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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Elasticity Page 1

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1. Deformation: If an applied force causes a change in the distance between the particles of a material body, it results in a change in size or shape or both. Such a change in size or shape is called deformation.

2. Deforming Force: A force causing deformation of a material body is called a deforming force.

3. Elasticity: The property possessed by a material body due to which it offers a resistance to the deforming forces and recovers its original size and shape after the deforming forces are removed is called elasticity.

4. Perfectly Elastic Body: A material body which completely recovers its original size and shape upon removal of deforming forces is called a perfectly elastic body. Example: Quartz fibre (almost perfectly elastic).

5. Plasticity: The property possessed by a material body due to which it offers no resistance to the deforming forces and remains in the deformed state even after removal of the deforming forces is called plasticity.

6. Perfectly Plastic Body: A material body that does not recover its original shape or size at all even after removal of the deforming forces is called a perfectly plastic body. Example: Clay.

7. Internal Restoring Force: When deforming forces are applied to an elastic body, internal opposing forces are set up which tend to restore the body to its original size and shape. These forces are called internal restoring forces.
Generally, Internal restoring force = Applied force.

8. Stress: The internal restoring force per unit area of the body is called the stress.

\ Stress =  

Internal restoring force

 =

Applied restoring force

Area

Area


Units of Stress:

S I Units - newton / metre2 (N / m2 )
CGS Units
- dyne / centimetre2 (dyne / cm2)


Dimensions of Stress: [M1 L
-1 T-2 ]

9. Types of Stress:

There are three types of stress.

  1. Longitudinal Stress or Tensile Stress: If the applied forces cause a change in the length of a body, the stress is called longitudinal or tensile stress.
    If a mass M is attached at the free end of a wire of length l and radius r suspended from a rigid support,

  1.  

Longitudinal stress =  

Applied force

 = 

Mg

Area of cross section

pr2

  1.  
  2. Volume Stress or Bulk Stress: If the applied forces cause a change in the volume of a
    body, the stress is called volume stress or bulk stress.
    Volume stress is represented by the increase in pressure resulting from the
    application of compressive forces normally on all sides of the body.

\ Volume stress  

= Change in pressure = D P
= Applied force per unit area.
= Normal force / surface area.

  1.  
  2. Shearing Stress: If the applied forces produce a change in the shape of the body, the
    stress is called shearing stress.
    Consider a tangential force F applied to the upper surface of a solid cube.
    Let A be the area of the surface.

\ Shearing stress

Tangential force

 = 

F

Surface area

A

  1.  

10. Strain: The ratio of the change in dimensions of the body to the original dimensions is called the strain.
Since strain is the ratio of two similar quantities, it has no unit and no dimensions.
There are three types of strain corresponding to the three types of stress.

11. Types of Strain

  1. Longitudinal or Tensile Strain: It is defined as the ratio of the change in length to the original length.

Longitudinal strain =  

Change in Length

 = 

l

Original Length

L


  1. Where
    l is the increase in length of a wire of length L due to the applied force.
  2. Volume Strain or Bulk Strain: It is defined as the ratio of the change in volume to the original volume.

\ Volume strain = 

Change in volume

 = 

dV

Original volume

V

  1. where dV is the change in the volume of a body of volume V due to a pressure change dP.
  2. Shearing Strain or Shear: When a tangential force is applied to a body, there is a lateral displacement of the different layers of the body.

  1.  

Shearing strain =  

Lateral displacement of a layer

its distance from a fixed layer

  1.  

=  

AA'

 = tan q = q (Since q is very small and in radian)

AB

12. Elastic Limit: The internal restoring force comes into play only when a deforming force acts on a body. It opposes and balances the deforming force. However, the internal restoring forces can only balance the deforming forces upto a certain limit. If the deforming force is increased beyond this limit, the internal restoring forces cannot balance the deforming forces and the body loses its elasticity.

Definition: The maximum value of stress upto which the body shows elastic behaviour is called the elastic limit.

13. Hooke's Law of Elasticity: Within the elastic limit, the stress developed in a body is directly proportional to the strain produced in it.

\ Stress strain.
Stress = E(strain), The constant of proportionality E is called the modulus of elasticity
or the coefficient of elasticity.

Modulus of elasticity, E =  

stress

strain

14. Modulus of Elasticity:
The modulus of elasticity is defined as the ratio of the stress to the corresponding strain within the elastic limit.

Units of Modulus of Elasticity (E):

E = 

stress

 . Since strain has no units, the units of E are the same as that of stress - 

strain

Unit: N/m2 in the SI system and dyne / cm2 in the CGS system.

15. Elastic Constants (Moduli of Elasticity):

There are three kinds of moduli of elasticity corresponding to the three types of stress and strain.

  1. Young's Modulus (Y): It is defined as the ratio of longitudinal stress to longitudinal strain within elastic limit.


  1. Consider a mass M attached to the free end of a wire of length L and radius r attached to a rigid support. Let the elongation of the wire due to the force Mg active downward be
    l.

\ Longitudinal stress =

Mg

pr2


  1. and

Longitudinal strain =  

l

L

  1.  

Young's modulus, Y =  

Longitudinal stress

 = 

Mg

 / 

l

Longitudinal strain

pr2

L

  1.  

i.e.  

Y =

M g L

p r2l

  1.  
  2. Bulk Modulus (K): It is defined as the ratio of the volume stress to the volume strain within elastic limit.

\ K =  

Volume stress

Volume strain

  1.  


  1. Volume stress = Change in pressure,
    D P.
    Volume strain =
    D v/v where D v is the change in the original volume v due to a change in pressure D P.

\ K =  

- DP

 = 

- VDP

DV/V

DV

  1.  

\ 

K =

- VDP

DV

[The negative sign is introduced to make K positive. An increase in pressure always causes a decrease in volume. Thus, if D P is positive, D V will be negative.]

Modulus of Rigidity (
h): It is defined as the ratio of the shearing stress to the shearing strain within elastic limit.

\ h =  

shearing stress

shearing strain

 

 

Shearing stress =  

F

 , where F is the tangential force applied to a surface of area A.

A

Shearing strain = q.

Hence, h =  

F / A

 = 

F

q

A.q

 

\  

h =

F

A.q

 

 

 

 

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