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 provide us with an idea about how 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 the 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.: Dimensional 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 for whatever system is used for measurement, i.e., magnitude = numeric value(n) multiplied by unit (u).
To check the dimensional correctness of a given physical relation: constant \[n_{1}u_{1} = n_{2}u_{2}\].
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 cannot be derived using this method. The formula containing more than one term which is added or subtracted likes \[s = ut + \frac{1}{2} at^{2} \] also cannot be derived.
The relation derived from this method gives no information about the dimensionless constants.
Elasticity
When an external force is applied to a material, the molecular structure of that substance deforms. When this force is removed, the material returns to its former shape; this is known as material elasticity. It is the ability to return to its original shape after the applied stress has been removed. Hooke's Law can be used to describe elasticity. To put it another way, the amount of compression or stretching is proportionate to the force applied. However, materials retain their original shapes until a certain point under an applied force.
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,
\[\frac{stress}{strain} = constant \]
Here the elasticity coefficient depends only on 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 know, Stress = Force × Area-1 . . . (2)
Also, 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 equations (3) and (4) into equation (2), we get
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 equations (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 that 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 which external force can be easily opposed.
Stress
What is the definition of stress?
When the body is deformed by the application of external forces, internal forces are activated. Internal restorative forces allow elastic bodies to restore their original shape. Internal and external forces are in opposition to one another. The stress is defined as the force per unit area when a force F is applied uniformly over a surface of area A.
Force/Area = Stress
Nm-2 is the SI unit for stress.
Stresses of various kinds:
There are three sorts of stress.
Longitudinal stress
Bulk stress or volume stress
Shear stress (or tangential stress)
Longitudinal Stress
Longitudinal Stress is a type of stress that occurs over time. Longitudinal stress occurs when the stress is normal for the body's surface area, but there is a change in the length of the body.
It is divided into two sorts once again.
Tensile stress
Compressive stress
Tensile stress occurs when longitudinal stress is produced as a result of an increase in the length of an object.
Compressive stress is a type of longitudinal stress that occurs when an object's length decreases.
Bulk Stress or Volume Stress
Volume stress occurs when equal normal forces are applied to the body, and the result is a change in the volume of the body.
Tangential Stress
Tangential or Shear stress occurs when the stress is tangential or parallel to the body's surface. The shape of the body changes (or twists) as a result of this.
FAQs on Dimensional Formula Of Coefficient Of Elasticity
1. What is elasticity?
When a material is subjected to an external force, its molecular structure deforms. When the force is released, the material returns to its original shape, which is referred to as material elasticity. It is the ability to revert to its original shape after applied stress is eliminated. Elasticity can be described using Hooke's Law. In other words, the amount of compression or stretching is proportional to the applied force. Materials, on the other hand, retain their original forms until a certain point under an applied force.
2. What are the factors affecting the coefficient of Elasticity?
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 which external force can be easily opposed.
3. Explain a few limitations of the Dimensional Formula.
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 like \[s = ut + \frac{1}{2} at^{2} \], also cannot be derived.
The relation derived from this method gives no information about the dimensionless constants.
4. What is stress?
Internal forces are activated when the body is distorted by the application of external forces. Internal restorative forces enable elastic bodies to return to their original shape. Internal and external forces are diametrically opposed. When a force F is applied evenly over an area A surface, the stress is defined as the force per unit area.
Stress = Force/Area
The SI unit for stress is Nm-2.
There are three different types of stress.
Longitudinal stress
Bulk stress or volume stress
Shear stress (or tangential stress)
5. What is Strain?
When a body is under stress, it deforms. Strain is defined as the fractional change in the dimension of a body caused by external stress acting on it. Strain is defined as the charge-to-original-dimension ratio of any dimension. Strain is only a number because it is the ratio of two similar physical qualities. It lacks both a unit and a dimension.
Strains are further categorized into three kinds.
Longitudinal strain
Volume strain
Shearing strain or tangential strain
6. What is Young's modulus?
The Young's modulus (E or Y) of a solid is a measurement of its stiffness or resistance to elastic deformation under load. It describes the relationship between stress (force per unit area) and strain (proportional deformation) along an axis or line. When a material is squeezed or extended, it undergoes elastic deformation and returns to its original shape when the load is released. When a flexible material deforms, it deforms more than when a stiff substance deforms. That is to say:
A solid that has a low Young's modulus value is elastic.
A solid with a high Young's modulus value is inelastic or stiff.
