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CBSE NOTES CLASS 11 CHAPTER 9

MECHANICAL PROPERTIES OF SOLIDS

Deforming Force

Elasticity

Restoring force

Spring ball model of elastic behavior

Plasticity

Elastic limit

Perfectly elastic bodies

Perfectly plastic bodies

Stress

Types of stress

Normal stress

Tensile stress

Compressive stress

Tangential stress

Hydraulic stress

Strain

Longitudinal strain

Shearing strain

Volumetric strain

Hooke’s law

Types of modulus of elasticity

Young’s modulus

Shear modulus

Bulk modulus of elasticity

Compressibility

Stress-strain curve

Limit of elasticity

Breaking stress

Elastic fatigue

Ductile materials

Brittle materials

Elastomers

Elastic potential energy in a stretched wire

Poisson’s ratio

Relation between different modulii

Thermal stress

Interatomic force constant

Applications of elastic behaviour of materials

How thick should the steel rope of the crane be?

Parameters of a bridge

Cantilever

Why a mountain cannot be higher than 10 km?

Hollow shaft vs solid shaft

CBSE NOTES CLASS 11 CHAPTER 9

MECHANICAL PROPERTIES OF SOLIDS

Deforming Force

A force, which when applied, produces a change in configuration of the object, is called a deforming force.

Elasticity

The property of a body, by virtue of which it tends to regain its original size and shape when the applied deforming force is removed, is known as elasticity and the deformation caused is known as elastic deformation.

Restoring force

If the system is disturbed from its equilibrium by a deforming force, the system applies a force equal in magnitude but opposite in direction to the deforming force. This is called restoring force. The restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle (i.e. extent of deformation).

Spring ball model of elastic behavior

We can visualize the restoring mechanism by taking a model of spring-ball system as shown in the figure. The balls represent atoms and springs represent interatomic forces. If any ball is displaced from its equilibrium position, the spring system tries to restore the ball back to its original position. This is the microscopic model of elasticity.

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Plasticity

The bodies which have no gross tendency to regain their previous shape, and they get permanently deformed on application of deforming force, are called plastic and the property is called plasticity.

Elastic Limit

Elastic limit is the upper limit of deforming force upto which, if deforming force is removed, the body regains its original form completely and beyond which if the deforming force is increased the body loses its property of elasticity and gets permanently deformed.

Perfectly Elastic Bodies

Those bodies which regain their original configuration immediately and completely after the removal of deforming force are called perfectly elastic bodies. e.g., quartz and phosphor bronze etc.

Perfectly Plastic Bodies

Those bodies which do not regain their original configuration at all on the removal of deforming force are called perfectly plastic bodies, e.g., putty, paraffin wax etc.

Stress

The internal restoring force acting per unit area of a deformed body is called stress.

 Stress =   Restoring force  Area 

SI unit is N/m2 or Pascal.

Types of Stress

Normal Stress

If deforming force is applied normal to the area, then the stress is called normal stress.

If there is an increase in length, then stress is called tensile stress.

If there is a decrease in length, then stress is called compressive stress.

Tangential Stress

The restoring force per unit area developed due to the applied tangential force is known as tangential or shearing stress.

Hydraulic Stress

When an object is immersed in a fluid, there is a normal force acting on the surface of the object. The internal restoring force per unit area in this case is known as hydraulic stress and its magnitude is equal to the hydraulic pressure (applied force per unit area).

Strain

The fractional change in configuration is called strain.

Strain =  Change in the configuretion   Original configuration

It has no unit and it is a dimensionless quantity.

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According to the change in configuration, the strain is of three types

  1. Longitudinal strain

    The change in the length ΔL to the original length L of the body (cylinder in this case) is known as longitudinal strain.

    Longitudinal strain = Change in lengthOriginal length= ΔLL


  2. Shearing strain

    It is defined as the ratio of relative displacement of the two faces, Δx, to the length of the cylinder L.

    Shearing strain = ΔxL=tanθ, 

    where θ is the angular displacement of the cylinder from the vertical (original position of the cylinder).

    For small θ,

    Shearing strain = tan θ ≈ θ


  3. Volumetric strain

    The strain produced by a hydraulic stress is called volume strain and is defined as the ratio of change in volume (ΔV) to the original volume (V).

    Volume strain = Change in volumeOriginl volume=ΔVV

Hooke’s Law

Within the elastic limits, the stress is proportional to the strain.

Stress ∝ Strain

or Stress = E × Strain

where, E is the modulus of elasticity of the material of the body.

Hooke’s law is an empirical law and is found to be valid for most materials.

TYPES OF MODULUS OF ELASTICITY

Young’s Modulus

The ratio of tensile (or compressive) stress (σ) to the longitudinal strain (ε) is defined as Young’s modulus and is denoted by the symbol Y.

Y =Normal stressLongitudinal strain

Y =σε=F/AΔL/L= FLAΔL

Its unit is N/m2 or Pascal

Shear Modulus

The ratio of shearing stress to the corresponding shearing strain is called the shear modulus of the material and is represented by G. It is also called the modulus of rigidity.

G =Shearing stressσsShearing strain

G =F/AΔx/L=F × LA × Δx

Or G =F/Aθ=FA × θ

The shearing stress σs can also be expressed as σs = G × θ

SI unit of shear modulus is N m–2 or Pa.

For most materials G Y/3.

Bulk Modulus of Elasticity

The ratio of hydraulic stress to the corresponding volume strain is called bulk modulus. It is denoted by symbol B.

B = ΔpΔV/V

The negative sign indicates the fact that with an increase in pressure, a decrease in volume occurs. That is, if Δp is positive, ΔV is negative. Thus for a system in equilibrium, the value of bulk modulus B is always positive.

SI unit of bulk modulus is the same as that of pressure i.e., N m–2 or Pa.

Compressibility

Compressibility of a material is the reciprocal of its bulk modulus of elasticity.

Compressibility is given by,

 k = 1B=  1Δp× ΔVV

Its SI unit is N-1m2 and CGS unit is dyne-1 cm2.

Stress-Strain Curve

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Limit of Elasticity

The maximum value of deforming force for which elasticity is present in the body is called its limit of elasticity.

Breaking Stress

The minimum value of stress required to break a wire, is called breaking stress.

Breaking stress is fixed for a material but breaking force varies with area of cross-section of the wire.

Safety factor = Breaking stressWorking stress

Elastic Fatigue

The property of an elastic body by virtue of which its behavior becomes less elastic under the action of repeated alternating deforming force is called elastic fatigue.

Ductile Materials

The materials which show large plastic range beyond elastic limit are called ductile materials, e.g., copper, silver, iron, aluminum, etc.

Ductile materials are used for making springs and sheets.

Brittle Materials

The materials which show very small plastic range beyond elastic limit are called brittle materials, e.g., glass, cast iron, etc.

Elastomers

The materials for which strain produced is much larger than the stress applied, with in the limit of elasticity are called elastomers, e.g., rubber, the elastic tissue of aorta, the large vessel carrying blood from heart.

Elastomers have no plastic range.

Elastic Potential Energy in a Stretched Wire

The work done in stretching a wire is stored in form of potential energy of the wire.

Potential energy U = Average force × Increase in length

Elastic potential energy per unit volume

Elastic potential energy of a stretched spring = 12 k x2,

where, k = Force constant of spring and x = Change in length.

Poisson’s Ratio

When a deforming force is applied at the free end of a suspended wire of length 1 and radius R, then its length increases by dL but its radius decreases by dR. Now two types of strains are produced by a single force.

Poisson's ratio (σ) = Lateral strainLongotudinal strain =  ΔR/RΔL/L

The theoretical value of Poisson’s ratio lies between – 1 and 0.5. Its practical value lies between 0 and 0.5.

Relation between Y, B, G and σ

i Y = 3B1  2σ

ii Y = 2 G 1 + σ

iii σ =3B  2G2G + 6B

iv  9Y=1B+3G   Y = 9 B GG + 3B

Thermal Stress

Applications of Elastic Behaviour of Materials

For δ to be small, L should be small, Y should be large, b and d should be large

Hollow shaft vs Solid Shaft

Torque τ required, to produce, a unit twist in a solid shaft of radius r, length L and of a material of modulus of rigidity G is given by,

τ = π G r42L

For a hollow cylindrical shaft

τ’ =π G r24- r142L

If the two shafts are made of same and equal amount of material, then there volumes will be same,

Or, r2 = r22 –r12

τ’τ =π G r24- r142L/π G r42L