Dimensions of a Physical Quantity || Dimensional Formula and Dimensional Equation || Applications of Dimensional Analysis

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Dimensions of a Physical Quantity

Dimensions of a physical quantity are powers (exponents) to which base quantities are raised to represent that quantity. They are represented by square brackets around the quantity.

Dimensions of the 7 base quantities are – Length [L], Mass [M], time [T], electric current [A], thermodynamic temperature [K], luminous intensity [cd] and amount of substance [mol].

Examples, Volume = Length x Breadth x Height = [L] x [L] x [L] = [L]³ = [L³]

Force = Mass x Acceleration = [M][L]/[T]² = [MLT^-2]

The other dimensions for a quantity are always 0. For example, for volume only length has 3 dimensions but the mass, time etc have 0 dimensions. Zero dimension is represented by superscript 0 like [M⁰].

Dimensions do not take into account the magnitude of a quantity

Dimensional Formula and Dimensional Equation

Dimensional Formula is the expression which shows how and which of the base quantities represent the dimensions of a physical quantity.

Dimensional Equation is an equation obtained by equating a physical quantity with its dimensional formula.

Physical Quantity	Dimensional Formula	Dimensional Equation
Volume	[M⁰ L³ T⁰]	[V] = [M⁰ L³ T⁰]
Speed	[M⁰ L T^-1]	[υ] = [M⁰ L T^-1]
Force	[M L T^-2]	[F] = [M L T^-2]
Mass Density	[M L^-3 T⁰]	[ρ] = [M L^-3 T⁰]

Dimensional Analysis

Only those physical quantities which have same dimensions can be added and subtracted. This is called principle of homogeneity of dimensions.
Dimensions can be multiplied and cancelled like normal algebraic methods.
In mathematical equations, quantities on both sides must always have same dimensions.
Arguments of special functions like trigonometric, logarithmic and ratio of similar physical quantities are dimensionless.
Equations are uncertain to the extent of dimensionless quantities.

Example Distance = Speed x Time. In Dimension terms, [L] = [LT-1] x [T]

Since, dimensions can be cancelled like algebra, dimension [T] gets cancelled and the equation becomes [L] = [L].

Applications of Dimensional Analysis

Checking Dimensional Consistency of equations

A dimensionally correct equation must have same dimensions on both sides of the equation.
A dimensionally correct equation need not be a correct equation but a dimensionally incorrect equation is always wrong. It can test dimensional validity but not find exact relationship between the physical quantities.

Example, x = x₀ + v₀t + (1/2) at²Or Dimensionally, [L] = [L] + [LT^-1][T] + [LT^-2][T²]

x – Distance travelled in time t, x₀ – starting position, v₀ – initial velocity, a – uniform acceleration.

Dimensions on both sides will be [L] as [T] gets cancelled out. Hence this is dimensionally correct equation.

Deducing relation among physical quantities

To deduce relation among physical quantities, we should know the dependence of one quantity over others (or independent variables) and consider it as product type of dependence.
Dimensionless constants cannot be obtained using this method.

Example, T = k l^xg^ym^z

Or [L⁰M⁰T¹] = [L¹]^x [L¹T^-2]^y [M¹]^z= [L^x+yT^-2yM^z]

Means, x+y = 0, -2y = 1 and z = 0. So, x = ½, y = -½ and z = 0

So the original equation reduces to T = k √l/g