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Acknowledgement

I take opportunity to express my deep sense of gratitude and thank to miss Purva Mishra teacher of physics of Shri Agresen Vidhalaya for giving guidance in this project work . Her enormous contribution towards my project ‘elasticity’. Her continuous guidance throughout project and very friendly and useful approach, I will never forget.

I also thank Mrs.. Mradula Patni for her encouragement for this work.

Student of class 11th (p.c.m)

Certificate

This is to certify that Ishita Agrawal of class 11thB in the session 2007-2008 of this school has completed the project ‘Elasticity’.

she has submitted a complete and satisfactory account of the same.

Subject teacher Principle

Miss Purva Mishra Mrs. Mradula Patni

Introduction

Stress

Types of stress

Strain

Types of Strain

Hooke’s law

Modulus of Elasticity

Young’s modulus

Bulk modulus

Modulus of rigidity

Poisson\'s ratio

Work done in stretching a wire

Some other terms related to Elasticity

1. Elastic limit

2.Elstic after effect

3. Elastic fatigue

4.Compressiblity

Some conceptual question’s

Bibliography

ContentIntroduction

Elasticity is a branch of physics which studies the properties of elastic materials. A material is said to be elastic if it deforms under stress (e.g., external forces), but then returns to its original shape when the stress is removed. The amount of deformation is called the strain.

Stress

- Stress, defined as force per unit area, is a measure of the intensity of the total internal forces acting within a body across imaginary internal surfaces, as a reaction to external applied forces and body forces. It was introduced into the theory of elasticity by Cauchy around 1822. Stress is a concept that is based on the concept of continuum.

Types of Stress

- Basically it is of three types
- Tensile stress:-If there is an increase in the length or extension of the body in the direction of the body in the direction of force applied.
- Compression stress:-If there is decrease in length of the wire or compression of the body due to force applied.
- Tangential stress:-When a deforming force, acting tangentially to the surface of a body produces a change in the shape of the body .

Strain

- Stress is defined as the ratio of change in dimension of the body to the original dimension . strain is the geometrical expression of deformation caused by the action of stress on a physical body. Strain is calculated by first assuming a change between two body states: the beginning state and the final state.

Types of Strain

- Basically Strain are of three types
- Longitudinal Strain:-It is defined as the increase in length (L) per unit original length (l) when deformed by the external force.

Longitudinal Strain=L

l

- Volumetric Strain:-It is defined as change in volume (V) per unit original volume (v), when deformed by external force.

Volume Strain=V

v

3. Shear Strain:-When change takes place in the shape of the body, the strain is called shear strain.

Hooke’s Law

- It states that the extension produced in a wire is directly proportional to the load attached to it.

i.e. Stress=E*Strain (where E is constant called modulus of Elasticity or coeffient of elasticity.)

Modulus of Elasticity

- It is defined as ratio of Stress to the corresponding Strain within the proper elastic limit. Its unit is n/m . There are three types of modulus of elasticity.

E=Stress

Strain

Young\'s Modulus

- the description of the elastic properties of linear objects like wires, rods, columns which are either stretched or compressed, a convenient parameter is the ratio of the stress to the strain, a parameter called the Young\'s modulus of the material. Young\'s modulus can be used to predict the elongation or compression of an object as long as the stress is less than the yield strength of the material.

Bulk Modulus

- The bulk elastic properties of a material determine how much it will compress under a given amount of external pressure. The ratio of the change in pressure to the fractional volume compression is called the bulk modulus of the material.

Modulus of rigidity

- It is defined as the ratio of tangential stress to shear strain . It is also called shear modulus . It is denoted by the Greek letter eta (ή) .

ή=tangential stress

shear strain

ή=F

Aθ

Poisson’s Ratio

- It is the ratio of lateral strain per unit longitudinal strain. It is denoted by σ . It is unit less and dimensionless quantity. Its value is in between 0 to 0.5 .

σ =lateral strain (β)

longitudinal strain (α)

σ = d*L

D l

σ = β

α

Work done in stretching a wire

Y=Normal stress

Longitudinal strain

Y=FL

Al

F=YAl

L

This is the expression for stretching force

∫(0->w)dW =∫(0->l)F.dl

W=∫(0->l)YAl/L [F=YAl/L]

W=YA/L∫(0->l)l.dl

W=YAl²

2L

W=1/2 YAl/L . l

This work done is converted into potential energy (U).

U=1/2 extension*stretching force

U=YAl² * L [f/A=Yl/L]

L l

U=1/2 .Yl/L .l/L.AL

U=i/2 stress*strain*volume

since volume = 1

U=1/2 stress*strain

Some other terms related to Elasticity

- Elastic limit: The maximum stress to which the body can regain its original status after the removal of the deforming force is called elastic limit.
- Elastic after effect: The delay in regaining the original state after the removal of the deforming force is called elastic after effect.
- Elastic fatigue: The loss of strength of the material caused due to repeated alternating strain to which the material is subjected is known as elastic fatigue.
- Compressibility: The reciprocal of bulk modulus is called compressibility.

Some conceptual Questions

Q.1 Explain why solids are more elastic than gases?

(Ans) The solids are more elastic than gases because for a given stress, the gases are more compressible than solids.

Q.2 Which is more elastic: water or air, why?

(Ans) Water is more elastic then air, because bulk modulus of elasticity is reciprocal of compressibility than water.

cont…………..

Q.3 Why does spring balance shows wrong reading after long use ?(Ans) The spring will lose its elastic character due to long use .Q.4 Why a spring is made of steel and not of copper ?(Ans) Under a given deforming force , the steel spring is stretched more than copper spring . Moreover the steel spring recovers its original state quicker than copper after the removal of deforming force.Q.5Crystalline solids are called true solids. why?(Ans)Bcoz they have a well defined, regularly repeated three dimensional arrangement of ions in crystalline solids.

Numericals

Q.1The young’s modulus of the material of a wire is 6X1012N/m2 and there is no transverse strain in it. Find its modulus of rigidity?

(Ans) Modulus of rigidity is given by

Cont…………….

Q.2The breaking stress of aluminum is 8.1X107Nm-2. Find the greatest length of aluminum wire that can hang vertically without breaking. Density of aluminum is 2.7X103 kgm-3.(Ans) Let a = area of cross section and l= length of wire which can be suspended Mass of the wire, m=volume X densityCont…………..

Q.3Two wires A and B of the same material have radii in the ratio 2:1 and lengths in the ratio 4:1. Find the ratio of the normal forces required to produce the same change in the lengths of the two wires?(Ans) Given radii are 2:1 and lengths are 4:1 Cont………………

Q.4Find the maximum length of steel wire that can hang without breaking. Breaking stress=7.9X1012dyne/cm2. Density of steel =7.9g/cc(Ans) Cont……………..

Q.5 A long spring is stretched by 2 cm and its potential energy is V. Find the potential energy of the spring if it is stretched by 10 cm?(Ans) Potential energy of the stretched springCont……………..

Bibliography

- www.google.com
- Pradeep’s
- Comprehensive
- Yogbodh
- Modern abc

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