴ N = mg + mau, i.e., orbit at that height.
we feel heavier. 5.8.3 Time Period of a Satellite:
N >

It should also be remembered that this is The time taken by a satellite to complete
not an apparent feeling. The weighing machine one revolution round the Earth is its time period.
really records a reading greater than mg.

Case III (a): Lift having net downward Consider a satellite of mass m projected to
acceleration a height h and provided horizontal velocity equal
to the critical velocity. The satellite revolves in
d a circular orbit of radius (R+h) = r.

This happens when the lift just starts The distance traced by satellite in one
moving downwards or is about to stop at a higher
floor during its upward motion (remember, revolution is equal to the circumference of the
while stopping during upward motion, the
acceleration must be downwards). circular orbit within periodic time T.

As the net acceleration is downwards, the ?Critical speed = Circumference of the orbit
downward force must be greater. Time period

∴ F = ma = mg -N ∴ N = mg - ma , i.e., vc = 2S r
N < mg, hence,dwe feel lighter. d T

It should be remembered that this is not an but we have, vc = Gm
apparent feeling. The weighing machine really r
records a reading less than mg.
? Gm 2S r
r T
Case III (b): State of free fall: This will be
possible if the cables of the lift are cut. In this or, GM 4S 2r2
case, the downward acceleration ad = g. r T2

If the downward acceleration becomes ? T2 4S 2r3
equal to the gravitational acceleration g, we get, GM
N = mg - mad = 0.
As π2, G and M are constant, T 2 ∝ r3, i.e.,
Thus, there will not be any feeling of the square of period of revolution of satellite is
weight. This is the state of total weightlessness directly proportional to the cube of the radius
and the weighing machine will record zero. of orbit.

In the case of a revolving satellite, the T = 2S r3
satellite is performing a circular motion. The GM
acceleration for this motion is centripetal, which
is provided by the gravitational acceleration ?T 2S (R + h)3 --- (5.36)
g at the location of the satellite. In this case, GM
ad = g, or the satellite (along with the astronaut)
is in the state of free fall. Obviously, the apparent This is an expression for period of satellite
weight will be zero, giving the feeling of total revolving in a circular orbit round the Earth.
weightlessness. Perhaps you might have seen Period of a satellite does not depend on its
in some videos that the astronauts are floating mass. It depends on mass of the Earth, radius
inside the satellite. It is really difficult for them of the Earth and the height of the satellite. If
to change their position. the height of projection is increased, period of
the satellite increases. Period of the satellite can
In spite of free fall, why is the satellite also be obtained in terms of acceleration due to

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gravity. density (U ) = mass (M )
volume (V )
As GM = gh (R+h)2

?T = 2S (R +h)3 ?M = UV --- (2)
gh(R + h)2
As planet is spherical in shape, volume of

?T = 2S R +h planet is given as
gh
V = 4 S R3
3

?T = 2S r --- (5.37) ? M = 4 S R3U --- (3)
gh 3



Special case : Substituting the values form eq. (2) and (3)

When satellite revolves close to the surface in Eq. (1), we get

of the Earth, R + h r≈evRoluatnidongihs≈ g. Hence the T = 2S R3
minimum period of
Gu 4 S R3U
3
R
(T)min = 2π --- (5.38) 3S
g ?T = GU

Example 5.9: Calculate the period of revolution
of a polar satellite orbiting close to the surface
of the Earth. Given R = 6400 km, g = 9.8 m/s2. 5.8.4 Binding Energy of an orbiting satellite:

The minimum energy required by a satellite

Solution : h is negligible as satellite is close to to escape from Earth’s gravitational influence is
the Earth surface.
the binding energy of the satellite.
... R + h ≈ R
Expression for Binding Energy of satellite

revolving in circular orbit round the Earth

gh ≈ g Consider a satellite of mass m revolving

at height h above the surface of the Earth in a

R = 6400 km = 6.4×106 m. circular orbit. It possesses potential energy as

T = 2S R well as kinetic energy. Let M be the mass of the
g
Earth, R be the Radius of the Earth, vc be critical
6.4 u106 velocity of satellite, r = (R+h) be the radius of
9.8
= 2u 3.14 the orbit.

∴Kinetic energy of satellite

= 5.075u103second = 1 mvc2
2
= 85 minute (approximately)
1 GMm
Example 5.10: An artificial satellite revolves = 2r --- (5.39)
around a planet in circular orbit close to its
surface. Obtain the formula for period of the The gravitational potential at a distance r
satellite in terms of density ρ and radius R of
planet. from the centre of the Earth is - GM
... Potential energy of satellite r
Solution : Period of satellite is given by,
= Gravitational

(R +h)3 potential × mass of satellite
GM
T = 2π --- (1) = - GMm --- (5.40)
r
Here, the satellite revolves close to the
The total energy of satellite is given as
surface of planet, hence h is negligible, hence
R+h  R T.E. = K.E. + P.E.

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= 1 GMm - GMm influence its total energy should become non-
2 r r
negative (zero or positive). Hence the minimum

= - 1 GMm energy to be supplied to unbind the satellite
2r
--- (5.41) is + 1 GMm This is the binding energy of a
2r
satellite.
Total energy of a circularly orbiting
satellite is negative. Negative sign indicates Internet my friend
that the satellite is bound to the Earth, due
to gravitational force of attraction. For the hyperphysics.phy-astr.gsu.edu/hbase/grav.
satellite to be free from the Earth’s gravitational html#grav

ExercisesExercises

1. Choose the correct option. 2. Answer the following questions.
i) The value of acceleration due to gravity is
i) State Kepler’s law equal of area.
maximum at
(A) the equator of the Earth . ii) State Kepler’s law of period.
(B) the centre of the Earth.
(C) the pole of the Earth. iii) What are the dimensions of the universal
(D) slightly above the surface of the gravitational constant?

Earth. iv) Define binding energy of a satellite.
ii) The weight of a particle at the centre of the
v) What do you mean by geostationary
Earth is satellite?
(A) infinite.
(B) zero. vi) State Newton’s law of gravitation.
(C) same as that at other places.
(D) greater than at the poles. vii) Define escape velocity of a satellite.
iii) The gravitational potential due to the Earth
viii)  What is the variation in acceleration due
is minimum at to gravity with altitude?
(A) the centre of the Earth.
(B) the surface of the Earth. ix) On which factors does the escape speed
(C) a points inside the Earth but not at of a body from the surface of Earth
its centre. depend?
(D) infinite distance.
iv) The binding energy of a satellite revolving x) As we go from one planet to another
planet, how will the mass and weight of
around planet in a circular orbit is 3×109 J. a body change?
Its kinetic energy is
(A) 6×109J xi) What is periodic time of a geostationary
(B) -3 ×109J satellite?
(C) -6 ×10+9J
(D) 3 ×10+9J xii) State Newton’s law of gravitation and
express it in vector form.

xiii) What do you mean by gravitational
constant? State its SI units.

xiv) Why is a minimum two stage rocket
necessary for launching of a satellite?

xv) State the conditions for various possible
orbits of a satellite depending upon the
tangential speed of projection.

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2. Answer the following questions in xiv) What is critical velocity? Obtain an
detail. expression for critical velocity of an
orbiting satellite. On what factors does it
i) Derive an expression for critical velocity depend?
of a satellite.

ii) State any four applications of a xv) Define escape speed. Derive an
communication satellite. expression for the escape speed of an
object from the surface of the each.
iii) Show that acceleration due to gravity
at height h above the Earth’s surface is xvi) Describe how an artificial satellite using
two stage rocket is launched in an orbit
gh g § R R h ·2 around the Earth.
©¨ ¸¹
iv) Drawalabelleddiagramtoshowdifferent 4. Solve the following problems.

trajectories of a satellite depending upon i) At what distance below the surface of
the Earth, the acceleration due to gravity
the tangential projection speed. decreases by 10% of its value at the
surface, given radius of Earth is 6400
v) Derive an expression for binding energy km.
of a body at rest on the Earth’s surface.

vi) Why do astronauts in an orbiting satellite [Ans: 640 km].
have a feeling of weightlessness?
ii) If the Earth were made of wood, the mass
vii) Draw a graph showing the variation
of gravitational acceleration due to of wooden Earth would have been 10%
the depth and altitude from the Earth’s
surface. as much as it is now (without change in

its diameter). Calculate escape speed

viii) At which place on the Earth’s surface from the surface of this Earth.
is the gravitational acceleration
maximum? Why? [Ans: 3.54 km/s]

ix) At which place on the Earth surface the iii) Calculate the kinetic energy, potential
gravitational acceleration minimum? energy, total energy and binding energy
Why? of an artificial satellite of mass 2000 kg
orbiting at a height of 3600 km above the
x) Define the binding energy of a satellite. surface of the Earth.
Obtain an expression for binding energy
of a satellite revolving around the Earth Given:- G = 6.67×10-11 Nm2/kg2
at certain attitude.
R = 6400 km

M = 6×1024 kg

xi) Obtain the formula for acceleration due [Ans: KE = 40.02×109J,
to gravity at the depth ‘d’ below the
Earth’s surface. PE = -80.09 ×109J,

xii) State Kepler’s three laws of planetary TE = 40.02 ×109J,
motion. BE = 40.02×109J]

xiii) State the formula for acceleration due iv) Two satellites A and B are revolving
around a planet. Their periods of
to gravity at depth ‘d’ and altitude ‘h’ revolution are 1 hour and 8 hours
respectively. The radius of orbit of
Hence show that their ratio is equal to satellite B is 4×104 km. find radius of
§ R d · orbit of satellite A .
¨© R 2h ¸¹ by assuming that the altitude [Ans: 1×104 km]

is very small as compared to the radius

of the Earth.

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v) Find the gravitational force between the x) Calculate the value of the universal

Sun and the Earth. gravitational constant from the given

Given Mass of the Sun = 1.99×1030 kg data. Mass of the Earth = 6×1024 kg,
Radius of the Earth = 6400 km and the
Mass of the Earth = 5.98×1024 kg acceleration due to gravity on the surface
= 9.8 m/s2
vi) The average distance between the Earth
[Ans : 6.69×10-11 N m2/kg2 ]
and the Sun = 1.5×1011 m.
A body weighs 5.6 kg wt on the surface
[Ans: 3.5×1022 N] xi) of the Earth. How much will be its
weight on a planet whose mass is 7 times
Calculate the acceleration due to gravity the mass of the Earth and radius twice
that of the Earth’s radius.
at a height of 300 km from the surface of

the Earth. (M = 5.98 ×1024 kg, R = 6400

km).

[Ans :- 8.889 m/s2] [Ans: 9.8 kg-wt]

vii) Calculate the speed of a satellite in an xii). What is the gravitational potential due to

orbit at a height of 1000 km from the the Earth at a point which is at a height

Earth’s surface. ME= 5.98×1024 kg, R = oMfa2sRs EofatbhoevEeatrhthe surface of the Earth,
6.4×106 m. is 6×1024 kg, radius of

[Ans : 7.34 ×103 m/s] the Earth = 6400 km and G = 6.67×10-11

viii) Calculate the value of acceleration due Nm2 kg-2.

to gravity on the surface of Mars if the [Ans: 2.08×107 J]

radius of Mars = 3.4×103 km and its

mass is 6.4×1023 kg. ***

[Ans : 3.69 m/s2]

ix) Aplanet has mass 6.4 ×1024 kg and radius

3.4×106 m. Calculate energy required to

remove on object of mass 800 kg from

the surface of the planet to infinity.

[Ans : 5.02 ×1010J]

99

6. Mechanical Properties of Solids

Can you recall?

1. Can you name a few objects which change their shape and size on application of a force
and regain their original shape and size when the force is removed ?

2. Can you name objects which do not regain their original shape and size when the external
force is removed?

6.1 Introduction: the larger is its deformation. Deformation
Solids are made up of atoms or a group could be in the form of change in length of a
wire, change in volume of an object or change
of atoms placed in a definite geometric in shape of a body.
arrangement. This arrangement is decided by
nature so that the resultant force acting on each We know that when a deforming force
constituent due to others is zero. This is the (e.g. stretching) is applied to a rubber band, it
equilibrium state of a solid at room temperature. gets deformed (elongated) but when the force
The given equilibrium arrangement does not is removed, it regains its original length. When
change with time. It can change only when an a similar force is applied to a dough, or clay
external stimulus, like compressive force from it also gets deformed but it does not regain its
all sides, is applied to a solid. The constituents original shape and size after removal of the
vibrate about their equilibrium positions even deforming force. These observations indicate
at very low temperatures but cannot leave their that rubber and clay are different in nature.
fixed positions. This fact provides the solids The property that decides this nature is called
a definite shape and size (allows the solids to elasticity/plasticity. We will learn more about
maintain a definite shape and size). these properties of solids in this Chapter .
6.2 Elastic Behavior of Solids:
If an external force is applied to a solid the
constituents are slightly displaced and restoring If a body regains its original shape and
forces are developed in it. These restoring size after removal of the deforming force, it
forces try to bring the constituents back to is called an elastic body and the property is
their equilibrium positions so that the solid can called elasticity. Here the restoring forces are
regain its shape. When the deforming forces are strong enough to bring the displaced molecules
removed, the interatomic forces tend to restore to their original positions. Examples of elastic
the original positions of the molecules and thus materials are metals, rubber, quartz, etc.
the body regains its original shape and size.
However, as we will see later, this is possible If a body regains its original shape and
only within certain limits. size completely and instantaneously upon
removal of the deforming force, then it is said
The form of a body is decided by its size to be perfectly elastic.
and shape, e.g., a tennis ball and a football
both are spherical, i.e., they have the same If a body does not regain its original
shape. But a tennis ball is smaller in size than shape and size and retains its altered shape
a football. When a force is applied to a solid or size upon removal of the deforming force,
(which is not free to move), the size or shape it is called a plastic body and the property is
or both change due to changes in the relative called plasticity. Here, the restoring forces are
positions of molecules. Such a force is called not strong enough to bring the molecules back
deforming force. to their original positions. Examples of plastic
materials are clay, putty, plasticine, thick mud,
The change in shape or size or both etc. There is no solid which is perfectly elastic
of a body due to an external force is called or perfectly plastic. The best example of a near
deformation. ideal elastic solid is quartz fibre and that of a
plastic body is putty.
The larger the deforming force on a body,

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6.3 Stress and Strain: Fig. 6.1 (a): Tensile stress.
The elastic properties of a body are

described in terms of stress and strain. When
a body gets deformed under an applied force,
restoring forces are set up internally. They
oppose change in shape or size of the body.
When body is in equilibrium in its altered shape
or size, deforming force and restoring force are
equal and opposite.
The internal restoring force per unit area of
a body is called stress.

--- (6.1) Fig. 6.1 (b): Compressive stress.
B) Tensile strain:
where F is internal restoring force (external
applied deforming force). SI unit of stress is The strain produced by a tensile deforming
N m-2 or pascal (Pa). The dimensions of a stress force is called tensile strain or longitudinal
are [ L-1 M1 T-2 ]. strain or linear strain.

Strain is a measure of the deformation of a If L is the original length and ∆l is the
body. When two equal and opposite forces are change in length due to the deforming force,
applied to an elastic body, there is a change in then
the dimensions of the body, Strain is defined
as the ratio of change in dimensions of the --- (6.5)
body to its original dimensions.
2 : When a deforming force acting on a body
--- (6.2) produces change in its volume, the stress is
called volume stress and the strain produced is
called volume strain.
It is the ratio of two similar quantities. A) Volume stress or hydraulic stress:
Hence strain is a dimensionless physical
quantity. It has no units. There are three types Let F be a force acting perpendicular to
of stress and corresponding strains. the entire surface of the body. It acts normally
1: Stress produced by a deforming force acting and uniformly all over the surface area A of the
along the length of a body or a rod is called body. Such a stress which produces change in
tensile stress or a longitudinal stress. The size but no change in shape is called volume
strain produced is called tensile strain. stress.
A) Tensile stress or compressive stress:
--- (6.6)
Suppose a force F is applied along the
length of a wire, or perpendicular to its cross
section A. This produces an elongation in Volume stress produces change in size
the wire and the length of the wire increases without change in shape of body, it is called
accordingly, as shown in Fig. 6.1 (a). hydraulic or hydrostatic volume stress as shown
in Fig. 6.2.
Tensile stress = | F | --- (6.3) B) Volume strain:

A A deforming force acting perpendicular to
the entire surface of a body produces a volume
When a rod is pushed at two ends with equal strain. Let V be the original volume and ∆V be
and opposite forces, its length decreases. the change in volume due to deforming force,
The restoring force per unit area is called then
compressive stress as shown in Fig. 6.1 (b).
--- (6.7)
Compressive stress | F | --- (6.4)
A

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tangential force. Tangential force is parallel to

the top and the bottom surface of the block.

The restoring force per unit area developed

due to the applied tangential force is called

shearing stress or tangential stress.

B) Shearing strain:

There is a relative displacement, ∆l, of the

Fig. 6.2 : Volume stress. bottom face and the top face of the cube. Such

Do you know ? relative displacement of two surfaces is called

When a balloon is filled with air at high shear strain. It can be calculated as follows,
pressure, its walls experience a force from
within. This is also volume stress. It tries whenSthheearreinlagtisvteradinisp∆llal ce=mtaenntθ∆=l θ --- (6.9)
to expand the balloon and change its size is very small.
without changing shape. When the volume
stress exceeds the limit of bulk elasticity, the 6.4 Hooke’s Law:
balloon explodes. Similarly, a gas cylinder
explodes when the pressure inside it exceeds Robert Hooke (1635-1703), an English
the limit of bulk elasticity of its material.
physicist, studied the tension in a wire and
A submarine when submerged under
water is under volume stress. strain produced in it. His study led to a law now

3 : When a deforming force acting on a body known as Hooke’s law.
produces change in the shape of a body, shearing
stress and shearing strain are produced. Statement: Within elastic limit, stress is
A) Shearing stress:
directly proportional to strain.
Let F be a tangential force acting on a
surface area A. This force produces change in Stress = constant
shape of the body without changing its size as Strain
shown in Fig. 6.3.
The constant is called the modulus
--- (6.8)
of elasticity. The modulus of elasticity

of a material is the ratio of stress to the

corresponding strain. It is defined as the

slope of the stress-strain curve in the elastic

deforming region and depends on the nature of

the material. The maximum value of stress up

to which stress is directly proportional to strain

is called the elastic limit. The stress-strain

curve within elastic limit is shown in Fig. 6.4

D

Fig. 6.3 : Tangential force produces Fig 6.4: Stress versus strain graph within
shearing stress. elastic limit for an elastic body.
6.5 Elastic modulus:
Suppose ABCD is the front face of a cube.
There are three types of stress and strain
A force F is applied to the cube so that the related to change in length, change in volume
bottom of the cube is fixed and only the top and change in shape. Hence, we have three
surface is slightly displaced. Such force is called moduli of elasticity corresponding to each type

102

of stress and strain. Table 6.1: Young's modulus of some
familiar materials
6.5.1 Young’s modulus (Y):

It is the modulus of elasticity related to Material Young's modulus Y
×1010 Pa (N/m2)
change in length of an object like a metal

wire, rod, beam, etc., due to the applied Lead 1.5
Glass (crown) 6.0
deforming force. Hence it is also called as 7.0
Aluminium 7.6
elasticity of length. It is named after the British Silver 8.1
Gold 9.0
physicist Thomas Young (1773-1829). Brass 11.0
Copper 21.0
Consider a metal wire of length L having Steel

radius r suspended from a rigid support. A load

Mg is attached to the free end of the wire. Due

to this, deforming force is applied at the free

end of the wire in downward direction. In its Example 6.1: A brass wire of length 4.5m with
crosssectional area of 3×10-5 m2 and a copper
equilibrium position, wire of length 5.0 m with cross sectional area
4×10-5 m2 are stretched by the same load. The
Longitudinal  stress = Applied  force same elongation is produced in both the wires.
Area Find the ratio of Young’s modulus of brass and
copper.
= F Solution: For brass,
A
LB= 4.5m, AB= 3×10-5 m2
Mg --- (6.10) lB= l, FB= F

Sr2

It produces a change in length of the wire. If

(L+l) is the new length of wire, then l is the

extension or elongation in wire.

Longitudinal  strain = change in length  
original  length
?YB F u 4.5
 Ll - -- (6.11) 3u10 5 u l
= For copper,

LC= 5m, AC= 4×10-5 m2
Young’s modulus is the ratio of longitudinal lC= l, FC=F

stress to longitudinal strain.

Young cs  modulus  lloonnggiittuuddiinnaall  stress -- (6.12)
 strain

?Yc F u 5.0
4 u10 5 u l
u10 5 u
YB F u 4.5 u 4 F u5 l
YC 3u10 5 u
MgL l
S r2l
Y ---(6.13) 18 u 10 5 1.2
15 u 10 5

SI unit of Young’s modulus is N/m2. Its Example 6.2: A wire of length 20 m and area

dimensions are [ L-1 M1 T-2 ].
Young’s modulus indicates the of cross section 1.25×10-4 m2 is subjected to a
resistance of an elastic solid to elongation or load of 2.5 kg. (1 kgwt = 9.8 N). The elongation
compression. Young’s modulus of a material is produced in wire is 1×10-4 m. Calculate Young’s
useful for characterization of an object subjected modulus of the material.
to compression or tension. Young's modulus is Solution: Given,
L = 20 m
the property of solids only.
A = 1.25 ×10-4 m2

103

F = mg = 2.5 × 9.8N Compressibility is the fractional decrease
in volume, -∆V/V per unit increase in pressure.
L = 10-4 m SI unit of compressibility is m2/ N or Pa-1 and its
dimensions are [ L1 M-1 T2].
To find: Y

Y = FL 2.5u9.8u20
Al 1.25u10 4 u10 4

= 3.92 × 1010 N m-2 Do you know ?

6.5.2 Bulk modulus (K): The bulk modules of water is 2.18×108 Pa
and its compressibility is 45.8×10-10 Pa-1.
It is the modulus of elasticity related to Materials with small bulk modulus and large
compressibility are easier to compress.
change in volume of an object due to applied

deforming force. Hence it is also called as

elasticity of volume. Bulk modulus of elasticity Example 6.3: A metal cube of side 1m is

is a property of solids, liquids and gases. subjected to a force. The force acts normally

If a sphere made from rubber is completely on the whole surface of cube and its volume

immersed in a liquid, it will be uniformly changes by 1.5×10-5 m3. The bulk modulus of

compressed from all sides. Suppose this metal is 6.6×1010 N/m2. Calculate the change in

compressive force is F. Let the change in pressure.

pressure on the sphere be dP and let the change Solution: Given,

in its volume be dV. If the original volume of volume of cube=V = l3 = (1)3 =1m3

the sphere is V, then volume strain is defined as Change in volume = dV = 1.5×10-5 m3

Bulk modulus = K = 6.6×1010 N/m2.

To find: Change in pressure dP

  dV --- (6.14) K =V dP
V dV

The negative sign indicates that there is a dP = K dV
V
decrease in volume. The magnitude of the

volume strain is dV dP 6.6 u1010 u1.5u10 5
V 1
Bulk modulus is defined as the ratio of
dP = 9.9×105 N/m2.
volume stress to volume strain.

Table 6.2: Bulk modulus of some familiar
materials

K dP V  dP --- (6.17) Material Bulk modulus K
§ dV · dV ×1010 Pa (N/m2)

¨© V ¹¸ Lead 4.1
Brass 6.0
SI unit of bulk modulus is N/m2. Dimensions of Glass (crown) 6.0
K are [ L-1 M1 T-2 ]. Aluminium 7.5
Silver 10.0
Table 6.2 gives bulk moduli of some Copper 14.0
familiar materials Steel 16.0
Gold 18.0
Bulk modulus measures the resistance
offered by gases, liquids or solids while an 6.5.3 Modulus of rigidity (η):
attempt is made to change their volume. The modulus of elasticity related to

The reciprocal of bulk modulus of elasticity change in shape of an object is called rigidity
is called compressibility of the material. modulus. It is the property of solids only as
they alone possess a definite shape.
--- (6.18)

104

The block shown in Fig. 6.5 is made of Table 6.3: Rigidity modulus η of some
a uniform isotropic material. It has a uniform familiar materials
crosssection area A and height l. A cross section
of the block is defined as any plane parallel Material Rigidity modulus η
to the top and the bottom surface and cuts the ×1010 Pa (N/m2)
block. Two forces of magnitude 'F' are applied
along top and bottom surface as shown in Fig. Lead 0.6
(6.5). They constitute a couple. The upper Aluminium 2.5
surface is displaced relative to the lower surface Glass (crown) 2.5
by a small distance ∆l and corresponding angles 2.7
change by a small amount θ = ∆l/l. Silver 2.9
Gold 3.5
Brass 4.4
Copper 8.3
Steel

Rigidity modulus indicates the resistance

offered by a solid to change in its shape.

Example 6.4: Calculate the modulus of rigidity

of a metal, if a metal cube of side 40 cm is

subjected to a shearing force of 2000 N. The

upper surface is displaced through 0.5cm with

Fig. 6.5: Modulus of rigidity, tangential respect to the bottom. Calculate the modulus of
force F and shear strin θ.
rigidity of the metal.
A couple is applied by pushing the top
and the bottom surfaces as shown in Fig. 6.5. Solution: Given,
Similar couple would be applied if the bottom
of the block is fixed and only the top is pushed. Length of side of cube = l= 40 cm = 0.40 m

The forces F and - F are parallel to the Shearing force = F= 2000 N = 2×103 N
cross section. This is different than the tensile
stress where the force is normal to the cross Displacement of top face = ∆l = 0.5cm = 0.005m
section. Area = A = l 2 = 0.16m2

As a result of the way in which the forces To find: modulus of rigidity, η
are applied the block is subjected to a shear
stress defined by shear stress = F/A. K F
AT
The SI unit of shear stress is N/m2 or Pa.
The block is distorted as a result of the shear T 'l 0.005 0.0125
stress. The top and bottom surface are relatively l 0.40
displaced by a small distance ∆l. The corner
angle changes by a small amount θ which is K 2.0 u103 N
called shear strain and is expressed in radian. (0.16m2 ) ˜ (0.0125)
Shear strain 'θ' is given by θ = ∆l/l, (for small
∆l). = 1.0 u106 N / m2
Shear modulus or modulus of rigidity: It is 6.5.4 Poisson’s ratio:
defined as the ratio of shear stress to shear strain
within elastic limits. Suppose a wire is fixed at one end and a
force is applied at its free end so that the wire
gets stretched. Length of the wire increases and
at the same time, its diameter decreases, i.e., the
wire becomes longer and thinner as shown in
Fig. 6.6 (a).

K = shear stress F/A F --- (6.17)
shear strain T AT

Table 6.3 gives values of rigidity Fig. 6.6 (a): When a wire is stretched its
modulus η of some familiar materials. length increases and its diameter decreases.

105

Do you know ?

Fig. 6.6 (b): When a wire is compressed its For most of the commonly used metals,
the value of σ is between 0.25 and 0.35.
length increases and its diameter increases. Many times we assume that volume is
constant while stretching a wire. However,
If equal and opposite forces are applied to in reality, its volume also increases. Using
approximations it can be shown that σmax ≈
an object along its length inwards, the object gets 0.5 if volume is unchanged. In practice, it
is much less. This shows that volume also
compressed (Fig. 6.6 (b)). There is a decrease increases while stretching.
6.6 Stress-Strain Curve:
in dimensions along its length and at the same
Suppose a metal wire is suspended
time there is an increase in its dimensions vertically from a rigid support and stretched
by applying load to its lower end. The load is
perpendicular to its length. When length of the gradually increased in small steps until the wire
breaks. The elongation produced in the wire is
wire decreases, its diameter increases. measured during each step. Stress and strain
is noted for each load and a graph is drawn by
The ratio of change in dimensions to taking tensile strain along x-axis and tensile
stress along y-axis. It is a stress-strain curve as
original dimensions in the direction of the shown in Fig. 6.7.

applied force is called linear strain while Fig. 6.7 : stress-strain curve.
The initial part of the graph is a straight
the ratio of change in dimensions to original line OA. This is the region in which Hooke's
law is obeyed and stress is directly proportional
dimensions in a direction perpendicular to the to strain. The straight line portion ends at A.
The stress at this point is called proportional
applied force is called lateral strain. Within limit. If the load is further increased till point
B is reached, stress and strain are no longer
elastic limit, the ratio of lateral strain to the proportional and Hooke's law is not valid. If the
load is gradually removed starting at any point
linear strain is called the Poisson’s ratio. between O and B. The curve is retraced until
the wire regains its original length. The change
If l is the original length of wire, ∆l is is reversible. The material of the wire shows
increase/decrease in length of wire, D is the elastic behaviour in the region OB. Point B is
original diameter and d is corresponding change called the yield point. The corresponding point
in diameter of wire then, Poisson’s ratio is given is called the elastic limit.

by

V Lateral strain
Linear strain

= d / D
l / L

d.L --- (6.18)
D.l
Poisson’s ratio has no unit. It is dimensionless.
Table 6.4 gives values of Poisson ratio, σ, of
some familiar materials.

Table 6.4: Poisson ratio, σ, of some familiar
materials

Material Poisson ratio σ

Glass (crown) 0.2

Steel 0.28

Aluminium 0.36

Brass 0.37

Copper 0.37
Silver 0.38
Gold 0.42

106

When the stress is increased beyond point the energy dissipated during deformation of a
B, the strain continues to increase. If the load is material.
removed at any point beyond B, C for example,
the material does not regain its original length.
It follows the line CE. Length of the wire when Fig. 6.8: Stress-stain curve for increasing
there is no stress is greater than the original and decreasing load.
length. The deformation is irreversible and the
material has acquired a permanent set. Can you tell?

Further increase in load causes a large Why does a rubber band become loose after
increase in strain for relatively small increase repeated use?
in stress, until a point D is reached at which
fracture takes place. 6.7 Strain Energy:

The material shows plastic flow or plastic The elastic potential energy gained by a
deformation from point B to point D. The
material does not regain its original state when wire during elongation by a stretching force
the stress is removed. The deformation is
called plastic deformation. is called as strain energy.

The curve described above shows all Consider a wire of original length L and
the possibilities for an elastic substance.
In particular, many metallic wires (copper, cross sectional area A stretched by a force F
aluminum, silver, etc) exhibit this type of
behavior. However, majority of materials in acting along its length. The wire gets stretched
every day life exhibit only some part of it.
and elongation l is produced in it. The stress and
Materials such as glass, ceramics, etc.,
break within the elastic limit. They are called the strain increase proportionately.
brittle.
Longitudinal stress = F
Metals such as copper, aluminum, wrought A
iron, etc. have large plastic range of extension. l
They lengthen considerably and undergo plastic Longitudinal strain = L
deformation till they break. They are called
ductile. Young’s modulus = longitudinal  stress
longitudinal  strain
Metals such as gold, silver which can be
hammered into thin sheets are called malleable. §F ·
©¨ A ¹¸
Rubber has large elastic region. It can be Y §l·  FALl
stretched so that its length becomes many times ¨© L ¹¸
its original length, after removal of the stress it
returns to its original state but the stress strain ?F YAl --- (6.19)
curve is not a straight line. A material that can L
be elastically stretched to a larger value of strain
is called an elastomer. The magnitude of stretching force increases

In case of some materials like vulcanized from zero to F during elongation of wire. At a
rubber, when the stress applied on a body
decreases to zero, the strain does not return to certain stage, let ‘f ’ be the force applied and ‘x’
zero immediately. The strain lags behind the
stress. This lagging of strain behind the stress is be the corresponding extension. The force at
called elastic hysteresis. Figure 6.8 shows the
stress-strain curve for increasing and decreasing this stage is given by Eq. (6.19) as
load. It encloses a loop. Area of loop gives
f = YAx
L

107

For further extension dx in the wire, the work Work done per unit volume
done is given by
Strain e=ne12rgy(sptreersusn).i(tsvtroaliunm) e
Work = (force).(displacement).

dW = f dx 1
= 2
∴ dW = YAx  dx (stress).(strain) --- (6.22)
L
When the wire gets stretched from x = 0 to = stress
As Y strain ,
x = l, the total work done is given as
Stress = Y. (strain) and
l
stress
W ³dW strain = Y

0 ∴ Strain energy per unit volume

?W l YAx dx 1
³ 0L 2 Y ˜ (strain)2 --- (6.23)

?W YA l xdx Also, strain energy per unit volume
³
L --- (6.24)
0
Thus Eq. (6.22), (6.23) and (6.24) give strain
?W YA ª x2 ºl energy per unit volume in various forms.
« » 6.8 Hardness:
L ¬ 2 ¼ 0
Hardness is the property of a material
?W YA ª l2 02 º which enables it to resist plastic deformation.
L « 2 2 » Hard materials have little ductility and they are
¬ ¼ brittle to some extent. The term hardness also
refers to stiffness or resistance to bending,
W = YAl 2 scratching abrasion or cutting. It is the
2L property of a material which gives it the ability
to resist permanent deformation when a load is
W = 1 YAl l applied to it. The greater the hardness, greater
2L is the resistance to deformation.

W = 1 Fl The most well-known example of the hard
2 materials is diamond. It is incredibly difficult
to scratch a diamond. Metal with very low
1 (load).(extension) --- (6.20) hardness is aluminium.
Work done = 2
Hardness of material is different from
This work done by stretching force is equal its strength and toughness.

to energy gained by the wire. This energy is If a force is applied to a body it produces
deformation in it. Higher is the force required
strain energy. for deformation, the stronger is the material,
i.e., the material has more strength.
Strain energy = 1 (load).(extension) --- (6.21)
2 Steel has high strength whereas plasticine
Strain energy per unit volume can be obtained clay is not strong because it gets easily deformed
even by a small force.
by using Eq. (6.20) and various formula of
Toughness is the ability of a material to
stress, strain and young’s modulus. resist fracturing when a force is applied to it.
Plasticine clay is relatively tough as it can be
Work done per unit volume stretched and deformed due to applied force
without breaking.
= work done in streching wire
volume of wire.

= 1 F .l
2 A.L

1§ F ·§ l ·
2 ©¨ A ¹¸ ©¨ L ¸¹

108

A single material may be hard, strong and In this section we are going to study friction in
tough, e.g., solids only.
1) Bulletproof glass is hard and tough but not 6.9.1 Origin of friction:

strong. If smooth surfaces are observed under
2) Drill bits must be hard, strong and tough powerful microscope, many irregularities and
projections are observed. Friction arises due
for their work. to interlocking of these irregularities between
3) Anvils are very tough and strong but they two surfaces in contact. The surfaces can be
made extremely smooth by polishing to avoid
are not hard. irregularities but it is noticed that in this case
6.9 Friction in Solids: also, friction does not decrease but may increase.
Hence the interlocking of irregularities is not
Whenever the surface of one body slides the real cause of friction.
over another, each body exerts a certain amount
of force on the other body. These forces are According to modern theory, cause of
tangential to the surfaces. The force on each friction is the force of attraction between
body is opposite to the direction of motion molecules of two surfaces in actual contact in
between the two bodies. It prevents or opposes addition to the force due to the interlocking
the relative motion between the two bodies. It is between the two surfaces. When one body is in
a common experience that an object placed on contact with another body, the real microscopic
any surface does not move easily when a small area in contact is very small due to irregularities
force is applied to it. This is because of certain in contact. Figure 6.9 shows the microscopic
force of opposition acting between the surface of view of two polished surfaces in contact.
the object and the surface on which it is placed.
Even a rolling ball comes to rest after covering Fig. 6.9: Microscopic view of polished
a finite distance on playground because of such surfaces in contact.
opposing force. Our foot ware is provided
with designs at the bottom of its sole so as to Due to small area, pressure at points of
produce force of opposition to avoid slipping. contact is very high. Hence there is a strong force
It is difficult to walk without such opposing of attraction between the surfaces in contact.
force. You know what happens when you try If both the surfaces are of the same material
to walk fast on polished flooring at home with the force of attraction is called cohesive force
soap water spread on it. There is a possibility of while if the surfaces are of different materials
slipping due to lack of force of opposition. To the force of attraction is called adhesive force.
initiate any motion between a pair of surfaces, When the surfaces in contact become more and
we need a certain minimum force. Also after the more smooth, the actual area of contact goes on
motion begins, it is constantly opposed by some increasing. Due to this, the force of attraction
natural force. This mechanical force between between the molecules increases and hence the
two solid surfaces in contact with each other is friction also increases. Putting some grease or
called as frictional force. The property which other lubricant (a different material) between
resists the relative motion between two the two surfaces reduces the friction.
surfaces in contact is called friction. 6.9.2 Types of friction:
1. Static friction:
In some cases it is necessary to avoid
friction, because friction causes dissipation of Suppose a wooden block is placed on
energy in machines due to which efficiency a horizontal surface as shown in Fig 6.10. A
of machines decreases. In such cases friction small horizontal force F is applied to it. The
should be reduced by using polished surfaces,
lubricants, etc. Relative motion between solids
and fluids (i.e. liquids and gases) is also naturally
opposed by friction, e.g., a boat on the surface
of water experiences opposition to its motion.

109

block does not move with this force as it cannot to the normal reaction. Table 6.4 gives
the coefficient of static fiction for some
overcome the frictional force between the block materials.
2] The limiting force of friction is independent
and horizontal surface. In this case, the force of the apparent area between the surfaces
in contact, so long as the normal reaction
of static friction is equal to F and balances it. remains the same.
3] The limiting force of friction depends upon
The frictional force which balances applied materials in contact and the nature of their
surfaces.
force when the body is static is called force
Table 6.4: Coefficient of static friction
forficsttiaotnicprfreivcetniotns ,slFids.inIgn other words, static
motion.

If we keep increasing F, a stage will come

when, for F = Fmax, the object will start moving. Material Coefficient of
For F < Fmax, the force of static friction is equal Teflon on Teflon static friction µs
to F. For F ≥ Fmax, the kinetic friction comes
into play. Static friction opposes impending 0.4

motion i.e. the motion that would take place

in absence of frictional force under the applied

force. Brass on steel 0.51
Copper on steel 0.53
Aluminium on steel 0.61

Steel on steel 0.74

Fig. 6.10: Static friction. Glass on glass 0.94
Rubber on concrete (dry) 1.0

The force of static friction is self adjusting Example 6.5: The coefficient of static friction
between a block of mass 0.25 kg and a horizontal
force. When the applied force F is very small, the surface is 0.4. Find the horizontal force applied
to it.
block remains at rest. Here the force of friction
Solution: Given,
is also small. When F is increased by a small µs = 0.4
m = 0.25 kg
value, the block remains still at rest as the force To find: Force

of friction is increased to balance the applied F = µs. N = µs. (mg)
F = 0.4 × 0.25 × 9.8
force. If applied force is increased, the friction
F = 0.98 N
also increases and reaches the maximum value. 2. Kinetic friction :

Just before the body starts sliding over another Once the sliding of block on the surface
starts, the force of friction decreases. The force
body, the value of frictional force is maximum, required to keep the body sliding steadily is
thus less than the force required to just start its
it is called limiting force of ifsrircetvioenrs,eFd,L .thIef sliding. The force of friction that comes into
the direction of applied force play when a body is in steady state of motion
over another surface is called force of kinetic
direction of static friction is also reversed, i.e., friction.

it adjusts its direction also. Friction between two surfaces in contact
when one body is actually sliding over the
Laws of static friction: other body, is called kinetic friction or
dynamic friction.
1] The limiting force of static friction is

directly proportional to the normal reaction

(N) between the two surfaces in contact.

FL ∝ N
... FL = µs N --- (6.25)
Where µs is constant of proportionality. It

is called as coefficient of static friction.

?µs FL --- (6.26)

N

The coefficient of static friction is defined

as the ratio of limiting force of friction

110

Laws of kinetic friction : friction while the force of kinetic friction is
1. The force of kinetic friction (Fk ) is directly greater than force of rolling friction. As rolling
friction is the minimum, ball bearings are
proportional to the normal reaction between used to reduce friction in parts of machines to
two surfaces in contact. increase its efficiency.

∴ Fk α N Advantages of friction:
∴ Fk = µk N --- (6.27) Friction is necessary in our daily life.
Where µk is constant of proportionality. It
• We can walk due to friction between
is called as coefficient of kinetic friction. ground and feet.

? Pk Fk   --- (6.28) • We can hold object in hand due to static
N friction.

The coefficient of kinetic friction is defined • Brakes of vehicles work due to friction;
as the ratio of force of kinetic friction to the hence we can reduce speed or stop
normal reaction between the two surfaces vehicles.
in contact. Table 6.5 gives the co-efficient
of kinetic friction for some materials. • Climbing on a tree is possible due to
friction.
2. Force of kinetic friction is independent of
shape and apparent area of the surfaces in Disadvantages of friction
contact. • Friction opposes motion.
• Friction produces heat in different parts
3. Force of kinetic friction depends upon of machines. It also produces noise.
the nature and material of the surfaces in • Automobile engines consume more fuel
contact. due to friction.

4. The magnitude of the force of kinetic Methods of reducing friction
friction is independent of the relative • Use of lubricants, oil and grease in
velocity between the object and the surface different parts of a machine.
provided that the relative velocity is neither • Use of ball bearings converts kinetic
too large nor too small. friction into rolling friction.

Table 6.5: Coefficient of kinetic friction

Material Coefficient of

kinetic friction µk Can you tell?

Rubber on concrete (dry) 0.25 1) It is difficult to run fast on sand.
2) It is easy to roll than pull a barrel
Glass on glass 0.40
along a road.
Brass on steel 0.40 3) An inflated tyre rolls easily than a

Copper on steel 0.44 flat tyre.
4) Friction is a necessary evil.
Aluminium on steel 0.47
Internet my friend
Steel on steel 0.57
1. https://opentextbc.ca>chapter>friction.
Teflon on Teflon 0.80 2. https://www.livescience.com
3. https://www.khanacdemy.org.physics
3 Rolling friction : 4. https://courses.lumenlearning.com>
Motion of a body over a surface is said to be elastiscitychapter>elasticity
5. https://www.toper.com>guides>physics
rolling motion if the point of contact of the body
with the surface keeps changing continuously.

Friction between two bodies in contact
when one body is rolling over the other, is
called rolling friction.

For same pair of surfaces, the force of
static friction is greater than the force of kinetic

111

ExercisesExercises

1. Choose the correct answer: and the softest material.

i) Change in dimensions is known as….. xiii) Define friction.

(A) deformation (B) formation xiv) Why force of static friction is known as
‘self-adjusting force’?
(C) contraction (D) strain.
xv) Name two factors on which the co-
ii) The point on stress-strain curve at which efficient of friction depends.
strain begins to increase even without
increase in stress is called…. 3. Answer in short:

(A) elastic point (B) yield point i) Distinguish between elasticity and
plasticity.
(C) breaking point (D) neck point
ii) State any four methods to reduce friction.
iii) Strain energy of a stretched wire is
18×10-3 J and strain energy per unit iii) What is rolling friction? How does it
volume of the same wire and same cross arise?
section is 6×10-3 J/m3. Its volume will
be.... iv) Explain how lubricants help in reducing
friction?
(A) 3cm3 (B) 3 m3
v) State the laws of static friction.
(C) 6 m3 (D) 6 cm3
vi) State the laws of kinetic friction.
iv) ----- is the property of a material which
enables it to resist plastic deformation. vii) State advantages of friction.

(A) elasticity (B) plasticity viii) State disadvantages of friction.

(C) hardness (D) ductility ix) What do you mean by a brittle substance?
Give any two examples.
v) The ability of a material to resist fracturing
when a force is applied to it, is called…… 4. Long answer type questions:

(A) toughness (B) hardness i) Distinguish between Young’s modulus,
bulk modulus and modulus of rigidity.
(C) elasticity (D) plasticity.
ii) Define stress and strain. What are their
2. Answer in one sentence: different types?

i) Define elasticity. iii) What is Young’s modulus? Describe an
experiment to find out Young’s modulus
ii) What do you mean by deformation? of material in the form of a long straight
wire.
iii) State the SI unit and dimensions of stress.
iv) Derive an expression for strain energy per
iv) Define strain. unit volume of the material of a wire.

v) What is Young’s modulus of a rigid body? v) What is friction? Define coefficient of
static friction and coefficient of kinetic
vi) Why bridges are unsafe after a very long friction. Give the necessary formula for
use? each.

vii) How should be a force applied on a body vi) State Hooke’s law. Draw a labeled graph
to produce shearing stress? of tensile stress against tensile strain for a
metal wire up to the breaking point. In this
viii) State the conditions under which Hooke’s graph show the region in which Hooke’s
law holds good. law is obeyed.

ix) Define Poisson’s ratio.

x) What is an elastomer?

xi) What do you mean by elastic hysteresis?

xii) State the names of the hardest material

112

5. Answer the following vii) A wire of mild steel has initial length
1.5 m and diameter 0.60 mm is extended
i) Calculate the coefficient of static friction by 6.3 mm when a certain force is applied
for an object of mass 50 kg placed on to it. If Young’s modulus of mild steel
horizontal table pulled by attaching a is 2.1 x 1011 N/m2, calculate the force
spring balance. The force is increased applied.
gradually it is observed that the object just
moves when spring balance shows 50N. [Ans: 250 N]

[Ans: µs = 0.102] viii) A composite wire is prepared by joining
a tungsten wire and steel wire end to end.
ii) A block of mass 37 kg rests on a rough Both the wires are of the same length
and the same area of cross section. If this
horizontal plane having coefficient of composite wire is suspended to a rigid
support and a force is applied to its free
static friction 0.3. Find out the least end, it gets extended by 3.25mm. Calculate
the increase in length of tungsten wire and
force required to just move the block steel wire separately.

horizontally.

[Ans: Fs = 108.8N]

iii) A body of mass 37 kg rests on a rough

horizontal surface. The minimum [Given: Ysteel = 2 × 1011N/m2,

horizontal force required to just start the YTungsten = 3.40 × 108 N/m2]

motion is 68.5 N. In order to keep the

body moving with constant velocity, a [Ans: extension in tungsten wire = 3.244 mm,

force of 43 N is needed. What is the value extension in steel wire = 0.0052 mm]

of a) coefficient of static friction? and b) ix) A steel wire having cross sectional area
1.2 mm2 is stretched by a force of 120 N.
coefficient of kinetic friction? If a lateral strain of 1.455 mm is produced
in the wire, calculate the Poisson’s ratio.
[Ans: a) µs = 0.188

b) µk = 0.118]

iv) A wire gets stretched by 4mm due to a [Ans: 0.291]

certain load. If the same load is applied x) A telephone wire 125m long and 1mm in
radius is stretched to a length 125.25m
to a wire of same material with half the when a force of 800N is applied. What is
the value of Young’s modulus for material
length and double the diameter of the of wire?
[Ans: 1.27×10 11N/m2]
first wire. What will be the change in its

length?

[Ans: 0.5mm]

v) Calculate the work done in stretching a xi) A rubber band originally 30cm long is
steel wire of length 2m and cross sectional
area 0.0225mm2 when a load of 100 N is stretched to a length of 32cm by certain
slowly applied to its free end. [Young’s
modulus of steel= 2×1011 N/m2 ] load. What is the strain produced?

[Ans: 6.667× 10-2 ]

xii) What is the stress in a wire which is 50m

[Ans: 2.222J] long and 0.01cm2 in cross section, if the

vi) A solid metal sphere of volume 0.31m3 wire bears a load of 100kg?
is dropped in an ocean where water
pressure is 2×107 N/m2. Calculate change [Ans: 9.8× 108 N/m2]
in volume of the sphere if bulk modulus
of the metal is 6.1×1010 N/m2 xiii) What is the strain in a cable of original
length 50m whose length increases by
2.5cm when a load is lifted?

[Ans: 10-4 m3] [Ans: 5× 10-4 ]

***

113

7. Thermal Properties of Matter

Can you recall?

1. Temperature of a body determines its 3. Solids, liquids and gases expand on
hotness while heat energy is its heat heating.
content.
4. Substances change their state from solid
2. Pressure is the force exerted per unit area to liquid or liquid to gas on heating up to
normally on the walls of a container by the specific temperature.
gas molecules due to collisions.


7.1 Introduction: quantitatively measure it. Scientific precision
requires measurement of a physical quantity in
In previous lessons, while describing the numerical terms. A thermometer is the device to
equilibrium states of a mechanical system measure the temperature.
or while studying the motion of bodies, only
three fundamental physical quantities namely In this chapter , we will learn properties of
length, mass and time were required. All other matter and various phenomena that are related
physical quantities in mechanics or related to heat. Phenomena or properties having to do
to mechanical properties can be expressed in with temperature changes and heat exchanges
terms of these three fundamental quantities. are termed as thermal phenomena or thermal
In this chapter, we will discuss properties or properties. You will understand why the
phenomena related to heat. These require a direction of wind near a sea shore changes
fourth fundamental quantity, the temperature, during day and night, why the metal lid of a
as mentioned in Chapter 1. glass bottle comes out easily on heating and
why two metal vessels locked together can be
The sensation of hot or cold is a matter of separated by providing heat to the outer vessel.
daily experience. A mother feels the temperature
of her child by touching its forehead. A cook 7.2 Temperature and Heat:
throws few drops of water on a frying pan to
know if it is hot enough to spread the dosa Heat is energy in transit. When two bodies
batter. Although not advisable, in our daily at different temperatures are brought in contact,
lives, we feel hotness or coldness of a body by they exchange heat. After some time, the heat
touching or we dip our fingers in water to check transfer stops and we say the two bodies are
if it is hot enough for taking bath. When we say in thermal equilibrium. The property or the
a body or water is hot, we actually mean that its deciding factor to determine the state of thermal
temperature is more than our hand. However, equilibrium is the temperature of the two bodies.
in this way, we can only compare the hotness Temperature is a physical quantity that defines
or coldness of two objects qualitatively. Hot the thermodynamic state of a system.
and cold are relative terms. You might recall
the example given in your science textbook of You might have experienced that a glass of
VIIIth standard. Lukewarm water seems colder ice-cold water when left on a table eventually
than hot water but hotter than cold water to our warms up whereas a cup of hot tea on the
hands. We ascribe a property ‘temperature’ to same table cools down. It means that when the
an object to determine its degree of hotness. temperature of a body, ice-cold water or hot
The higher the temperature, the hotter is the tea in the above examples, is different from its
body. However, the precise temperature of surrounding medium, heat transfer takes place
a body can be known only when we have between the body and the surrounding medium
an accurate and easily reproducible way to until the body and the surrounding medium
are at the same temperature. We then say that
the body and its surroundings have reached

114

a state of thermal equilibrium and there is no of a gas. Hence gases neither have a definite
net transfer of heat from one to the other. In volume nor shape. Interatomic spacing in
fact, whenever two bodies are in contact, there solids is ~ 10-10 m while the average spacing
is a transfer of heat owing to their temperature in liquids is almost twice that in solids. The
difference. average inter molecular spacing in gases at
normal temperature and pressure (NTP) is
Matter in any state - solid, liquid or gas- ~10-9 m.
consists of particles (ions, atoms or molecules).
In solids, these particles are vibrating about their From the above discussion, we understand
fixed equilibrium positions and possess kinetic that heat supplied to the substance increases
energy due to motion at the given temperature. the kinetic energy of molecules or atoms of
The particles possess potential energy due to the substance. The average kinetic energy
the interatomic forces that hold the particles per particle of a substance defines the
together at some mean fixed positions. Solids temperature. Temperature measures the degree
therefore have definite volume and shape. When of hotness of an object and not the amount of its
we heat a solid, we provide energy to the solid. thermal energy.
The particles then vibrate with higher energy
and we can see that the temperature of the A glass of water, a gas enclosed in a
solid increases (except near its melting point). container, a block of copper metal are all
Thus the energy supplied to the solid (does examples of a 'system'. We can say that heat
not disappear!) becomes the internal energy in in the form of energy is transferred between
the form of increased kinetic energy of atoms/ two (or more) systems or a system and its
molecules and raises the temperature of the surroundings by virtue of their temperature
solid. The temperature is therefore a measure difference. SI unit of heat energy is joule (J)
of the average kinetic energy of the atoms/ and that of temperature is kelvin (K) or celcius
molecules of the body. The greater the kinetic (°C). The CGS unit of heat energy is erg.
energy is, the faster the molecules will move (1J = 107 erg). The other unit of heat energy,
and higher will be the temperature of the body. that you have learnt in VIIIth standard, is calorie
If we continue heating till the solid starts to melt, (cal) and the relation with J is 1 cal = 4.184 J.
the heat supplied is used to weaken the bonds Heat being energy has dimension [L2M1T-2K°]
between the constituent particles. The average while dimension of temperature is [L°M°T°K1].
kinetic energy of the constituent particles does
not change further. The order of magnitude of 7.3 Measurement of Temperature:
the average distance between the molecules
of the melt remains almost the same as that of In order to isolate two liquids or gases from
solid. Due to weakened bonds liquids do not each other and from the surroundings, we use
possess definite shape but have definite volume. containers and partitions made of materials like
The mean distance between the particles and wood, plastic, glass wool, etc. An ideal wall or
hence the density of liquid is more or less the partition (not available in practice) separating
same as that of the solid. On heating further, two systems is one that does not allow any flow
the atoms/molecules in liquid gain kinetic or exchange of heat energy from one system to
energy and temperature of the liquid increases. the other. Such a perfect thermal insulator is
If we continue heating the liquid further, at the called an adiabatic wall and is generally shown
boiling point, the constituents can move freely as a thick cross-shaded (slanting lines) region.
overcoming the interatomic/molecular forces When we wish to allow exchange of heat energy
and the mean distance between the constituents between two systems, we use a partition like a
increases so that the particles are farther apart. thin sheet of copper. It is termed as a diathermic
wall and is represented as a thin dark region.
As per kinetic theory of gases, for an ideal
gas, there are no forces between the molecules Let us consider two sections of a container
separated by an adiabatic wall. Let them contain
two different gases. Let us call them system A

115

and system B. We independently bring systems temperatures just as we select the standard
A and B in thermal equilibrium with a system C. of length (metre) to be the distance between
Now if we remove the adiabatic wall separating two fixed marks. The fact that substances
systems A and B, there will be no transfer of change state from solid to liquid to gas at
heat from system A to system B or vice versa. fixed temperatures is used to define reference
This indicates that systems A and B are also temperature called fixed point. The two fixed
in thermal equilibrium. Overall conclusion of temperatures selected for this purpose are the
this activity can be summarized as follows: melting point of ice or freezing point of water
If systems A and B are separately in thermal and the boiling point of water. The next step is
equilibrium with a system C, then A and B are to sub-divide this standard temperature interval
also mutually in thermal equilibrium. When into sub-intervals by utilizing some physical
two or more systems/ bodies are in thermal property that changes with temperature and
equilibrium, their temperatures are same. This call each sub-interval a degree of temperature.
principle is used to measure the temperature of This procedure sets up an empirical scale for
a system by using a thermometer. temperature.
* The temperature at which pure water
Do you know ?
freezes at one standard atmospheric
If TA = TB and TB = TC, then TA = TC is not pressure is called ice point/ freezing point
a mathematical statement, if TX represents the of water. This is also the melting point of
temperature of system X. It is the zeroth law ice.
* The temperature at which pure water boils
of thermodynamics and makes the science of and vaporizes into steam at one standard
atmospheric pressure is called steam point/
Thermometry possible. boiling point. This is also the temperature
at which steam changes to liquid water.
Do you remember that to know the Having decided the fixed point phenomena,
temperature of our body, doctor brings the it remains to assign numerical values to these
mercury in the thermometer down to indicate fixed points and the number of divisions
some low temperature. We are then asked to between them. In 1750, conventions were
keep the thermometer in our mouth. We have adopted to assign (i) a temperature at which
to wait for some time before the thermometer is pure ice melts at one atmosphere pressure
taken out to know the temperature of our body. (the ice point) to be 0º and (ii) a temperature
There is transfer of heat energy from our body at which pure water boils at one atmosphere
to the thermometer since initially our body is (the steam point) to be 100º so that there are
at a higher temperature. When the temperature 100 degrees between the fixed points. This was
on the thermometer is same as that of our body, the centigrade scale (centi meaning hundred in
thermal equilibrium is said to be attained and Latin). This was redefined as celcius scale after
heat transfer stops. the Swedish scientist Anders Celcius (1701-
1744). It is a convention to express temperature
As mentioned above, to precisely know as degree celcius (ºC).
the thermodynamic state of any system, we
need to know its temperature. The device used To measure temperature quantitatively,
to measure temperature is a thermometer. generally two different scales of temperature
Thermometry is the science of temperature are used. They are describe below.
and its measurement. For measurement of
temperature, we need to establish a temperature 1) Celsius scale:- On this scale, the ice point is
scale and adopt a set of rules for assigning marked as 0 and the steam point is marked
numbers (with corresponding units). as 100, both taken at normal atmospheric
pressure (105 Pa or N/m2). The interval
For the calibration of a thermometer, between these points is divided into 100
a standard temperature interval is selected
between two easily reproducible fixed

116

equal parts. Each of these is known as Given TF = 98.4 qF,

degree Celsius and is written as ºC. 100 (98.4-32)
180
2) Fahrenheit scale :- On this scale, the ice TC

point is market as 32 and the steam point 100 (66.4)
180
is marked as 212, both taken at normal

atmospheric pressure. The interval between = 36.89 qC

these points is divided into 180 equal A device used to measure temperature, is
based on the principle of thermal equilibrium.
parts. Each division is known as degree To measure the temperature, we use different
measurable properties of materials which
Fahrenheit and is written as °F. change with temperature. Some of them are
length of a rod, volume of a liquid, electrical
A relationship for conversion between resistance of a metal wire, pressure of a gas at
constant volume etc. Such changes in physical
the two scales may be obtained from a graph properties with temperature are used to design
a thermometer. Physical property that is used in
of fahrenheit temperature (TF) versus celsius the thermometer for measuring the temperature
temperature (TC). The graph is a straight line is called the thermometric property and the
(Fig. 7.1) whose equation is material employed for the purpose is termed
as the thermometric substance. Temperature
TF 32 TC --- (7.1) is measured by exploiting the continuous
monotonic variation of the chosen property
180 100 with temperature. A calibration, however, is
required to define the temperature scale.

Fig. 7.1: A plot of fahrenheit temperature There are different kinds of thermometers
each type being more suitable than others for a
Ex(aTmF)pvleersu7s.1c:elsAiuvsetreamgeperraotoumre (TC). certain job. In each type, the physical property
temperature used to measure the temperature must vary
continuously over a wide range of temperature.
on a normal day is 27 °C. What is the room It must be accurately measurable with simple
apparatus.
temperature in °F?

Solution: We have An important characteristic of a
thermometer is its sensitivity, i.e., a change in the
TF 32 = TC thermometric property for a very small change
180 100 in temperature. Two other characteristics
are accuracy and reproducibility. Also it is
?TF 180 TC + 32 important that the system attains thermal
100 equilibrium with the thermometer quickly.

Given TC = 27 qC,

TF 180 u 27 + 32
100

= 48.6 + 32 If the values of a thermometric property
are P1 and P2 at the ice point (0 ºC) and steam
= 80.6 qF point (100 ºC) respectively and the value of this
property is PT at unknown temperature T, then
Example 7.2: Normal human body temperature T is given by the following equation
in feherenheit is 98.4 °F. What is the body
temperature in °C? 100 PT P1
Solution: We have P2 P1

TC TF 32 T --- (7.2)

100 180

?TC 100 (TF -32) Ideally, there should be no difference
180

117

in temperatures recorded on two different 27 100 R 95.2 ,
thermometers. This is seen for thermometers 138.6 95.2
based on gases as thermometric substances. In a
constant volume gas thermometer, the pressure ?R 27 u 138.6 95.2 95.2
of a fixed volume of gas (measured by the
difference in height) is used as the thermometric 100
property. It is an accurate but bulky instrument.
11.72 95.2 106.92 :

Liquid-in-glass thermometer depends Normally in research laboratories,
on the change in volume of the liquid with a thermocouple is used to measure the
temperature. The liquid in a glass bulb expands temperature. A thermocouple is a junction of
up a capillary tube when the bulb is heated. The two different metals or alloys e.g., copper and
liquid must be easily seen and must expand (or iron joined together. When two such junctions
contract) rapidly and by a large amount for a at the two ends of two dissimilar metal rods
small change in temperature over a wide range are kept at two different temperatures, an
of temperature. Most commonly used liquids electromotive force is generated that can be
are mercury and alcohol as they remain in liquid calibrated to measure the temperature.
state over a wide range. Mercury freezes at -39
°C and boils at 357 °C; alcohol freezes at -115 °C Thermistor is another device used to
and boils at 78 °C. Thermochromic liquids are measure temperature based on the change in
ones which change colour with temperature but resistance of a semiconductor materials i.e., the
have a limited range around room temperatures. resistance is the thermometric property. You
For example, titanium dioxide and zinc oxide will learn more about this device in Chapter
are white at room temperature but when heated 14 on Semiconductors.
change to yellow.
7.4 Absolute Temperature and Ideal Gas
Equation:

Example 7.3: The length of a mercury column 7.4.1 Absolute zero and absolute temperature
in a mercury-in-glass thermometer is 25 mm at
the ice point and 180 mm at the steam point. Experiments carried out with gases at
What is the temperature when the length is 60 low densities indicate that while pressure is
mm? held constant, the volume of a given quantity
of gas is directly proportional to temperature
Solution: Here the thermometric property P is (measured in ºC). Similarly, if the volume of
the length of the mercury column. Using Eq. a given quantity of gas is held constant, the
(7.2), we get pressure of the gas is directly proportional to
temperature (measured in ºC). These relations
T 100 60 25 22.58 qC are graphically shown in Fig. 7.2 (a) and (b).
180 25 Mathematically, this relationship can be written
as PV ∝ TC. Thus the volume-temperature
Resistance thermometer uses the change or pressure-temperature graphs for a gas are
of electrical resistance of a metal wire straight lines. They show that gases expand
with temperature. It measures temperature linearly with temperature on a mercury
accurately in the range -2000 °C to 1200 °C but thermometer i.e., equal temperature increase
it is bulky and is best for steady temperatures. causes equal volume or pressure increase. The
similar thermal behavior of all gases suggests
Example 7.4: A resistance thermometer has that this relationship of gases can be used to
resistance 95.2 Ω at the ice point and 138.6 Ω measure temperature in a constant-volume gas
at the steam point. What resistance would be thermometer in terms of pressure of the gas.
obtained if the actual temperature is 27 ºC?
Although actual experimental
Solution: Here the thermometric property P
is the resistance. Using Eq. (7.2), if R is the measurements might differ a little from the
resistance at 27 ºC, we have
ideal linear relationship, the linear relationship

118

holds over a wide temperature range. temperature is not possible in practice. It may
be noted that the point of zero pressure or zero
Fig. 7.2 (a): Graph of volume versus volume does not depend on any specific gas.
temperature (in °C) at constant pressure.
The two fixed point scale, described in
Fig. 7.2 (b): Graph of pressure versus Section 7.3, had a practical shortcoming for
temperature (in °C) at constant volume. calibrating the scale. It was difficult to precisely
control the pressure and identify the fixed
It may be noted that the lines do not points, especially for the boiling point as the
pass through the origin i.e., have non-zero boiling temperature is very sensitive to changes
intercept along the y-axis. The straight lines in pressure. Hence, a one fixed point scale
have different slopes for different gases. If we was adopted in 1954 to define a temperature
assume that the gases do not liquefy even if scale. This scale is called the absolute scale or
we lower the temperature, we can extend the thermodynamic scale. It is named as the kelvin
straight lines backwards for low temperatures. scale after Lord Kelvin (1824-1907).
Is it possible to reach a temperature where the
gases stop exerting any pressure i.e., pressure is It is possible for all the three phases - solid,
zero? In a constant pressure thermometer, as the liquid and gas/vapour of a material - to coexist
temperature is lowered, the volume decreases. in equilibrium. This is known as the triple point.
Suppose the gas does not liquefy even at very To know the triple point one has to see that three
low temperature, at what temperature, will phases coexist in equilibrium and no one phase
its volume become zero? Practically it is not is dominating. This occurs for each substance at
possible to keep a material in gaseous state for a single unique combination of temperature and
very low temperature and without exerting any pressure. Thus if three phases of water - solid
pressure. If we extrapolate the graph of pressure ice, liquid water and water vapour- coexist,
P versus temperature TC (in ºC), the temperature the pressure and temperature are automatically
at which the pressure of a gas would be zero fixed. This is termed as the triple point of water
is -273.15 ºC. It is seen that all the lines for and is a single fixed point to define a temperature
different gases cut the temperature axis at the scale.
same point at i.e., -273.15 ºC. This point is
termed as the absolute zero of temperature. The absolute scale of temperature, is so
It is not possible to attain a temperature lower termed since it is based on the properties of an
than this value. Even to achieve absolute zero ideal gas and does not depend on the property
of any particular substance. The zero of this
scale is ideally the lowest temperature possible
although it has not been achieved in practice. It
is termed as Kelvin scale with its zero at -273.15
°C and temperature intervals same as that on
the celsius scale. It is written as K (without °).
Internationally, triple point of water has been
assigned as 273.16 K at pressure equal to 6.11 ×
102 Pa or 6.11 × 10-3 atmosphere, as the standard
fixed point for calibration of thermometers. Size
of one kelvin is thus 1/273.16 of the difference
between the absolute zero and triple point of
water. It is same as one celcius. On celcius scale,
the triple point of water is 0.01 ºC and not zero.

Three identical thermometers, marked in
kelvin, celcius and fahrenheit, placed in a fixed
temperature bath, each thermometer showing

119

the same rise in the level of mercury for human 7.4.2 Ideal Gas Equation:
body temperature, are depicted in Fig. 7.3.
The relation between three properties of
The relation between the three scales of a gas i.e., pressure, volume and temperature is
temperature is as given in Eq. (7.3) . called ideal gas equation. You will learn more
about the properties of gases in chemistry.
TC TF 32 TK 273.15 --- (7.3)
100 180 100 Using absolute temperatures, the gas laws
can be stated as given below.

1) Charles’ law- In Fig. 7.2 (b), the volume-
temperature graph passes through the
origin if temperatures are measured on the
kelvin scale, that is if we take 0 K as the
origin. In that case the volume V is directly
proportional to the absolute temperature T.

Thus V∝T

Fig. 7.3: Comparison of the kelvin, or, V = constant --- (7.4)
celsius and fahrenheit temperature scales Thus Charles'Tlaw can be stated as, the
(Thermometer reading are not to the scale).
volume of a fixed mass of gas is directly

Example 7.5: Express T = 24.57 K in celsius proportional to its absolute temperature if
and fahrenheit.
Solution: We have the pressure is kept constant.

2) Pressure (Gay Lussac's) law- From

TF 32 TC = TK - 273.15 Fig.7.2, it can be seen that the pressure-

180 100 100 temperature graph is similar to the volume-

?TC = TK -273.15 temperature graph.

= 24.57-273.15 Thus P ∝ T

= - 248.58qC or, P = constant --- (7.5)
T
TF 32 TK - 273.15 Pressure law can be stated as the pressure of

180 100 a fixed mass of gas is directly proportional

?TF 180 (TK - 273.15) + 32 to its absolute temperature if the volume is
100

9 (24.57 273.15) 32 kept constant.
5
3) Boyle’s law- For fixed mass of gas at
= - 447.44 + 32 constant temperature, pressure is inversely
proportional to volume.
= - 415.44qF
Thus P ∝ 1
Example 7.6: Calculate the temperature which V
has the same value on fahernheit scale and PV = constant --- (7.6)
kelvin scale.
Solution: Let the required temperature be y. Combining above three equations, we get
i.e., TF = TK = y then we have
PV = constant --- (7.7)
y 32 y 273.15 T

180 100 For one mole of a gas, the constant of

or, 5y 160 9 y 2458.35 proportionality is written as R

or, 4 y 160 2458.35 ∴ PV =R or PV = RT --- (7.8)
T
? y 574.59
If given mass of a gas consists of n moles,
Thus 574.59 °F and 574.59 K are equivalent
temperatures. then Eq. (7.8) can be written as

PV= nRT --- (7.9)

120

This relation is called ideal gas equation. If the substance is in the form of a long
rod of length l, then for small change ∆T, in
The value of constant R is same for all gases. temperature, the fractional change ∆l/l, in length
(shown in Fig.7.4), is directly proportional to
Therefore, it is known as universal gas constant. ∆T.

Its numerical value is 8.31 J K-1 mol-1.

Example 7.7: The pressure reading in a

thermometer at steam point is 1.367 × 103 Pa. 'l v 'T
l
What is pressure reading at triple point knowing

the linear relationship between temperature and or 'l D 'T --- (7.10)
l
pressure?

Solution: We have Pprtreipsles=ur2e7s3a.1t 6te×m§¨©pTPera¹¸·twurheeoref where α is called the coefficient of linear
Ptriple and P are the expansion of solid. Its value depends upon
triple point (273.16 K) and T respectively. We nature of the material. Rearranging Eq. (7.10),
we get
are given that P = 1.367×103 Pa at steam point

i.e., at 273.15 + 100 = 373.15 K. D 'l
l'T
§ 1.367u103 ·
∴ Ptriple = 273.16 × ¨ ¸ = lT l0
= 1.000×103 Pa © 373.15 ¹ --- (7.11)
l0 (T T0 )

7.5 Thermal Expansion: where l0 = length of rod at 0 °C
lT = length of rod when heated to T °C
When matter is heated, it normally expands T0 = 0 °C is initial temperature
and when cooled, it normally contracts. The T = final temperature
atoms in a solid vibrate about their mean
positions. When heated, they vibrate faster and ∆l =lT - l0 = change in length
force each other to move a little farther apart. ∆T =T - T0= rise in temperature
This results into expansion. The molecules in Referring to Eq. (7.11), if l0=1 and T- T0=1 °C,
a liquid or gas move with certain speed. When then
heated, they move faster and force each other
to move a little farther apart. This results in α = lT - l0 (numerically).
expansion of liquids and gases on heating. The Coefficient of linear expansion of a solid
expansion is more in liquids than in solids;
gases expand even more. is thus defined as increase in the length per
unit original length at 0 °C for one degree
A change in the temperature of a body centigrade rise in temperature.
causes change in its dimensions. The increase
in the dimensions of a body due to an increase The unit of coefficient of linear expansion is
in its temperature is called thermal expansion. per degree celcius or per kelvin. The magnitude
There are three types of thermal expansion: of α is very small and it varies only a little with
1) Linear expansion, 2) Areal expansion, temperature. For most practical purposes, α
3) Volume expansion. can be assumed to be constant for a particular
7.5.1 Linear Expansion: material. Therefore, it is not necessary that
initial temperature be taken as 0 °C. Equation
The expansion in length due to thermal (7.11) can be rewritten as
energy is called linear expansion.
D l2 l1
l1(T2 T1) --- (7.12)

Fig. 7.4: Linear expansion ∆l is exaggerated where l1 = initial length at temperature T1 °C
for explanation. l2 = final length at temperature T2 °C.

Table 7.1 lists average values of coefficient
of linear expansion for some materials in the

121

temperature range 0°C to 100 °C. We have

Table 7.1: Values of coefficient of linear D = l2 l1 l2 l1
expansion for some common materials. l1(T2 T1) l1T2 l1T1

Materials a (K-1) ? l1T2 l1T1 l2 l1
D
Carbon (diamond) 0.1×10-5
Glass 0.85×10-5 T2 1 ª«l1T1 l2 l1 º
Iron 1.2×10-5 l1 ¬ D »
Steel 1.3×10-5 ¼
Gold 1.4×10-5
Copper 1.7×10-5 1 ¬ª«(4.256 u 27) 4.268 4.256 º
Silver 1.9×10-5 4.256 1.2 u10 5 ¼»
2.5×10-5
Aluminium 6.1×10-5 1 «¬ª114.912 0.012 º
Sulphur 6.1×10-5 4.256 1.2 u10 5 »¼
Mercury 6.9×10-5
Water 8.8×10-5 1 ¬ª114.912 1000º¼
4.256
Carbon (graphite)
261.96 qC

7.5.2 Areal Expansion:

The increase ∆A, in the surface area, on
heating is called areal expansion or superficial

Example 7.8: The length of a metal rod at 27 °C expansion.
is 4 cm. The length increases to 4.02 cm when
the metal rod is heated upto 387 °C. Determine 'A v 'T
the coefficient of linear expansion of the metal A
rod.
or 'A E 'T
A
Solution: Given

T1 = 27 °C Fig. 7.5: Areal expansion ∆A is exaggerated
for explanation.
T2 = 387 °C
If a substance is in the form of a plate of
l1 = 4 cm = 4×10-2 m
l2 = 4.02 cm = 4.02×10-2 m area A, then for small change ∆T in temperature,
the fractional change in area, ∆A/A (as shown
We have in Fig. 7.5), is directly proportional to ∆T.

D l2 l1 'A v 'T
l1(T2 T1) A

4.02 4.0 u10 2 or 'A E 'T --- (7.13)
A


4 u10 2 (387 27) where β is called the coefficient of areal
expansion of solid. It depends on the material
0.02 u10 2
4 u10 2 u 360 of the solid. Rearranging Eq. (7.13), we get
1.39u10 5 / qC
E 'A AT A0 --- (7.14)
Example 7.9: Length of an iron rod at A'T
temperature 27 °C is 4.256 m. Find the A0 (T T0 )
temperature at which the length of the same rod
increases to 4.268 m.(α for iron = 1.2×10-5 K-1) where A0= area of plate at 0 °C
AT = area of plate when heated to T °C
Solution: Given T0 = 0 °C is initial temperature
T = final temperature

T1 = 27 °C, l1 = 4.256 m, ∆A = AT - A0 = change in area
l2 = 4.268m, α = 1.2×10-5 K-1 ∆T =T - T0= rise in temperature.

If A0 = 1 m2 and T - T0 = 1 °C, then
β = AT - A0 (numerically).

122

Therefore, coefficient of areal expansion of a If the substance is in the form of a cube
solid is defined as the increase in the area per
unit original area at 0°C for one degree rise of volume V, then for small change ∆T in
in temperature.
temperature, the fractional change, ∆V/V

(as shown in Fig.7.6), in volume is directly

The unit of β is per degree celcius or per proportional to ∆T.
kelvin.
'V v 'T
V
As in the case of α, β also does not vary
'V J 'T --- (7.16)
much with temperature. Hence, if A1 is the area or V
of a metal plate at T1 °C and A2 is the area at
higher temperature T2 °C, then where γ is called coefficient of cubical or

A2 A1 volume expansion. It depends upon the nature
A1(T2 T1)
E --- (7.15) of the material. Its unit is per degree celcius or

per kelvin. From Eq.(7.16), we can write

Example 7.10: A thin aluminium plate has an J 'V VT V0 --- (7.17)
area 286 cm2 at 20 °C. Find its area when it is V 'T V0 (T - T0 )
heated to 180 °C.

where V0 = volume at 0 °C

(β for aluminium = 4.9×10-5 /°C) VT = volume when heated to T °C

Solution: Given T0 = 0 °C is initial temperature

T1 = 20 °C T = final temperature

T2 = 180 °C ∆V = VT - V0= change in volume
A1 = 286 cm2 ∆T =T - T0= rise in temperature.
β = 4.9×10-5 /°C If V0 = 1 m3, T - T0=1 °C, then

We have γ =VT - V0 (numerically).
The coefficient of cubical expansion of
E A2 A1 a solid is therefore defined as increase in
A1(T2 T1) volume per unit original volume at 0°C for
one degree rise in the temperature.
∴ A2= A1 [1 + β (T2-T1)]
= 286 [1 + 4.9×10-5 (180-20)] If V1 is the volume of a body at T1 °C and
V2 is the volume at higher temperature T2 °C,
= 286 [1 + 4.9×10-5×160] then
= 286 [1 + 784.0×10-5]
= 286 [1 + 0.00784] J1 V2 V1 --- (7.18)

= 286 [1.00784] V1(T2 T1)
... A2 = 288.24 cm2
7.5.3 Volume expansion γ1 is the coefficient of volume expansion at
temperature T1 °C.
The increase in volume due to heating is
called volume expansion or cubical expansion. Since fluids possess definite volume and

take the shape of the container, only change

in volume is significant. Equations (7.17) and

(7.18) are valid for cubical or volume expansion

of fluids. It is to be noted that since fluids are kept

'V v 'T in containers, when one deals with the volume
V
expansion of fluids, expansion of the container

or 'V J 'T is also to be considered. If expansion of fluid
V
results in a volume greater than the volume

Fig. 7.6: Volume expansion ∆V is of the container, the fluid will overflow if the
exaggerated for explanation.
container is open. If the container is closed,

volume expansion of fluid will cause additional

123

pressure on the walls of the container. Can you Example 7.11 : A liquid at 0 °C is poured
now tell why the balloon bursts sometimes on in a glass beaker of volume 600 cm3 to fill it
its own on a hot day? completely. The beaker is then heated to 90 °C.
How much liquid will overflow?
Normally solids and liquids expand on
heating. Hence their volume increases on (γliquid = 1.75×10-4 /°C, γglass = 2.75×10-5 /°C)
heating. Since the mass is constant, it results in Solution: Given
a decrease in the density on heating. You have
learnt about the anomalous behaviour of water. V1= 600 cm3
Water expands on cooling from 4°C to 0°C.
Hence its density decreases on cooling in this T1 = 0 °C
temperature range.
T2 = 90 °C
In Table 7.2 are given typical average We have J V2 V1
values of the coefficient of volume expansion
γ for some materials in the temperature range V1(T2 T1)
0°C to 100°C.
... increase is volume = V2 - V1= γ V1 (T2- T1)
Table 7.2: Values of coefficient of volume
expansion for some common materials. Increase in volume of beaker

Materials γ (K-1) = γglass× V1 (T2- T1)
= 2.75×10-5×600×(90-0)

Invar 2×10-6 = 2.75×10-5×600×90
Glass (ordinary) 2.5×10-5
(3.3-3.9)×10-5 = 148500×10-5 cm3
Steel 3.55×10-5 ... increase in volume of beaker = 1.485 cm3
Iron 4.2×10-5
Gold 5.7×10-5 Increase in volume of liquid
Brass 6.9×10-5
Aluminium 18.2×10-5 = γliquid × V1 (T2- T1)
Mercury 20.7×10-5 = 1.75×10-4×600×(90-0)
Water 58.8×10-5
Paraffin 95.0×10-5 = 1.75×10-4×600×90
Gasoline 110×10-5
Alcohol (ethyl) = 94500×10-4 cm3
... increase in volume of liquid = 9.45 cm3
γ is also characteristic of the substance but ... volume of liquid which overflows
is not strictly a constant. It depends in general
on temperature as shown in Fig.7.7. It is seen = (9.45-1.485) cm3
that γ becomes constant only at very high
temperatures. = 7.965 cm3

7.5.4 R elation between Coefficients of

Expansion:

i) Relation between β and α:

Consider a square plate of side l0 at 0 °C
and at T °C.
... lT lT l0 (1+αT) from Eq. (7.11).
=

If area of plate at 0 °C is A0, A0 = l02.

If area of plate at T °C is AT,

AT = lT2 = l02 (1+αT)2

or AT = A0 (1+αT)2 --- (7.19)
Also from Eq. (7.14),

Fig. 7.7: Coefficient of volume expansion AT = A0 (1+βT) --- (7.20)
of copper as a function of temperature.

124

Using Eqs. (7.19) and (7.20), we get Solution: Given

A0 (1+αT)2 = A0 (1+βT) T1 = 0 °C
T2 = 100 °C
or 1+ 2αT +α2T2 =1+βT A1 = 50×8 = 400 cm2
A2 = 401.57 cm2
Since the values of α are very small, the
We have
term α2T2 is very small and may be neglected.

∴ β = 2α --- (7.21) A2 A1
A1(T2 T1)
E 2D

Can you tell? (401.57 400) cm2

1. Why the metal wires for electrical 400 cm2 u (100 0) qC
transmission lines sag?
1.57 0.3925u10 4 qC 1
2. Why a railway track is not a continuous 400 u100
piece but is made up of segments
separated by gaps? ?D 0.1962 u10 4 qC 1

3. How a steel wheel is mounted on an 1.962 u10 5 qC 1
axle to fit exactly?
∴ Coefficient of linear expansion of brass
4. Why lakes freeze first at the surface? is 1.962×10-5 /°C.

The result is general because any solid can

be regarded as a collection of small squares. Do you know ?

ii) Relation between γ and α: * When pressure is held constant, due to
change in temperature, the volume of
Consider a cube of side l0 at 0 °C a liquid or solid changes very little in
comparison to the volume of a gas.
and lT at T °C. from Eq. (7.11).
... lT = l0 (1+αT) * The coefficient of volume expansion, γ, is
generally an order of magnitude larger for
If volume of the cube at 0 °C is V0, V0 = l03. liquids than for solids.

If volume of the cube at T °C is VT , * Metals have high values for the coefficient
VT = lT3 = l03 (1+αT)3 for linear expansion, α, than non-metals.

or VT = V0 (1+αT)3 --- (7.22) * γ changes more with temperature than α
and β.
Also from Eq. (7.17),
* We know that water expands on freezing
VT = V0 (1+γT) --- (7.23) from 4 ºC to 0 ºC. Other two substances,
Using Eqs. (7.22) and (7.23), we get that expnad on freezing are metals bismuth
(Bi) and antimony (Sb). Thus the density
V0 (1+αT)3 = V0 (1+γT) of liquid is more than corresponding solid
and hence solid Bi or Sb float on their
or 1+ 3αT +3α2T 2 + α3T 3 =1+γT liquids like ice floats on water.

Since the values of α are very small, the 7.6 Specific Heat Capacity:

terms with higher powers of α may be neglected. 7.6.1 Specific Heat Capacity of Solids and

∴ γ = 3α --- (7.24) Liquids

Again the result is general because any If 1 kg of water and 1 kg of paraffin are
heated in turn for the same time by the same
solid can be regarded as a collection of small heater, the temperature rise of paraffin is about
twice that of water. Since the heater gives
cubes. equal amounts of heat energy to each liquid, it
seems that different substances require different
Finally, the relation between α, β and γ is

D E J --- (7.25)
2 3

Example 7.12: A sheet of brass is 50 cm long

and 8 cm broad at 0 °C. If the surface area at

100 °C is 401.57cm2, find the coefficient of

linear expansion of brass.

125

amounts of heat to cause the same temperature C 1 'Q --- (7.27)
rise of 1°C in the same mass of 1 kg. P 'T

If ∆Q stands for the amount of heat
absorbed or given out by a substance of mass The SI unit of molar specific heat capacity
m when it undergoes a temperature change ∆T, is J/mol °C or J/mol K. Like specific heat, molar
then the specific heat capacity of that substance specific heat also depends on the nature of the
is given by substance and its temperature. Table 7.3 lists
the values of specific heat capacity for some
s 'Q common materials.
m'T --- (7.26)
If m = 1 kg and ∆T = 1°C then s = ∆Q. From Table 7.3, it can be seen that water
Thus specific heat capacity is defined as has the highest specific heat capacity compared
the amount of heat per unit mass absorbed to other substances. For this reason, water is
or given out by the substance to change its used as a coolant in automobile radiators as well
temperature by one unit (one degree) 1 °C as for fomentation using hot water bags.
or 1K.
7.6.2 Specific Heat Capacity of Gas:
Table 7.3: Specific heat capacity of some
substances at room temperature and In case of a gas, slight change in
atmospheric pressure. temperature is accompanied with considerable
changes in both, the volume and the pressure.
Substance Specific heat capacity If gas is heated at constant pressure, volume
(J kg-1 K-1) changes and therefore some work is done on
the surroundings during expansion requiring
Steel 120 additional heat. As a result, specific heat at
Lead 128 constant pressure (Sp) is greater than specific
Gold 129 heat at constant volume (Sv). It is thus necessary
Tungsten 134.4 to define two principal specific heat capacities
Silver 234 for a gas.
Copper 387
Iron 448 Principal specific heat capacities of gases:
Carbon 506.5
Glass 837 a) The principal specific heat capacity of a
Aluminium 903.0
Kerosene 2118 gas at constant hveoaltumabeso(Srbv)edis defined as
Paraffin oil 2130 the quantity of or released
Alcohol (ethyl) 2400
Ethanol 2500 for the rise or fall of temperature of unit
Water 4186.0
mass of a gas through 1 K (or 1°C) when

its volume is kept constant.

b) The principal specific heat capacity of a

The SI unit of specific heat capacity is J/ gas at constant hpereastsaubreso(rSbp)edisodrerfeinleeadseads
kg °C or J/kg K and C.G.S. unit is erg/g °C or the quantity of
erg/g K. The specific heat capacity is a property
of the substance and weakly depends on its for the rise or fall of temperature of unit
temperature. Except for very low temperatures,
the specific heat capacity is almost constant for mass of a gas through 1K (1°C) when its
all practical purposes.
pressure is kept constant.

Molar specific heat capacities of gases:

If the amount of substance is specified in a) Molar specific heat capacity of a gas at
terms of moles µ instead of mass m in kg, then constant volume (C ) is defined as the
the specific heat is called molar specific heat (C) quantity of heat absov rbed or released for
and is given by the rise or fall of temperature of one mole
of the gas through 1K (or 1°C), when its
volume is kept constant.

126

b) Molar specific heat capacity of a gas at = 4×10 × 300 J
... Q = 12000 J
constant opfrehsesautreab(sCopr)beisd defined as the
quantity or released for 7.6.4 Heat Capacity (Thermal Capacity):

the rise or fall of temperature of one mole Heat capacity or thermal capacity of a
body is the quantity of heat needed to raise or
of the gas through 1K (or 1°C), when its lower the temperature of the whole body by
1°C (or 1K).
pressure is kept constant.

Relation between Principal and Molar ∴ Thermal heat capacity can be written as
Specific Heat Capacities: Heat received or given out

A relation between principal specific heat = mass × 1 × specific heat capacity
capacity and molar specific heat capacity is
given by the following expression Heat capacity = Q = m × s --- (7.29)

Molar specific heat capacity = Molecular Heat capacity (thermal capacity) is measured
weight × principal specific heat capacity. in J/°C.

i.e. Cp= M × Sp and Cv= M × Sv Example 7.14: Find thermal capacity for a
where M is the molecular weight of the gas. copper block of mass 0.2 kg, if specific heat
capacity of copper is 290 J/kg °C.
Table 7.4 lists values of molar specific Solution: Given
heat capacity for some commonly known gases.
m = 0.2 kg
Table 7.4: Molar specific heat capacity of s = 290 J/kg °C
some gases.

Gas CP CV Thermal capacity = m × s = 0.2 kg×290 J/kg °C
(J mol-1 K-1) (J mol-1 K-1)
=58 J/ °C
He 20.8 12.5
H2 28.8 20.4 7.7 Calorimetry:
N2 29.1 20.8
O2 29.4 21.1 Calorimetry is an experimental technique
CO2 37.0 28.5
for the quantitative measurement of heat

exchange. To make such measurement a

calorimeter is used. Figure 7.8 shows a simple

7.6.3 Heat Equation: water calorimeter.

If a substance has a specific heat capacity It consists of cylindrical vessel made of
of 1000 J/kg °C, it means that heat energy of
1000 J raises the temperature of 1 kg of that copper or aluminium and provided with a stirrer
substance by 1°C or 6000 J will raise the
temperature of 2 kg of the substance by 3 °C. If and a lid. The calorimeter is well-insulated to
the temperature of 2 kg mass of the substance
falls by 3 °C, the heat given out would also prohibit any transfer of heat into or out of the
be 6000 J. In general we can write the heat
equation as calorimeter.

Heat received or given out (Q) = mass

(m) × temperature change (∆t) × specific heat

capacity (s).

or Q = m × DT × s --- (7.28)

Example 7.13: If the temperature of 4 kg Fig. 7.8: Calorimeter.
mass of a material of specific heat capacity One important use of calorimeter is to
300 J/ kg °C rises from 20 °C to 30 °C. Find the determine the specific heat of a substance using
heat received.
Solution:

Q = 4 kg × (30-20) °C × 300 J/kg °C

127

the principle of conservation of energy. s1 (m2s2 m3s3 )(T T2 ) --- (7.31)
Here we are dealing with heat energy and the
system is isolated from surroundings. Therefore, m1(T1 T )
heat gained is equal to the heat lost.
Also, one can find specific heat capacity
In the technique known as the “method
of mixtures”, a sample 'A' of the substance is of water or any liquid using the following
heated to a high temperature which is accurately
measured. The sample 'A' is then placed quickly expression, it the specific heat capacity of the
in the calorimeter containing water. The contents
are stirred constantly until the mixture attains a material of calorimeter and sample is known
final common temperature. The heat lost by the
sample 'A' will be gained by the water and the s3 m1s1(T1 T ) m2 s2 --- (7.32)
calorimeter. The specific heat of the sample 'A' m3(T T2 ) m3
of the substance can be calculated as under:

Let Note - In the experiment, the heat from the solid
sample 'A' is given to the liquid and therefore
m1 = mass of the sample 'A' the sample should be denser than the liquid, so
m2 = mass of the calorimeter and the stirrer that sample does not float on the liquid.
m3 = mass of the water in calorimeter
s1 = specific heat capacity of the substance of Example 7.15: A sphere of aluminium of 0.06
sample 'A' kg is placed for sufficient time in a vessel
containing boiling water so that the sphere is
s2 = specific heat capacity of the material of at 100 °C. It is then immediately transferred to
calorimeter (and stirrer) 0.12 kg copper calorimeter containing 0.30 kg
of water at 25 °C. The temperature of water rises
s3 = specific heat capacity of water and attains a steady state at 28 °C. Calculate
T1 = initial temperature of the sample 'A' the specific heat capacity of aluminium.
T2 = initial temperature of the calorimeter (Specific heat capacity of water, sw = 4.18 ×
stirrer and water 103J kg-1 K-1, specific heat capacity of copper
sCu = 0.387×103 J kg-1 K-1)
T = final temperature of the combined system Solution : Given

We have the data as follows: Mass of aluminium sphere = m1 = 0.06 kg
Mass of copper calorimeter = m2= 0.12 kg
Heat lost by the sample 'A' = m1s1 (T1- T) Mass of water in calorimeter
Heat gained by the calorimeter and the stirrer
= m3= 0.30 kg
= m2s2 (T - T2) Specific heat capacity of copper
Heat gained by the water = m3s3 (T - T2)
= sCu = s2 = 0.387×103 J kg-1 K-1
Assuming no loss of heat to the Specific heat capacity of water
surroundings, the heat lost by the sample goes
into the calorimeter, stirrer and water. Thus = sw = s3= 4.18×103 J kg-1 K-1
writing heat equation as, Initial temperature of aluminium sphere

m1s1(T1- T) = T1 =100°C
= m2s 2(T - T2) + m3s3(T - T2) ---(7.30) Initial temperature of calorimeter and
water = T2= 25°C
Knowing the specific heat capacity of Final temperature of the mixture
water (s3 = 4186 J kg-1 K-1) and copper (s2 = 387
J kg-1 K-1) being the material of the calorimeter = T = 28°C
and the stirrer, one can calculate specific heat
capacity (s1) of material of sample 'A', from Eq. We have
(7.30) as

128

s1 (m2s2 m3s3 )(T T2 ) (ice) to liquid (water).
m1(T1 T )

= ª¬ 0.12 u 387 0.30u 4180 º¼ 28 25 Temperature T (0C)
............................
(0.06)(100 28) C
D
46.44 1254 u 3 3901.32
= = 4.32
(0.06) u 72
A
= 903.08 J kg-1 K-1 (0,0) B Time (t) (s)

∴ Specific heat capacity of aluminium is Fig. 7.9 : Variation of temperature with time.

903.08 J kg-1 K-1. a) The change of state from solid to liquid is
called melting and from liquid to solid is
7.8 Change of State: called solidification.

Matter normally exists in three states: solid, b) Both the solid and liquid states of the
liquid and gas. A transition from one of these substance co-exist in thermal equilibrium
states to another is called a change of state. Two during the change of states from solid to
common changes of states are solid to liquid and liquid or vice versa.
liquid to gas (and vice versa). These changes
can occur when exchange of heat takes place c) The temperature at which the solid and
between the substance and its surroundings. the liquid states of the substance are in
thermal equilibrium with each other is
Activity called the melting point of solid (here ice)
or freezing point of liquid (here water). It
To understand the process of change of is characteristic of the substance and also
state depends on pressure.

Take some cubes of ice in a beaker. d) The melting point of a substance at one
Note the temperature of ice (0 °C). Start standard atmospheric pressure is called its
heating it slowly on a constant heat source. normal melting point.
Note the temperature after every minute.
Continuously stir the mixture of water and e) At one standard atmospheric pressure, the
ice. Observe the change in temperature. freezing point of water and melting point
Continue heating even after the whole of of ice is 0 °C or 32°F. The freezing point
ice gets converted into water. Observe describes the liquid to solid transition
the change in temperature as before till while melting point describes solid-to-
vapours start coming out. Plot the graph liquid transition.
of temperature (along y-axis) versus time
(along x-axis). You will obtain a graph 2) From point B to D:
of temperature versus time as shown in The temperature begins to rise from point
Fig. 7.9.
B to point C, i.e., after the whole of ice gets
Analysis of observations : converted into water and we continue further
heating. We see that temperature begins to
1) From point A to B: rise. The temperature keeps on rising till it
reaches point C i.e., nearly 100 °C. Then it
There is no change in temperature from again becomes steady. It is observed that the
point A to point B, this means the temperature temperature remains constant until the entire
of the ice bath does not change even though amount of the liquid is converted into vapour.
heat is being continuously supplied. That is the The heat supplied is now being utilized to
temperature remains constant until the entire change water from liquid state to vapour or
amount of the ice melts. The heat supplied is gaseous state.
being utilised in changing the state from solid a) The change of state from liquid to vapour

129

is called vapourisation while that from is more. Hence the rate of losing such molecules
vapour to liquid is called condensation. to atmosphere will be larger. Thus, higher is the
b) Both the liquid and vapour states of the temperature of the liquid, greater is the rate of
substance coexist in thermal equilibrium evaporation. Since faster molecules are lost, the
during the change of state from liquid to average kinetic energy of the liquid is reduced
vapour. and hence the temperature of the liquid is
c) The temperature at which the liquid and lowered. Hence the phenomenon of evaporation
the vapour states of the substance coexist gives a cooling effect to the remaining liquid.
is called the boiling point of liquid, Since evaporation takes place from the surface
here water or steam point. This is also of a liquid, the rate of evaporation is more if the
the temperature at which water vapour area exposed is more and if the temperature of
condenses to form water. the liquid is higher.
d) The boiling point of a substance at one
standard atmospheric pressure is called its You might have seen that if your mother
normal boiling point. wants her sari/clothes to dry faster, she does
not fold them. More is the area exposed, faster
Can you tell? is the drying because the water gets evaporated
faster. The presence of wind or strong breeze
1. What after point D in graph ? Can steam and content of water vapour in the atmosphere
be hotter than 100 °C ? are two other important factors determining the
drying of clothes but we do not refer to them here.
2. Why steam at 100 °C causes more harm
to our skin than water at 100 °C? Before giving an injection to a patient,
normally a spirit swab is used to disinfect the
Do you know ? region. We feel a cooling effect on our skin due
to evaporation of the spirit as explained before.
You must have seen that water spilled
on floor dries up after some time. Where does Activity
the water disappear? It is converted into water
vapour and mixes with air. We say that water Activity to understand the dependence
has evaporated. You also know that water can be of boiling point on pressure
converted into water vapour if you heat the water
till its boiling point. What is then the difference Take a round bottom flask, more than
between boiling and evaporation? half filled with water. Keep it over a burner
and fix a thermometer and steam outlet
Both evaporation and boiling involve through the cork of the flask
change of state, evaporation can occur at any as shown in figure. As water
temperature but boiling takes place at a fixed in the flask gets heated, note
temperature for a given pressure, unique for each that first the air, which was
liquid. Evaporation takes place from the surface dissolved in the water comes
of liquid while boiling occurs in the whole liquid. out as small bubbles. Later
bubbles of steam form at the
As you know, molecules in a liquid are bottom but as they rise to the
moving about randomly. The average kinetic cooler water near the top, they
energy of the molecules decides the temperature condense and disappear. Finally, as the
of the liquid. However, all molecules do not temperature of the entire mass of the water
move with the same speed. One with higher reaches 100 °C, bubbles of steam reach the
kinetic energy may escape from the surface surface and boiling is said to occur. The
region by overcoming the interatomic forces. steam in the flask may not be visible but as
This process can take place at any temperature. it comes out of the flask, it condenses as tiny
This is evaporation. If the temperature of the droplets of water giving a foggy appearance.
liquid is higher, more is the average kinetic
energy. Since the number of molecules is fixed, it If now the steam outlet is closed for a
implies that the number of fast moving molecules few seconds to increase the pressure in the

130

flask, you will notice that boiling stops. arrangement of carbon atoms is different in the
More heat would be required to raise the two cases. Figure 7.10 shows the phase diagram
temperature (depending on the increase in of water and CO2. Let us try to understand the
pressure) before boiling starts again. Thus diagram.
boiling point increases with increase in
pressure. Fig. 7.10 (a): Phase diagram of water (not
to scale).
Let us now remove the burner. Allow
water to cool to about 80°C. Remove the
thermometers and steam outlet. Close the
flask with a air tight cork. Keep the flask
turned upside down on a stand. Pour ice-
cold water on the flask. Water vapours in the
flask condense reducing the pressure on the
water surface inside the flask. Water begins
to boil again, now at a lower temperature.
Thus boiling point decreases with decrease
in pressure and increases with increase in
pressure.

Can you tell?

1. Why cooking is difficult at high altitude?
2. Why cooking is faster in pressure cooker?

7.8.1 Sublimation: Fig. 7.10 (b): Phase diagram of CO2 (not to
scale).
Have you seen what happens when
camphor is burnt? All substances do not pass i) Vapourisation curve l - v: The curve labelled
through the three states: solid-liquid-gas. l - v represents those points where the liquid and
There are certain substances which normally vapour phases are in equilibrium. Thus it is a
pass from the solid to the vapour state directly graph of boiling point versus pressure. Note
and vice versa. The change from solid state to that the curves correctly show that at a pressure
vapour state without passing through the liquid of 1 atmosphere, the boiling points of water is
state is called sublimation and the substance is 100°C and that the boiling point is lowered for
said to sublime. Dry ice (solid CO2) and iodine a decreased pressure.
sublime. During the sublimation process, both
the solid and vapour states of a substance ii) Fusion curve l - s: The curve l - s represents
coexist in thermal equilibrium. Most substances the points where the solid and liquid phases
sublime at very low pressures. coexist in equilibrium. Thus it is a graph of the
freezing point versus pressure. At one standard
7.8.2 Phase Diagram: atmosphere pressure, the freezing point of water
is 0 °C as shown in Fig. 7.10 (a). Also notice
A pressure - temperature (PT) diagram that at a pressure of one standard atmosphere
often called a phase diagram, is particularly water is in the liquid phase if the temperature is
convenient for comparing different phases of a between 0 °C and 100 °C but is in the solid or
substance. vapour phase if the temperature is below 0 °C
or above 100 °C. Note that l - s curve for water
A phase is a homogeneous composition of slopes upward to the left i.e., fusion curve of
a material. A substance can exist in different water has a slightly negative slope. This is true
phases in solid state, e.g., you are familiar with
two phases of carbon- graphite and diamond.
Both are solids but the regular geometric

131

only of substances that expand upon freezing. Gas and vapour can thus be defined as-
However, for most materials like CO2, the 1) A substance which is in the gaseous phase
l - s curve slopes upwards to the right i.e., fusion and is above its critical temperature is called
curve has a positive slope. The melting point of a gas.
CO2 is -56 °C at higher pressure of 5.11 atm. 2) A substance which is in the gaseous phase
iii) Sublimation curve s - v: The curve and is below its critical temperature is called
labelled s - v is the sublimation point versus a vapour.
pressure curve. Water sublimates at pressure
less than 0.0060 atmosphere, while carbon Vapour can be liquefied simply by
dioxide, which in the solid state is called dry increasing the pressure, while gas cannot.
ice, sublimates even at atmospheric pressure at Vapour also exerts pressure like a gas.
temperature as low as -78 °C.
7.8.4 Latent Heat:
iv) Triple point: The temperature and pressure
at which the fusion curve, the vapourisation Whenever there is a change in the state of
curve and the sublimation curve meet and all the a substance, heat is either absorbed or given out
three phases of a substance coexist is called the but there is no change in the temperature of the
triple point of the substance. That is, the triple substance.
point of water is that point where water in solid,
liquid and gaseous states coexist in equilibrium Latent heat of a substance is the quantity
and this occurs only at a unique temperature and of heat required to change the state of unit
pressure. The triple point of water is 273.16 K mass of the substance without changing its
and 6.11×10-3 Pa and that of CO2 is -56.6 °C and temperature.
5.1×10-5 Pa.
Thus if mass m of a substance undergoes
7.8.3 Gas and Vapour:
a change from one state to the other then the

quantity of heat absorbed or released is given

by Q = mL --- (7.33)

The terms gas and vapour are sometimes where L is known as latent heat and is
used quite randomly. Therefore, it is important characteristic of the substance. Its SI unit is J
to understand the difference between the two. kg-1. The value of L depends on the pressure and
A gas cannot be liquefied by pressure alone, is usually quoted at one standard atmospheric
no matter how high the pressure is. In order pressure.
to liquefy a gas, it must be cooled to a certain
temperature. This temperature is called critical The quantity of heat required to convert
temperature.
unit mass of a substance from its solid state to
Critical temperatures for some common gases
and water vapour are given in Table 7.5. Thus, the liquid state, at its melting point, without
nitrogen must be cooled below -147 °C to
liquefy it by pressure. any change in its temperature is called its

latent heat of fusion (Lf ).
The quantity of heat required to convert

unit mass of a substance from its liquid state

Table 7.5: Critical Temperatures of some to vapour state, at its boiling point without
common gases and water vapour.
any change in its temperature is called its

Gas Critical Temperature latent heat of vteampopuerraiztuarteionve(rLsuv )s. heat energy
Air (°C) (K) A plot of
-190 83
for a given quantity of water is shown is Fig.

7.11.

N2 -147 126 From Fig. 7.11, we see that when heat is
O2 -118 155
CO2 31.1 241.9 added (or removed) during a change of state,
Water vapour 374 647
the temperature remains constant. Also the

slopes of the phase lines are not all the same,

which indicates that specific heats of the various

states are not equal. For water the latent heat

132

of fusion and 2v2a.6p×or1i0sa5Jtikogn-1arreespLef c=tiv3e.3ly3.×T1h0a5Jt
iksg-31.3a3nÍd L10v 5=J of heat is needed to melt 1kg of

ice at 0 °C and 22.6×105J of heat is needed to

convert 1 kg of water to steam at 100 °C. Hence,

steam at 100°C carries 22.6×105J kg-1 more heat

than water at 100 °C. This is why burns from

steam are usually more serious than those from

boiling water. Melting points, boiling point and Fig. 7.11: Temperature versus heat for
water at one standard atmospheric pressure
latent heats for various substances are given in (not to scale).

Table 7.6.

Table 7.6 : Temperature of change of state and latent heats for various substances at one
standard atmosphere pressure.

Substance Melting point Lf Boiling point Lv
(°C) (×105 Jkg-1) (°C) (×105 Jkg-1)
Gold
Lead 1063 0.645 2660 15.8
Water 328 0.25 1744 8.67
Ethyl alcohol 3.33 100 22.6
Mercury 0 1.0 78 8.5
Nitrogen -114 0.12 357 2.7
Oxygen -39 0.26 -196 2.0
-210 0.14 -183 2.1
-219

Example 7.16: When 0.1 kg of ice at 0 °C = mice s (T - T)
is mixed with 0.32 kg of water at 35 °C in a ice 
container. The resulting temperature of the
mixture is 7.8 °C. Calculate the heat of fusion = 0.1 kg×4186 J×(7.8 - 0) C°
of ice (swater = 4186 J kg-1 K-1).
Solution: Given = 3265.08 J

Head lost = Heat gained

36434.944 0.1 Lf 3265.08

mice = 0.1 kg Lf 36434.944 3265.08 = 3316.9864
0.1
mwater = 0.32 kg
= 3.31698u105 J kg-1
Tice = 0 °C

Twater = 35 °C Do you know ?

TF = 7.8 °C The latent heat of vapourization is
much larger than the latent heat of fusion.
swater = 4186 kg K-1 The energy required to completely separate
the molecules or atoms is greater than the
Heat lost by water energy needed to break the rigidity (rigid
bonds between the molecules or atoms) in
= mwater swater (TF- Twater) solids. Also when the liquid is converted
= 0.32 kg × 4186 J × (7.8 - 35) °C into vapour, it expands. Work has to be done
against the surrounding atmosphere to allow
= - 36434.944 J (here negative sign this expansion.
indicates loss of heat energy)

Heat required to melt ice = mice Lf = 0.1×Lf
Heat required to raise temperature of water
(from ice) to final temperature

133

7.9 Heat Transfer: the rod. This method of heat transfer is called
conduction.
Heat may be transferred from one point
of body to another in three different ways- by Those solid substances which conduct heat
conduction, convection and radiation. Heat easily are called good conductors of heat e.g.
transferes through solids by conduction. In this silver, copper, aluminium, brass etc. All metals
process, heat is passed on from one molecule are good conductors of heat. Those substances
to other molecule but the molecules do not which do not conduct heat easily are called bad
leave their mean positions. Liquids and gases conductors of heat e.g. wood, cloth, air, paper,
are heated by convection. In this process, there etc. In general, good conductors of heat are
is a bodily movement of the heated molecules. also good conductors of electricity. Similarly
In order to transfer heat by conduction and bad conductors of heat are bad conductors of
convection a material medium is required. electricity also.
However transfer of heat by radiation does not
need any medium. Radiation of heat energy 7.9.1.1 Thermal Conductivity:
takes place by electromagnetic (EM) waves that
travel with a speed of 3×108 ms-1 in the space/ Thermal conductivity of a solid is a
vacuum . The energy from the Sun comes to us measure of the ability of the solid to conduct
by radiation. It may be noted that conduction is heat through it. Thus good conductors of heat
a slow process of heat transfer while convection have higher thermal conductivity than bad
is a rapid process. However radiation is the conductors.
fastest process because the transfer of heat takes
place at the speed of light. Suppose that one end of a metal rod is

7.9.1 Conduction: heated (see Fig 7.12 (a)). The heat flows by

Conduction is the process by which heat conduction from hot end to the cold end. As a
flows from the hot end to the cold end of a solid
body without any net bodily movement of the result the temperature of every section of the
particles of the body.
rod starts increasing. Under this condition, the
Heat passes through solids by conduction
only. When one end of a metal rod is placed in rod is said to be in a variable temperature state.
a flame while the other end is held in hand, the
end held in hand slowly gets hotter, although After some time the temperature at each section
it itself is not in direct contact with flame. We
say that heat has been conducted from the hot of the rod becomes steady i.e. does not change.
end to the cold end. When one end of the rod
is heated, the molecules there vibrate faster. As Note that temperature of each cross-section of
they collide with their slow moving neighbours,
they transfer some of their energy by collision the rod is constant but not the same. This is
to these molecules which in turn transfer energy
to their neighbouring molecules still farther called steady state condition. Under steady state
down the length of the rod. Thus the energy
of thermal motion is transferred by molecular condition, the temperature at points within the
collisions down the rod. The transfer of heat
continues till the two ends of the rod are at the rod decreases uniformly with distance from the
same temperature in principle but this will take
infinite time. Normally various sections of the hot end to the cold end. The fall of temperature
rod will attain a temperature which remains
constant but not same through out the length of with distance between the ends of the rod in the

direction of flow of heat, is called temperature

gradient.

?Temperature gradient T1 T2
x

where T1= temperature of hot end
T2= temperature of cold end
x = length of the rod

7.9.1.2 Coefficient of Thermal Conductivity:

Consider a cube of each side x and each face
of cross-sectional area A. Suppose its opposite
faces are maintained at temperatures T1 and T2
(T1 > T2) as shown in Fig. 7.12 (b). Experiments
show that under steady state condition, the

134

quantity of heat ‘Q’ that flows from the hot face SI unit of coefficient of thermal
to the cold face is conductivity k is J s-1 m-1 °C-1 or J s-1 m-1 K-1 and
its dimensions are [L1 M1 T-3 K-1].
i) directly proportional to the cross-sectional
area A of the face. i.e., Q ∝ A From Eq. (7.34), we also have

ii) directly proportional to the temperature Q kA(T1 T2 ) --- (7.36)
difference between the two faces i.e., Q ∝
(T1- T2) t x

iii) directly proportional to time t (in seconds) The quantity Q/t, denoted by Pcond , is the
for which heat flows i.e. Q ∝ t time rate of heat flow (i.e. heat flow per second)
from the hotter face to the colder face, at right
iv) inversely proportional to the perpendicular angles to the faces. Its SI unit is watt (W). SI
distance x between hot and cold faces i.e., unit of k can therefore be written as W m-1 °C-1
Q ∝ 1/x or W m-1 K-1.

Combining the above four factors, we Using calculus, Eq. (7.36) may be written

have the quantity of heat as

Q v A(T1 T2 )t dQ kA dT
x dt dx

?Q kA(T1 T2 )t --- (7.34) where dT is the temperature gradient.
x dx

where k is a constant of proportionality and is The negative sign indicates that heat flow

called coefficient of thermal conductivity. Its is in the direction of decreasing temperature.

value depends upon the nature of the tm=a1te sriaanl.d=I(fnuAmer1i=cma2llayn).d ddTx 1, then dQ = k
If A = l m2, T1-T2= 1 °C (or 1 K), dt

x = 1 m, then from Eq. (7.34), Q = k. Hence the coefficient of thermal
conductivity of a material may also be
Fig 7.12 (a): defined as the rate of flow of heat per unit
Section of a area per unit temperature gradient when the
metal bar in the heat flow is at right angles to the faces of a
steady state.

thin parallel-sided slab of material.

The coefficients of thermal conductivity of
some materials are given in Table 7.7.

Fig 7.12 (b): Section 7.9.1.3 Thermal Resistance (RT):
of a cube in the Conduction rate Pcond is the amount of
steady state.
energy transferred per unit time through a slab
Thus the coefficient of thermal of area A and thickness x, the two sides of the
slab being at temperatures T1 and T2(T1 >T2), and
conductivity of a material is defined as the is given by Eq. (7.36)

quantity of heat that flows in one second Pcond Q kA T1 T2 --- (7.37)
t x
between the opposite faces of a cube of side

1 m, the faces being kept at a temperature As discussed earlier, k depends on the
material of the slab. A material that readily
difference of 1°C (or 1 K). transfers heat energy by conduction is a good
thermal conductor and has high value of k.
From Eq. (7.34), we have

k Qx
A(T1 T2 )t --- (7.35)

135

Table 7.7: Coefficient of thermal or °C s/J and its dimensional formula is
conductivity (k). [M-1 L-2 T3 K1].

Substance Coefficient of thermal The lower the thermal conductivity k,
conductivity (J s-1 m-1 K-1)
the higher is the thermal resistance. RT A
material with high value is a poor thermal
Silver 406 conductor and is aRT good thermal insulator.
Copper 385
Aluminium 205 Thermal resistivity ρ is the reciprocal of
Steel 50.2 T
Insulating brick 0.15
Glass 0.8 thermal conductivity k and is characteristic of a
Brick and concrete 0.8
Water 0.8 material while thermal resistance is that of slab
Wood 0.04-0.12
Air at 0 °C 0.024 (or of rod) and depends on the material and on

the thickness of slab (or length of rod).

In western countries , where the temperature Example 7.17: What is the rate of energy
loss in watt per square metre through a glass
falls below 0 ºC in winter season, insulating the window 5 mm thick if outside temperature
is -20 ºC and inside temperature is 25 ºC?
house from the surroundings is very important. (kglass = 1 W/m K)
Solution : Given
In our country, if we wish to carry cold drinks
kglass = 1 W/m K
with us for picnic or wish to bring ice-cream T1 = 25 ºC
T2 = -20 ºC
from the shop to our house, we need to keep x = 5 mm = 5 × 10-3 m

them in containers (made up of say thermocol)

that are poor thermal conductors. Hence the

concept of threesrmistaalncree,sisistanincetrodRuTc, eds.imTilhaer ∴ T1 -T2 = 25 – (-20) ºC = 45 K
to electrical
Q T1 T2
opposition of a body, to the flow of heat through We have Pcond t kA x .

it, is called thermal resistance. The greater the ∴The rate of energy loss per square metre is

thermal conductivity of a material, the smaller

is its thermal resistance and vice versa. Thus Pcond k T1 T2

bad thermal conductors are those which have A x

high thermal resistance. = 1W m-1 K-1 × 45 K / (5 × 10-3 m)
= 9 × 103 W/m2
From Eq. (7.37) 7.9.1.4Applications of Thermal Conductivity:
i) Cooking utensils are made of metals
(T1 T2 ) x --- (7.38) but are provided with handles of bad
Pcond kA conductors.
Since metals are good conductors of
We know that when a current flows through a heat, heat can be easily conducted through
the base of the utensils. The handles of
conductor, the ratio V/I is called the electrical utensils are made of bad conductors of
heat (e.g., wood, ebonite etc.) so that they
resistance of the conductor where V is the can not conduct heat from the utensils to
our hands.
electrical potential difference between the ends ii) Thick walls are used in the construction
of cold storage rooms. Brick is a bad
of the conductor and I is the current or rate conductor of heat so that it reduces the
flow of heat from the surroundings to
of flow of charge. In Eq. (7.38), (T1-T2) is the the rooms. Still better heat insulation
temperature difference between the ends of the is obtained by using hollow bricks. Air

conductor and Pcond is the rate of flow of heat.
Therefore in analogy with electrical resistance,

(T1-T2)/ Pcond is called thermal resistance RT of
the material i.e.,

Thermal resistance RT = x
kA
The SI unit of thermal resistance is °C s/ kcal

136

being a poorer conductor than a brick, it Solution : Given
further avoids the conduction of heat from A = 1000 cm2 = 1000 ×10-4 m2
outside. k = 0.022 cal/ s cm °C = 0.022 ×102 cal/m °C
iii) To prevent ice from melting it is wrapped x = 4 mm = 0.4 ×10-2 m
in a gunny bag. A gunny bag is a poor T1 = 27 °C, T2 = -5 °C
conductor of heat and reduces the heat
flow from outside to ice. Moreover, the From Eq. (7.34), we have
air filled in the interspaces of a gunny
bag, being very bad conductor of heat, Q kA(T1 T2 )t
further avoids the conduction of heat from x
outside.
Low thermal conductivity can also be a ? Q kA(T1 T2 )
disadvantage. When hot water is poured in t x
a glass beaker the inner surface of the glass
expands on heating. Since glass is a bad 0.022u102 u1000u10 4 u (27 ( 5))
conductor of heat, the heat from inside does not
reach the outside surface so quickly. Hence the 0.4 u10 2
outer surface does not expand thereby causing a
crack in the glass. 1.76 u103 cal / s 1.76 kcal / s

Example 7.18: The temperature difference 7.9.2 Convection:
between two sides of an iron plate, 2 cm thick,
is 10 °C. Heat is transmitted through the plate at We have seen that heat is transmitted
the rate of 600 kcal per minute per square metre through solids by conduction wherein energy
at steady state. Find the thermal conductivity is transferred from one molecule to another but
of iron. the molecules themselves vibrating with larger
Solution: Given amplitude do not leave their mean positions.
But in convection, heat is transmitted from one
Q 600 kcal / min m2 600 10 kcal / s m2 point to another by the actual bodily movement
At 60 of the heated (energised) molecules within the
fluid.
x 2 cm 2 u10 2 m
In liquids and gases heat is transmitted by
T1 T2 10 qC convection because their molecules are quite
From Eq.(7.34), we have free to move about. The mechanism of heat
transfer by convection in liquids and gases is
Q kA(T1 T2 )t described below.
x
Consider water being heated in a vessel
?k Qx from below. The water at the bottom of the
At T1 T2 vessel is heated first and consequently its
density decreases i.e., water molecules at the
10 kcal / s m2 u 2u10 2 m bottom are separated farther apart. These hot
molecules have high kinetic energy and rise
10qC upward to cold region while the molecules
0.02 kcal / m s qC from cold region come down to take their place.
Thus each molecule at the bottom gets heated
Example 7.19: Calculate the rate of loss of and rises then cools and descends. This action
heat through a glass window of area 1000 sets up the flow of water molecules called
cm2 and thickness of 4 mm, when temperature convection currents. The convection currents
inside is 27 °C and outside is -5 °C. Coefficient transfer heat to the entire mass of water. Note
of thermal conductivity of glass is 0.022 cal/ s that transfer of heat is by the bodily/ physical
cm °C. movement of the water molecules.

137

Always remember: circulation of fresh air. This is called forced
convection. Example in section 7.9.2.1
The process by which heat is transmitted are of forced convection, namely, heat
through a substance from one point to convector, air conditioner, heat radiators
another due to the actual bodily movement in IC engine etc.
of the heated particles of the substance is
called convection. 7.9.3 Radiation:

7.9.2.1 Applications of Convection: The transfer of heat energy from one
place to another via emission of EM energy
i) Heating and cooling of rooms (in a straight line with the speed of light)
The mechanism of heating a room by without heating the intervening medium is
called radiation.
a heat convector or heater is entirely
based on convection. The air molecules For transfer of heat by radiation, molecules
in immediate contact with the heater are are not needed i.e. medium is not required. The
heated up. These air molecules acquire fact that Earth receives large quantities of heat
sufficient energy and rise upward. The cool form the Sun shows that heat can pass through
air at the top being denser moves down to empty space (i.e., vacuum) between the Sun and
take their place. This cool air in turn gets the atmosphere that surrounds the Earth . In
heated and moves upward. In this way, fact, transfer of heat by radiation has the same
convection currents are set up in the room properties as light (or EM wave).
which transfer heat to different parts of the
room. The same principle but in opposite A natural question arises as to how heat
direction is used to cool a room by an air- transfer occurs is the absence of a medium
conditioner. (i.e., molecules). All objects possess thermal
energy due to their temperature T(T > 0 K).
ii) Cooling of transformers The rapidly moving molecules of a hot body
Due to current flowing in the windings of emit EM waves travelling with the velocity
of light. These are called thermal radiations.
the transformer, enormous heat is produced. These carry energy with them and transfer it
Therefore, transformer is always kept in to the low-speed molecules of a cold body on
a tank containing oil. The oil in contact which they fall. This results in an increase in
with transformer body heats up, creating the molecular motion of the cold body and
convection currents. The warm oil comes its temperature rises. Thus transfer of heat by
in contact with the cooler tank, gives heat radiation is a two-fold process- the conversion
to it and descends to the bottom. It again of thermal energy into waves and reconversion
warms up to rise upward. This process is of waves into thermal energy by the body on
repeated again and again. The heat of the which they fall. We will learn about EM waves
transformer body is thus carried away by in Chapter 13.
convection to the cooler tank. The cooler
tank, in turn loses its heat by convection to 7.10 Newton’s Laws of Cooling:
the surrounding air.
If hot water in a vessel is kept on table,
7.9.2.2 Free and Forced Convection: it begins to cool gradually. To study how a
i) When a hot body is in contact with air given body can cool on exchanging heat with
its surroundings, following experiment is
under ordinary conditions, like air around performed.
a firewood, the air removes heat from the
body by a process called free or natural A calorimeter is filled up to two third of
convection. Land and sea breezes are its capacity with boiling water and is covered.
also formed as a result of free convection A thermometer is fixed through a hole in the
currents in air. lid and its position is adjusted so that the bulb
of the thermometer is fully immersed in water.
ii) The convection process can be accelerated
by employing a fan to create a rapid

138

The calorimeter vessel is kept in a constant According to Newton’s law of cooling the
temperature enclosure or just in open air since
room temperature will not change much rate of loss of heat dT/dt of the body is directly
during experiment. The temperature on the
thermometer is noted at one minute interval proportional to the difference of temperature
until the temperature of water decreases by
about 25 °C. A graph of temperature T (along (T -T0) of the body and the surroundings
y-axis) is plotted against time t (along x-axis). provided the difference in temperatures is small.
This graph is called cooling curve (Fig 7.13 (a)).
From this graph you can infer how the cooling Mathematically this may be expressed as
of hot water depends on the difference of its
temperature from that of its surroundings. You dT v (T -T0 )
will also notice that initially the rate of cooling dt
is higher and it decreases as the temperature
of the water falls. A tangent is drawn to the ? dT C (T -T0 ) --- (7.39)
curve at suitable points on the curve. The slope dt
of each tangent (dT/dt) gives the rate of fall of
temperature at that temperature. Taking (0,0) where C is constant of proportionality.
as the origin, if a graph of dT/dt is plotted
against corresponding temperature difference Example 7.20: A metal sphere cools at the rate
(T-T0), the curve is a straight line as shown in of 1.6 °C/min when its temperature is 70°C. At
Fig 7.13 (b). what rate will it cool when its temperature is
40°C. The temperature of surroundings is 30°C.
Fig 7.13 (a):
Temperature versus Solution: Given T1 = 70° C
time graph. lim 'T
T2 = 40° C
'to0 't T0 = 30° C
gives the slope of the § dT ·
tangent drawn to the ©¨ dt ¸¹1 1.6 qC / min
curve at point A and
indicates the rate of According to Newton’s law of cooling, if
fall of temperature. C is the constant of proportionality

§ dT · C (T1 -T0 )
¨© dt ¸¹1 C (70 30)
or, 1.6

?C 1.6 0.04 / min
40

Also § dT · C (T2 -T0 )
©¨ dt ¹¸2

0.04(40 30) 0.4 qC/ min

Fig 7.13 (b): Thus the rate of cooling drops by a factor of
Rate of change of four when the difference in temperature of the
temperature versus metal sphere and its surroundings drops by a
time graph. factor of four.

Internet my friend

The above activity shows that a hot body 1. https://hyperphysics.phy-astr.gsu.edu/
loses heat to its surroundings in the form of heat hbase/hframe.html
radiation. The rate of loss of heat depends on
the difference in the temperature of the body 2. https://youtu.be/7ZKHc5J6R5Q
and its surroundings. Newton was the first to 3. https://physics.info/expansion
study the relation between the heat lost by a
body in a given enclosure and its temperature
in a systematic manner.

139

ExercisesExercises

1. Choose the correct option. iv) What is absolute zero?
v) Derive the relation between three
i) Range of temperature in a clinical
coefficients of thermal expansion.
thermometer, which measures the vi) State applications of thermal expansion.
vii) Why do we generally consider two
temperature of human body, is
specific heats for a gas?
(A)70 ºC to 100 ºC viii) Are freezing point and melting point

(B) 34 ºC to 42 ºC same with respect to change of state ?
Comment.
(C) 0 ºF to 100 ºF ix) Define (i) Sublimation (ii) Triple point.
x) Explain the term 'steady state'.
(D) 34 ºF to 80 ºF xi) Define coefficient of thermal
conductivity. Derive its expression.
ii) A glass bottle completely filled with water xii) Give any four applications of thermal
conductivity in every day life.
is kept in the freezer. Why does it crack? xiii) Explain the term thermal resistance.
State its SI unit and dimensions.
(A) Bottle gets contracted xiv) How heat transfer occurs through
radiation in absence of a medium?
(B) Bottle is expanded xv) State Newton’s law of cooling and
explain how it can be experimentally
(C) Water expands on freezing verified.
xvi) What is thermal stress? Give an example
(D) Water contracts on freezing of disadvantages of thermal stress in
practical use?
iii) If two temperatures differ by 25 °C on xvii) Which materials can be used as thermal
insulators and why?
Celsius scale, the difference in temperature

on Fahrenheit scale is

(A) 65° (B) 45°

(C) 38° (D) 25°

iv) If α, β and γ are coefficients of linear, area

l and volume expansion of a solid then

(A) α:β:γ 1:3:2 (B) α:β:γ 1:2:3

(C) α:β:γ 2:3:1 (D) α:β:γ 3:1:2

v) Consider the following statements-

(I) The coefficient of linear expansion has

dimension K -1

(II) The coefficient of volume expansion

has dimension K -1 3. Solve the following problems.

(A) I and II are both correct i) A glass flask has volume 1×10-4 m3.

(B) I is correct but II is wrong It is filled with a liquid at 30 ºC.

(C) II is correct but I is wrong If the temperature of the system is raised

(D) I and II are both wrong to 100 ºC, how much of the liquid

vi) Water falls from a height of 200 m. What is will overflow. (Coefficient of volume

the difference in temperature between the expansion of glass is 1.2×10-5 (ºC)-1

water at the top and bottom of a water fall while that of the liquid is 75×10-5 (ºC)-   1).

given that specific heat of water is 4200 J

kg-1 °C-1? [Ans : 516.6 × 10-8 m3]

(A) 0.96°C (B) 1.02°C ii) Which will require more energy, heating

(C) 0.46°C (D) 1.16°C a 2.0 kg block of lead by 30 K or heating

2. Answer the following questions. a 4.0 kg block of copper by 5 K? (slead =

i) Clearly state the difference between heat 128 J kg-1 K-1, scopper = 387 J kg-1 K-1)

and temperature? [Ans : copper]

ii) How a thermometer is calibrated ? iii) Specific latent heat of vaporization of

iii) What are different scales of temperature? water is 2.26 × 106 J/kg. Calculate the

What is the relation between them? energy needed to change 5.0 g of water

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into steam at 100 ºC. x) An aluminium rod and iron rod show 1.5

[Ans : 11.3 × 103 J] m difference in their lengths when heated

iv) A metal sphere cools at the rate of at all temperature. What are their lengths

0.05 ºC/s when its temperature is 70 at 0 °C if coefficient of linear expansion

ºC and at the rate of 0.025 ºC/s when for aluminium is 24.5×10-6 /°C and for

its temperature is 50 ºC. Determine the iron is 11.9×10-6 /°C

temperature of the surroundings and find [Ans: 1.417m, 2.917m]

the rate of cooling when the temperature xi) What is the specific heat of a metal if 50

of the metal sphere is 40 ºC. cal of heat is needed to raise 6 kg of the

[Ans : 30 ºC, 0.0125 ºC/s] metal from 20°C to 62 °C ?

v) The volume of a gas varied linearly with [Ans: s = 0.198 cal/kg °C]

absolute temperature if its pressure is xii) The rate of flow of heat through a copper

held constant. Suppose the gas does not rod with temperature difference 30 °C is

liquefy even at very low temperatures, at 1500 cal/s. Find the thermal resistance

what temperature the volume of the gas of copper rod.

will be ideally zero? [Ans: 0.02 °C s cal]

[Ans : -273.15 ºC] xiii) An electric kettle takes 20 minutes to

vi) In olden days, while laying the rails for heat a certain quantity of water from 0°C

trains, small gaps used to be left between to its boiling point. It requires 90 minutes

the rail sections to allow for thermal to turn all the water at 100°C into steam.

expansion. Suppose the rails are laid at Find the latent heat of vaporisation.

room temperature 27 ºC. If maximum (Specific heat of water = 1cal/g°C)

temperature in the region is 45 ºC and [Ans: 450 cal/g]

the length of each rail section is 10 m, xiv) Find the temperature difference between

what should be the gap left given that two sides of a steel plate 4 cm thick,

α = 1.2 × 10-5 K-1 for the material of the when heat is transmitted through the

rail section? plate at the rate of 400 k cal per minute

[Ans : 2.16 mm] per square metre at steady state. Thermal

vii) A blacksmith fixes iron ring on the rim of conductivity of steel is 0.026 kcal/m s K.

the wooden wheel of a bullock cart. The [Ans:10.26°C or 10.26 K]

diameter of the wooden rim and the iron xv) A metal sphere cools from 80 °C to 60

ring are 1.5 m and 1.47 m respectively °C in 6 min. How much time with it take

at room temperature of 27 ºC. To what to cool from 60 °C to 40 °C if the room

temperature the iron ring should be temperature is 30°C?

heated so that it can fit the rim of the [Ans: 10 min]

wheel (αiron = 1.2×10-5 K-1).
[Ans: 1727.7 °C ] ***

viii) In a random temperature scale X, water

boils at 200 °X and freezes at 20 °X.

Find the boiling point of a liquid in this

scale if it boils at 62 °C.

[Ans: 131.6°X]

ix) A gas at 900°C is cooled until both

its pressure and volume are halved.

Calculate its final temperature.

[Ans: 293.29K]

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