Elasticity Maharashtra Board Additional Questions

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Elastic Limit:

  • It is the maximum stress to which an elastic body can be subjected without causing permanent deformation is called the elastic limit.

Effects of Applied Force on a Body:

  • If the forces acting on a body are unbalanced i.e. the resultant force acting on the body is not zero, then body sets into translational motion.
  • If the forces acting on a body are balanced but the net moment is not zero i.e. the resultant force acting on the body is zero but the resultant moment is not zero, then body sets into rotational motion.
  • If the forces on the body are balanced and there is no net moment for the body then the force may produce a change in size, shape or both in the body.

Characteristics of Elasticity:

  • The body exhibiting elasticity regains its shape or size after removal of the external force.
  • There is a temporary change in dimensions of the body on application of the deforming force.
  • Internal restoring force is set up inside the body.
  • The ratio of stress to corresponding strain produced is constant.

Characteristics of Plasticity:

  • The body exhibiting plasticity retains its new shape or size after removal of the external force.
  • There is a permanent change in dimensions of the body on application of the deforming force.
  • No Internal restoring force is set up inside the body. Or they are very negligible.
  • The ratio of stress to corresponding strain produced is not constant.

Characteristics of Rigidity:

  • There is no change in the shape of the body on the application of deforming force.
  • There is a no change in dimensions of the body on application of the deforming force.
  • As there is no change in the shape of the body, there is no question of internal restoring force.
  • As there is no change in the shape of the body the strain is zero.

Characteristics of Deforming Force:

  • It is an externally applied force
  • It tries to change the size, shape or both of the body. i.e. it produces deformation.
  • It is a vector quantity.
  • Its S.I. unit is N and its dimensions are [L1M1T-2].

Characteristics of Stress:

  • Stress is an internal restoring force per unit area.
  • It opposes the change in the size, shape or both of the body. i.e. it opposes deformation.
  • It is a tensor quantity.
  • Its S.I. unit is Nm-2 and its dimensions are [L-1M1T-2].

Characteristics of Strain:

  • The strain is defined as the ratio of change in dimensions of a body to its original dimensions when subjected to deformation.
  • For longitudinal loading, both the longitudinal and the lateral strain are produced.
  • It is a scalar quantity.
  • it is unit less dimension less quantity.

Characteristics of Modulus of Elasticity:

  • The ratio of the stress produced in a body to corresponding stress produced in it is called the modulus of elasticity of the material of the body.
  • It is the characteristic property of the material of the body.
  • Its S.I. unit is Nm-2 and its dimensions are [L-1M1T-2].
  • Depending upon loading there are three types of elasticity constant. If there is a change in the length of a body then the corresponding elastic constant is called Young’s modulus of elasticity.  If there is a change in the volume of a body then the corresponding elastic constant is called the bulk modulus of elasticity.  If there is a change in the shape of a body then the corresponding elastic constant is called modulus of rigidity.

Characteristics of Young’s Modulus of Elasticity:

  • Within the elastic limit, it is the ratio of longitudinal stress to longitudinal strain
  • It is associated with the change in the length of a body.
  • It exists in solid material bodies
  • It is a measure of the stiffness of a solid material.
  • Young’s modulus of the material of a wire is given by

Characteristics of Bulk Modulus of Elasticity:

  • Within the elastic limit, it is the ratio of volumetric stress to volumetric strain.
  • It is associated with the change in the volume of a body.
  • It exists in solids, liquids, and gases.
  • It determines how much the body will compress under a given amount of external pressure.
  • The bulk modulus of a material of a body is given by

Characteristics of Modulus of Rigidity or Shear Modulus:

  • Within the elastic limit, it is the ratio of shear stress to shear strain
  • It is associated with the change in the shape of a body.
  • It exists in solids only.
  • It describes an object’s tendency to shear
  • The shear modulus of a material of a body is given by

Hooke’s Law of Elasticity:

Statement: 

  • Within the elastic limit, the stress is directly proportional to the strain.

Explanation:

By Hooke’s Law,

Stress  ∝  Strain



∴ Stress  = Constant  x  Strain

∴ Stress / Strain  = Constant

  • This constant of proportionality is called the modulus of elasticity or coefficient of elasticity or elastic constant.
  • The ratio of the stress produced in a body to corresponding stress produced in it is called the modulus of elasticity of the material of the body.

Graphical Representation:

Elastic Limit

Compressibility:

  • The reciprocal of bulk modulus of elasticity is called as compressibility.
  • Its S.I. unit is m2 N-1 or Pa-1 and its dimensions are  [L-1M-1T2].

The Expression for Thermal Stress:

  • If a bar which is heated and prevented from expansion or heated rod is prevented from contraction as it cools, then stress is produced in the bar. This stress is called a thermal stress. If the bar is prevented from expansion or contraction, the change in length is given by

Δl = l ∝ Δθ



Where,  Δl = change in length prevented.

l = original length of the rod

Δθ = Change in temperature.

 ∝ = coefficient of linear expansion of the material of the rod.

Shear Stress, Shear Strain and Modulus of Rigidity



This is the expression for thermal stress produced in the body

Where Y = Young’s modulus of elasticity.

If A is the area of the cross-section of the bar then the force acting on the rod is

F = Thermal stress x area of cross-section

Force = Y α θ A



Scientific Reasons:

Pressure and stress have the same dimensions but are two different physical quantities.

  • Both the pressure and the stress has the same dimensions [L-1M1T-2], but pressure is a scalar quantity while stress is a tensor quantity. Hence they are two different physical quantities.

Steel is more elastic than rubber.

  • The elastic properties depend on Young’s modulus of elasticity. Greater the value of Young’s modulus, more the elastic material is.
  • The value of Young’s modulus of elasticity is more for steel than that for rubber. Hence steel is more elastic than rubber.
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