Dimensional Formula Of Coefficient Of Elasticity

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Dimensions

Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity.  Dimensions of any given quantity tell us about how and which way different physical quantities are related. Finding dimensions of different physical quantities has many real-life applications and is helpful in finding units and measurements. Imagine a physical quantity X which depends mainly on base mass(m), length(L), and time(T) with their respective powers, then we can represent dimensional formula as [MaLbTc]


Dimensional Formula 

The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e ( ).

E.g: Dimension formula of mass is: (M)


Dimensional Equation

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. 


Application of Dimensional Analysis 

  • To convert a physical quantity from one system of the unit to the other.

  • It is based on the fact that the magnitude of a physical quantity remains the same whatever system is used for measurement i.e magnitude = numeric value(n) multiplied by unit (u) 

  • To check dimensional correctness of a given physical relation:  constant n1u1= n1u2.

  • If in a given relation, the terms of both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions. 

  • To derive a relationship between different physical quantities.

  • Using the principle of homogeneity of dimension, the new relation among physical quantities can be derived if the dependent quantities are known. 


Limitation of This Method

  • This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. The formula containing more than one term which is added or subtracted likes s = ut+ ½ at2 also cannot be derived.

  • The relation derived from this method gives no information about the dimensionless constants. 


Elasticity

A body is said to be rigid if the relative position of its constituent particle remains unchanged when external deforming forces are applied to it. The nearest approach to a rigid body is diamond or carborundum. Actually nobody is perfectly rigid and everybody can be deformed more or less by the application of suitable forces. All these deformed bodies, however, regain their original shape or size, when the deforming forces are removed. 

The property of matter by virtue of which a body tends to regain its original shape and size after the removal of deforming forces is called elasticity. 


Definition of Some Important Terms Related to Elasticity

Deforming Force: External force that tries to change in the length, volume or shape of the body is called deforming force. 

Elasticity: Elasticity is the property of the material of a body by virtue of which the body opposes any change in its shape and size when deforming forces are applied to it and recover its original state as soon as the deforming force is removed.

Perfectly Elastic Body: The body which perfectly regains its original form on removing the external deforming force is defined as a perfectly elastic body. Example: quartz (very nearly a perfectly elastic body).

Plastic Body: The body which does not have the property of opposing the deforming force is known as a plastic body.


Dimensional Formula of the Coefficient of Elasticity

The internal restoring force acting per unit area of the cross-section of the deformed body is called the coefficient of elasticity. 

As stress is directly proportional to strain, therefore we can say that stress by strain leads to the constant term. 

Therefore,

stress/strain= constant

Here elasticity coefficient depends only one the type of material used and it does not depend upon the value of stress and strain.

The dimensional formula coefficient of elasticity is given by, [M1 L-1 T-2]

Where, M = Mass, L = Length and T = Time


Types of Elasticity Coefficient

The elasticity coefficient is of three types:

1. Young's Modulus Elasticity

2. Bulk Modulus Elasticity

3. Modulus of Rigidity


Derivation of the Dimension of Elasticity of Coefficient

We know that Coefficient of Elasticity = Stress × [Strain]-1 . . . . (1)

As we known, Stress = Force × [Area]-1 . . . (2)

And, Force = Mass × acceleration = [M × LT-1 × T-1]

Therefore dimensions of force = [M1 L1 T-2] . . . . (3)

The dimensional formula of area = [M0 L2 T0] . . . . (4)

On putting equation (3) and (4) in equation (2) we get,

 As, Stress = Force × [Area]-1 = [M1 L1 T-2] × [M0 L2 T0]-1

Therefore, the dimensional formula of stress = [M1 L-1 T-2] . . . . (5)

And, Strain = ΔL × L-1

∴ The dimensions of Strain = [M0 L0 T0] . . . . (6)

On putting equation (5) and (6) in equation (1) we get,

Coefficient of Elasticity = Stress × [Strain]-1

Or, Elasticity = [M1 L-1 T-2] × [M0 L0 T0]-1 = [M1 L-1 T-2].

Therefore, the coefficient of elasticity is dimensionally represented as [M1 L-1 T-2].


Factors Affecting the Elasticity Coefficient

There are two factors which affects the elasticity coefficient:

Effect of Temperature: when the temperature is increased, the elastic properties in general decrease i.e elastic constant decreases whereas plasticity increases with temperature.

Example: At ordinary temperature carbon is elastic but at high-temperature carbon becomes plastic. 

Effect of Impurity: Y slightly increases by impurity. The intermolecular attraction force inside the wire effectively increases by impurity due to this external force can be easily opposed.

FAQ (Frequently Asked Questions)

1. Define Elasticity?

Ans. A body is said to be rigid if the relative position of its constituent particle remains unchanged when external deforming forces are applied to it. The nearest approach to a rigid body is diamond or carborundum. Actually nobody is perfectly rigid and everybody can be deformed more or less by the application of suitable forces. All these deformed bodies, however, regain their original shape or size, when the deforming forces are removed. 

The property of matter by virtue of which a body tends to regain its original shape and size after the removal of deforming forces is called elasticity.

2. Factors affecting the coefficient of Elasticity?

Ans.

  • Effect of Temperature: when the temperature is increased, the elastic properties in general decrease i.e elastic constant decreases whereas plasticity increases with temperature. Example: At ordinary temperature carbon is elastic but at high-temperature carbon becomes Plastic. 

  • Effect of Impurity: Y slightly increases by impurity. The intermolecular attraction force inside the wire effectively increases by impurity due to this external force can be easily opposed.

3. Explain a few limitations of Dimensional Formula?

Ans.

  • This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. The formula containing more than one term which is added or subtracted likes s = ut+ ½ at2 also cannot be derived.

  • The relation derived from this method gives no information about the dimensionless constants.