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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Elasticity Page 2

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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

pr2l


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 r2

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 stress-strain 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 stress-strain 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 stress-strain 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 cross-section 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 = dW =

 

 fdx =  

YA

 xdx

L

 

\W =  

 = 

 

 

\

 = 

 

But Y =  

F / A

 = 

FL

 

 

 

 

e / L

Ae

 

 

 \ F =  

[

YAe

L

]

 

 

\   

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

 

=

1

2

 

(longitudinal stress )2

 

 

Y

 

 

 

 

 

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