16. Lateral
Strain: When a wire is stretched, the area of cross section
(and therefore the diameter) of the wire decreases. Thus, there is a
contraction in the dimensions in a direction perpendicular to the
direction of the deforming force.
The change in the dimension per unit original dimension perpendicular
to the direction of the deforming force is called the lateral strain.
17. Poisson's Ratio (s): Poisson's
ratio (s) is defined as the ratio of the lateral strain to the
corresponding longitudinal strain.
If L and D are the original length and diameter respectively of a
wire which is stretched and if DL and DD are the
changes in the length and diameter respectively due to the deforming
force,
Longitudinal strain =

DL


L

Lateral strain =

DD


D

Poisson's ratio s =

DD

/

DL



D

L


s =

LDD


DDL



Poisson's ratio is a pure number without any unit.
18. Y by Searle's Method

P,

Q  Rectangular metal frames;
R  A frame connecting P and Q;

A,

B  Two long identical wires of the same material
A  Reference wire
B  Experimental wire
L  Spirit level;
S  Micrometer screw;
H  Hanger with weights;
W  A dead load;

Principle: A long wire, fixed to a rigid support, is stretched
by attaching a known load to its free end. The extension of the wire
is accurately measured. Knowing the longitudinal stress and the
longitudinal strain, the Young's modulus of the material of the wire
can be calculated within elastic limits, stress a strain.
\ Young's modulus =

Longitudinal stress



Longitudinal strain

Apparatus:
 Two rectangular frames P and Q, loosely fitted together
by a third frame R.
 Spirit level, L  fixed to P. One end of L rests on the tip
of a micrometer screw fixed to Q.
 Two long identical wires, A and B fixed to P and Q
respectively.
 A heavy load W suspended from P  used to straighten
the wire A.
 Hanger H, suspended from Q.
Procedure:
 The original length L of the wire is measured.
 The diameter of the wire is measured using a micrometer
screw gauge. Its mean radius r is determined.
 A load of half kg is added to the hanger. This is
called the zero load and is used to
straighten out any links in the wire B.
 The micrometer screw is adjusted and the spirit level
is made horizontal.
 The micrometer reading is noted. This is the zero
reading.
 The load on the hanger is increased in steps of half a
kilogram.
 Each time the spirit level is made horizontal and the
micrometer reading is noted. Six such readings are taken.
 The load is then decreased in steps of half a kilogram
and the above procedure is repeated.
Formula: Young's modulus

Y =

MgL


pr^{2}l



Graph: A graph of the extension (l) versus the load M is plotted. The
slope of this graph is
(

l

)


M

Hence,

Y =

gL

(

1

)



p r^{2}

slope



Sources of Error:
 Error due to yielding of the support.
 Error due to change in room temperature.
 Error due to slipping of the wire from the chuck.
Elimination
of Errors: The errors due to yielding of the support and change in
temperature are minimised due to the second wire (reference wire) A attached in the apparatus. Both wires are
equally lowered due to the above sources of error hence the errors
are minimised and the correct extension of wire B is obtained.
19.
Behaviour of a Wire Under Increasing Load
 The graph represents stress versus strain.
 OA is a straight line. In this region Hooke's law is
obeyed. A is the elastic limit.
 Between A and B, Hooke's law is not obeyed. The
behaviour of the wire is partly elastic and partly plastic. Even
if the stress is removed, some residual strain remains.
 The tangent drawn at the point B to the stressstrain
graph is parallel to the strain axis. Here strain increases
without any increase in stress. The point at which this begins
to occur is called the yield point.
 Beyond yield point, the area of cross section of the
wire decreases till a neck is formed and the wire breaks. This
point is called the breaking point and the corresponding stress
is the breaking stress.
Yield
Point:
The point on the stressstrain curve, beyond which the strain
increases without any increase in the stress and at which the wire
begins to extend of its own accord is called the yield point.
Breaking Stress:
The maximum stress which can be applied to a wire is called the
breaking stress.
Breaking Point:
The point on the stressstrain curve at which the wire breaks is
called the breaking point.
20. Work Done in Stretching a Wire:
Consider a wire of length L and area of crosssection A. Let it be
stretched by a force that increases uniformly from
O to F. e is the total extension of the wire.
At some stage, during the extension, let the applied force be f and the
elongation be x.
\ Longitudinal stress =

f


A

Longitudinal strain =

x


L

\ Young's modulus, (Y) =

f / A

=

fL



x / L

xA

\ f =

YAx


l

The work done against the internal restoring force in stretching the
wire through
dx is given by dW
= f dx =

YAx

dx.


L

\ Total work done in producing extension e
= W = ò dW =



fdx =

YA


xdx.


L

\W =


=



\ W

=


But Y =

F / A

=

FL








e / L

Ae


\

W =

1

F
´ e


2



\ Work done =

1

´
stretching force ´ elongation = Average force x
elongation


2

Strain Energy: The work done by the applied force in deforming
the body is stored in it in the form of potential energy. This energy
is called the strain energy.
Strain energy =

1

´ F ´ e


2


= Average force x extension

\ Strain energy per unit volume =

1

F
´ e


2



A ´ L

=

1

´

F

´

e

=

1

(Longitudinal stress ´ longitudinal strain)





2

A

L

2

Since Y =

longitudinal
stress


longitudinal
strain

Longitudinal stress = Y ´
longitudinal strain.
\ Strain energy per unit volume =

1

´ Y ´
(longitudinal strain)^{2}


2


=

1

´

(

longitudinal
stress

)

2

x Y



2

Y


=


(longitudinal
stress )^{2}




Y

