ELASTICITY NOTES 1.pdf

1. ELASTICITY
Rigid body: Any object which doesn’t change its shape or size on application of external
force is called a rigid body.
In the universe there is no perfect rigid body but near to rigid body are quartz and diamond.
Deforming force: The external force which tries to change the shape or size of an object
is called deforming force.
Restoring force: The force developed inside a body due to change in molecular distance
and try to bring back the body to its original shape or size is called restoring force.
* In steady state deforming forces are equal and opposite to restoring forces
Elasticity: The property of a body by virtue of which it regains its original shape or size
after removal of deforming forces is called elasticity. A body which obeys the
property of elasticity is called elastic body
Example: Rubber, metal, diamond etc.
Plasticity: The property of a body by virtue of which it doesn’t regain its original shape or
size even after removal of deforming forces is called plasticity. A body which obeys
the property of plasticity is called plastic body.
Example: Paraffin wax, clay, chewing gum etc
Stress: The deforming forces applied per unit area of object is called stress
Stress =
Deforming Force Restoring force
area area

Unit of stress = 2
or pascal
N
m

2. DF of stress: [ M1
L-1
T-2
]
Types of deforming forces
1. Normal (perpendicular to the surface ) (changes size)
2. Tangential (parallel to the surface) (changes shape)
Types of Stress : (3 types)
1. Longitudinal Stress: The normal force per unit area acting on an object which changes
its length is called longitudinal stress.
Longitudinal Stress =
Normal Force
area
There are 2 types of longitudinal stress
a) Tensile (which increases length)

3. b) Compressive (which decrease
2. Volume or Bulk Stress : (Hydraulic stress)
The normal force per unit area which changes volume of
3. Shearing or Tangential Stress
The tangential force per unit area acting on an object which cha
shearing stress
Shear stress =
decreases length)
(Hydraulic stress)
The normal force per unit area which changes volume of an object is called bulk stress
Shearing or Tangential Stress
The tangential force per unit area acting on an object which changes its shape is
Shear stress =
Tangential force
area
is called bulk stress.
nges its shape is called

4. STRAIN : It is defined as the ratio of change in dimension or configuration to the original
dimension or configuration
Strain =
Change in dimension
Original dimension
Strain has no unit and no dimensional formula
Types of strains (3 types)
1. Longitudinal Strain =
Change in length
Original length
L
L


2. Bulk / Volume Strain =
Change in volume
Original volume
V
V


3. Shearing strain ( ) : It is defined as the ratio of relative displacement of the layers to the
distance between the layers
x
L
 

5. Note1 : When a cylinder is twisted on one side, then shearing strain is given by
Note 2. In case of spring there is shearing strain but not longitudinal strain
Elastic Limit: It is the maximum stress below which the material is elastic and above which
it is plastic.
Hooke’s Law: Within the elastic Limit stress acting on a material is directly proportional to
strain produced in it
Stress  strain
Stress = E Strain
Where E is a constant called as Modules of Elasticity
*
Stress
E
Strain

* Unit modules of Elasticity = 2
( )
N
or
m
pascal
* D.F of modules of Elasticity is [M1
L-1
T-2
]

6. * Modules of Elasticity is Independent of stress, strain and dimensions( like length and
area of cross section) depends on temperature & nature of material
* E-large means material is more elastic.
* Within the elastic limit, stress verses strain graph is a straight line passing through
origin.
 
stress
tanθ = slope = = modulus of elasticity
strain
E
strain 1
tanθ = slope = =
stress modulus of elasticity
 
modulus of elasticity cot
or E 

Prob 1. Among steel and rubber which is more elastic. Explain.
Steel is more elastic than rubber because for the same stress, strain is more in case of
rubber which means
rubber Steel
Stress
E E E
Strain
 
 
 
 

7. Prob 2. Stress verses strain graphs are given for 2 different materials P & Q state which is
more elastic
2 1
2 1
tan tan
Q is more elastic than P
Q P
E E
 
 




Types of Modulus of Elasticity
1. Young’s Modulus (Y): Within the elastic limit, the ratio of longitudinal stress to
longitudinal strain is called young’s modulus.
Longitudinal stress
Y
Longitudinal strain
F
FL
A
Y where L e elongationor compression
L A L
L

    
 
Note 1. Elongation or compression in the rod is given by
FL
L
AY
 
Note 2. Elongation in a wire due to a load connected at one end is
F L
L
AY
 
2
Mg L
L
r Y

  Here F = Mg and A=
2
r

Where r = radius of cross section of the wire

8. Note 3. Elongation of a wire due to its own weight.
F L
L
AY
 
2
L
Mg
L
AY
 
 
 
 
 
2
L
AL g
L where M volume density
AY

 
 
 
   
  2
2
2
L
L g
gL
L
Y Y


 
 
 
  
Elongation in a wire due to its own weight is independent of area of cross section
Prob 1. What mass must be suspended from steel wire of 2m long and 1mm diameter to
stretch its by 1mm. Ysteel =
12 2 2
2 10 / . 10
dyne cm g ms
  .

9. Prob 2. A wire of Young’s Modulus (Y) is stretched to double its length. What is the value
of longitudinal strain and longitudinal stress?
Prob 3. A wire increases by 1
1000
th
of its length when a stress of 108
N/m2
is applied to it.
What is the value of Young’s modulus?
Prob 4. When the tension in a wire is T1 its length is L1. When the tension is T2 its length is
L2. Find the natural length of the wire.

10. Prob 5. The length of an elastic string is ‘a’ m, when the tension is 4N and ‘b’ m, when the
tension is 5N. Calculate the length in m, when the tension is 9N.