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Dimensions:

Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity.

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:

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

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

n1u1= n1u2

2. To check dimensional correctness of a given physical relation:

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.

3. To develop a relationship between different given 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:

1. 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.

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

The Dimension of Acceleration Due to Gravity:

The dimensional formula of Acceleration Due To Gravity is written as: M0L1T-2

Where M = Mass; L = Length; T = Time all these units are standard units of a given quantity.

Derivation of Dimension Formula of Acceleration Due to Gravity:

We know that, Force = Mass × Acceleration due to gravity

∴ Acceleration due to gravity (g) = Force × [Mass]-1 takes it as an equation . . . . . (1)

The dimensional formula of the mass = [M1 L0 T0] takes it as an equation . . . . . (2)

At the same time, the dimensional formula of Force = [M1 L1 T-2] takes it as an equation . . . . (3)

On putting equation (2) and (3) in equation (1) we get,

Acceleration due to gravity = Force × [Mass]-1

Or, g = [M1 L1 T-2] × [M1 L0 T0]-1 = [M0 L1 T-2].

Dimensional representation of acceleration due to gravity is dimensionally given as [M0 L1 T-2].

Acceleration Due to Gravity:

Acceleration due to gravitational force in an object is called acceleration due to gravity. S.I unit of acceleration due to gravity is written as m/s2. It is a vector quantity as it has both magnitude and direction. It is denoted by symbol g and its value is approximately 9.80665 m/s2. However, the actual acceleration of a body in free fall varies with location.

Formula for acceleration due to gravity is: G = k.M1.M2/ r2.

FAQ (Frequently Asked Questions)

Q1. Explain a Few Limitations of Dimensional Formula?

1. 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.

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

2. Write Two Applications of Dimensional Formula Along with Dimensional Formula of Acceleration Due to Gravity?

Two applications of dimensional formula are:

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

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

n1u1= n1u2

2. To check dimensional correctness of a given physical relation:

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.

Dimensional formula of acceleration due to gravity:

The dimensional formula of Acceleration Due To Gravity is written as: M⁰L¹T⁻²

Where, M = Mass; L = Length; T = Time, all these are standard units of given quantity.

3. Write a Few Sets that Have the Same Dimension Formula.

Some of the sets having the same dimension formula that are discussed below:

Strain, refractive index, relative density, distance gradient, relative permeability, angle of content.

Mass and inertia.

Momentum and impulse.

Thrust, force, weight, tension, energy gradient.

Angular momentum and Planck’s constant.

Surface tension, surface area, force gradient, spring constant.

Latent heat and gravitational potential.

Thermal capacity, Boltzman constant, entropy.