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24 February 2010 Bharat Bhushan/s Physics Classes

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Elasticity

Deforming force. :A force acting on a body, if produces a change in the

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shape of the body. Such a force is called deforming force.

Elastic body. A body that returns to its original shape and size on the

removal of the deforming force (when deformed within elastic limit), is

called an elastic body.

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Elasticity. The property of matter by virtue of which it regains its original

shape and size, when the deforming forces have been removed is called

elasticity.

Plastic body. A body that does not return to its original shape and size on

the removal of deforming force, however small the magnitude of

deforming force may be, is called a plastic body. Putty paraffin wax, etc

are the examples of nearly plastic bodies.

HOOKE’S LAW

It is the basic law in elasticity It states that the extension produced in a

wire is directly proportional to the load attached to it. Thus, according to

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Hooke’s law, extension ∞ load . However, this proportionality holds good

upto a certain limit, called the elastic limit.

According to Thomas Young the load and the extension are more

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scientifically described in terms of stress and strain respectively.

Thus, Hooke’s law may be stated that stress is directly proportional to strain.

According to the modified form of Hooke’s law,

stress strain

or

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This constant of proportionality is called modulus of elasticity or coefficient of

elasticity of the material. Its value depends upon the nature of the material of

the body and the manner in which the body is deformed.

There are three modulii of elasticity namely

Young’s modulus (Y),

 Bulk modulus (K)

Modulus of rigidity

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Stress. It is defined as the restoring force per unit area set up in the body,

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when deformed by the external force. Thus,

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As the restoring force set up in the body is equal and opposite to the

external deforming force (so long as no permanent deformation is produced

i.e. within elastic limit), The stress may be measured as the external force

acting per unit area i.e.

Stress is of the following two types :

(i) Normal Stress.

(ii) Tangential stress. 4

(i) Normal Stress. The deforming force acting per unit area normal to the

surface of the body is called normal stress. For example, when a wire is

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pulled by a force, the force acts along the length of the wire and normal to

its cross-section. The stress so produced is called the normal stress.

ii) Tangential stress. The deforming force acting per unit area tangential to the

surface is called tangential stress. For example, a body being sheared (when

force is applied parallel to the surface of the body) is under the tangential

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stress.

SI UNITS :(N /m2) i The dimensional formula of stress is [M L-1 T-2].

STRAIN: The ratio of change in dimension of the body to its original

dimension is called strain.

Since a body can have three types of deformations i.e. in length, in volume or

in shape, likewise there are following three types of strains :

(i) Longitudinal strain. It is defined as the increase in length per unit original

length, when deformed by the external force. It is also called linear strain or

tensile strain.

Thus, longitudinal strain =

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(ii) Volumetric strain. It is defined as change in volume per unit original volume,

when deformed by the external force.

Thus, volumetric strain =

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(iii) Shear strain. When change takes place in the shape of the body, the strain

is called the shear strain.

It is defined as the angle θ (in radian), through which a line originally perpendicular

to the fixed face gets turned on applying tangential deforming force. This angle,

through which the reference line turns, is called the angle of shear.

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When force is applied parallel to the surface of a solid body, then the change takes

place only in the shape of the body. It can happen only in case of a solid. If a force

is applied parallel to the surface of a fluid, it will begin to flow in the direction of

applied force.

All the three types of strains have no units,

YOUNG’S MODULUS (Stretching of a wire)

It is defined as the ratio of normal stress to the longitudinal strain. It is denoted by Y.

Thus

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Consider a wire (or a rod) of length L and area of cross-section a fixed at one

end Suppose that a normal force F is applied to the free end of the wire and

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its length increases by ℓ. Then,

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The units of Young’s modulus ,are pascal (Pa) or N m-2in SI

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STRESS VERSUS STRAIN

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(i) The portion OP shows that stress is

directly proportional to strain. It shows

that Hooke’s law is obeyed up to the

value of stress corresponding to the

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point P. The point P is sometimes called

proportional limit.

(ii) Beyond the point P, the graph between

stress and strain is not found to be

straight line as indicated by the part PE

of the graph. If the wire is unloaded at

point E and the graph between stress

and strain is obtained in the reverse

direction along EPO, then the point E is

called elastic limit. The portion of the

graph between O and E is called elastic

region. Hooke’s law is not obeyed

between the points P and E.

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(iii) If the wire is loaded beyond the point E,

the strain increases much more rapidly

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with the stress.. If the wire is unloaded

A, the graph between stress and strain will

be as shown by the dotted line AO ‘ .

Therefore, even when the wire is

completely unloaded, its length increases

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permanently by some amount

corresponding to OO’. It is called

permanent set.

(iv) Beyond the point A, the length of the wire starts increasing virtually for no

increase in stress. Thus, wire begins to flow after the point A and it continues

upto point C. The point A, at which the wire begins to flow is called yield

point. The increase in length of the wire for virtually no increase in stress is

called plastic behavior of the wire.

(v) Beyond the point C, the graph indicates that the length of the wire increases,

even if the wire is unloaded. In this region, the constrictions (called necks and

waists) develop at few points along the length of the wire and as a result of it, the

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wire breaks ultimately corresponding to the point B, called breaking point of the

wire. The portion the graph between the points E and B is called plastic region.

The stress corresponding to the point B is

called breaking stress or ultimate stress.

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The product of the breaking stress and the

area of cross-section is equal to breaking

load for the wire.

The materials of the wire, which break as

soon as stress is increased beyond the

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elastic limit are called brittle. Graphically,

for such materials, the portion of graph

between E and B is almost zero.

The materials of the wire, which have quite

a good plastic range (large portion of graph

between E and B) are called ductile. Such

materials can be easily changed into

different shapes and can be drawn into thin

wires.

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STRETCHING OF RUBBER

When load is applied to stretch a metallic wire and a rubber band, the results

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of stretching in the two cases are quite different from each other the

comparison between the two is as given below:

1. In order to produce even a small strain in a metallic wire, the force to be

applied is very large. In case of a rubber band, even a feeble force produces a

very large strain.

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2. A metallic wire can regain its original length upto a strain of 0.001(or when

the increase in length is 0.1% of the original length).

A rubber band can regain its original length upto a strain as large as 10 (or

when the increase in length is 10 times its original length).

3. The value of force that can be applied to a metallic wire to stretch it within

elastic limit is very large. In case of a rubber band, the elastic limit is crossed

even on applying a feeble force.

The rubber comes to its original length, even when its length is increased

several times its original length. Thus, rubber has large elastic limit but it does

not obey Hooke’s law. Materials like rubber, which can be greatly stretched are

called elastomers. Further rubber does not have plastic range i.e. it just

breaks, when stretched beyond certain limit.

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WORK DONE IN STRETCHING A WIRE

Consider a wire of length L and area of cross-section a suspended from a rigid

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support [ Suppose that a normal force F is applied at its free end and its length

increases by ℓ. Then, Young’s modulus of the material of the wire is given by

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Suppose that the length of the wire is increased by an infinitesimally small

amount dl under the action of a constant force F. Then, the small amount of

work done

Therefore, amount of work done in stretching the wire by a length dℓ is

given by

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This work done in stretching (or deforming) the wire gets stored in the wire

the form of its elastic potential energy. It is sometimes also called as strain

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energy. Since volume of the wire is a x L,

work done per unit volume (or strain energy per unit volume) of the wire

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BULK MODULUS

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It is defined as the ratio of the normal stress to the volumetric strain. It is

denoted by K. Thus,

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The units of bulk modulus are Pa or N m-2in SI.

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Compressibility. The reciprocal of the bulk modulus of a material is called its

compressibility. Therefore,

The units of compressibility are reciprocal of those of the bulk modulus i.e.

the units of compressibility are Pa1 or N m- 2 in SI.

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MODULUS OF RIGIDITY. it ie defined as the ratio of tangential stress to

shear strain. It is also called shear modulus. It is denoted by the Greek letter

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η (eta). Thus,

Let a be the area of the each face and AF

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= L be the perpendicular distance between

the two faces. The tangential force will

shear the rectangular block into a

parallelepiped by displacing the upper face

through a distance FF’ = x (say). If angle

FAF’ = θ, then θ is the angle of shear

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In practice, for solids, the angle of shear is very small. Therefore, θ= tan θ

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The distance x through which the upper face has been displaced is called

lateral displacement. Therefore, the shear strain may also be defined as

the ratio of the lateral displacement of a layer to its distance from the fixed

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layer. The units of modulus of rigidity are also Pa or N m-2 in SI.

Note.

A solid develops restoring force whenever. it .is deformed in size

(length or volume) or shape , whereas in a fluid, the restoring force

is set up only when its volume is changed.

That is why a solid exhibits all the three types of

elasticity and the fluids possess only volume

elasticity

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APPLICATIONS OF ELASTICITY

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1. Any metallic part of a machinery is never subjected to a stress beyond the

elastic limit of the material otherwise it will get permanently deformed

2. The thickness of metallic ropes used in cranes to lift and move heavy

weights is decided on the basis of the elastic limit of the material of the

rope and the factor of safety.

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3. The bridges are declared unsafe after long use.

4. In designing a beam for its use to support a load (in construction of roofs

and bridges), it is advantageous to increase its depth rather than the

breadth of the beam.

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5.Ahollow shaft is stronger than a solid shaft made of same and equal

material. It can be calculated that the torque required to produce a unit

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twist in a solid shaft (bar) of radius r is given by

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where ℓ is length and η, the modulus of rigidity of the material of the

shaft. Further, torque required to produce a unit twist in a hollow

shaft of internal and external radii r1 and r2 is given by

If the amounts of materials used to make the two shafts are equal, then

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i.e. the torque required to twist a hollow shaft is greater than the torque

necessary to twist a solid shaft of the same mass, length and material

through the same angle. Hence, a hollow shaft will be stronger than a

solid shaft.

The electric poles are given hollow structure for the same reason.

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ELASTIC AFTER EFFECT

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When the deforming force is removed, the bodies tend to return to their

respective original state. While some bodies return to their original states

immediately, others take appreciably long time to do so.

For example, a quartz fibre returns immediately to its normal state, when

the twisting torque acting on it ceases to act.

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On the other hand, a glass fibre will take hours to return to its original

state.

This delay in regaining the original state by a body after the removal of the

deforming force is called elastic after effect.

In galvanometers and electrometers, the suspensions made from quartz

and phosphor-bronze are used as the elastic after effect is negligible in

wires of these materials.

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ELASTIC FATIGUE

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Due to continuous alternating strains, a wire is said to have been tired or

fatigued. This phenomenon as elastic fatigue.

The elastic fatigue is defined as the loss in the strength of a material caused

due to repeated alternating strains to which the material is subjected.

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If a body is subjected to repeated strains beyond its elastic limit, it ultimately

breaks.

For example,

A hard wire is usually broken by bending it repeatedly in opposite directions.

The railway bridges are declared unfit after their use for a reasonably long

period.

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Q. 1.01. What is a deforming force?

Ans. A force that produces a change in the shape or size of a body is called

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the deforming force.

Q. 1.02. Which is the property of a body that opposes its deformation ?

Ans. It is the property of elasticity.

Q. 1.03. Under what condition, the restoring forces are equal and opposite to

the external deforming force?

Ans. When the body is deformed within its elastic limit.

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Q. 1.04. Which of the two forces-deforming or restoring is responsible for

elastic behaviour of a substance?

Ans. Restoring force.

Q. 1.05. State Hooke’s law.

Ans. Within elastic limit, stress is directly proportional to strain.

Q. 1.06. What is the limitation of the Hooke’s law?

Ans. It holds good, when a wire is loaded within its elastic limit.

Q. 1.07. Define stress and give its SI unit.

Ans. It is defined as the restoring force set up in the body on being deformed.

The SI unit of stress is N m-2.

Q. 1.08. What are the factors on which modulus of elasticity of a material

depends?

Ans. Nature of the material and the manner in which it is deformed. 23

Q. 1.09. A heavy wire is suspended from a roof but no weight is attached to

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its lower end. Is it under stress? Justify your answer.

Ans. A heavy wire (even when no weight is attached to it) is under stress,

when it is suspended from a roof. It is because, the weight of the heavy wire

acts as the deforming

Q. 1.10. Why do we prefer steel to copper in the manufacture of spring.

Ans. It is because, Young’s modulus of steel is large as compared to that of

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copper.

Q. 1.11. In stretching a wire, work has to be performed. Why?

Ans. When a wire is stretched, interatomic forces come into play and these

forces oppose the increase in length of the wire. Therefore, in order to stretch

the wire, work has to be done against the interatomic forces.

Q.1.13. A wire fixed at the upper end stretches by length ℓ by applying a force

F. What is the work done in stretching the wire?

Ans. Work done in stretching the wire,

W = 1/2 stretching force x increase in length

=-Fℓ

Q. 1.12. When a wire is stretched, work has to be done. What happens to the

work done during the stretching of the wire?

Ans. The work done in stretching the wire is stored in it in the form of the 24

elastic potential energy.

Q. 1.14. Which of the three types of elasticity (Y, K and η possessed by all the

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three states of matter (solid, liquid and gas)?

Ans. The volume elasticity (K) is possessed by all the three states of the

matter.

Q. 1.15. A hard wire is broken by bending it repeatedly in opposite direction.

Why?

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Ans. It is because of the loss of the strength of the material due to repeated

alternating strains, to which the wire is subjected.

Q. 1.16. Why any metallic part of a machinery is never subjected to a stress

beyond the elastic limit of the material?

Ans. A permanent deformation will be set up in that metallic part of the

machinery.

Q. 1.17. What is an elastomer?

Ans. The elastic substances, which can be subjected to large values of strain,

are called elastomers.

Q. 1.19. Why are electric poles given hollow structure?

Ans. A hollow shaft is stronger than a solid shaft made from the same and

equal amounts of material.

Q. 1.20. Name the factors, which affect the property of elasticity of a solid.

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Ans. (i) Presence of impurities.

(ii) Change of temperature.

(iii) Effect of hammering, rolling and annealing.

Q. 1.02. Elasticity has different meaning in physics and in our daily life.

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Comment.

Ans. In daily life, a body is said to be more elastic, if large deformation or strain

is produced on subjecting the material to a given stress. However, in physics, it

is exactly opposite. A body is said to be more elastic, if a small strain is

produced applying the given stress.

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Q. 1.06. The stress versus strain graphs for wires of two materials A and B

are as shown in Fig. (a) Which material is more ductile ? (b) Which material

is more brittle ?

Ans. (a) Material A is more ductile. It is

because, the material A has greater plastic

range (portion of graph between the elastic

limit and breaking point).

(b) Material B is more brittle. It is because,

the material A has lesser plastic range.

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Q. 1.07. Why a spring is made of steel and not of copper ?

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Ans. A better spring will be the one, in which a large restoring force is

developed on being deformed. This, in turn, depends upon the elasticity of the

material of the spring. As Young’s modulus of steel is greater than that of

copper, steel is preferred to manufacture a spring.

Q. 1.08. The length of a wire increases by 8 mm, when a weight of 3 kg is

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hung. If all conditions are the same, what will be the increase in its length,

when diameter is doubled?

Therefore, if diameter of the wire is doubled, the increase in length will

become one fourth i.e. ℓ = 8/4 = 2 mm

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