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Topics Covered

• Dimensions of Physical Quantities
• Dimensional Formulae
• Dimensional Equations
• Checking the Dimensional Consistency of Equations
• Deducing Relation among the Physical Quantities

Dimensions of Physical Quantities

• The dimension of a physical quantity is defined as the powers to which the fundamental quantities are raised in order to represent that quantity. The seven fundamental quantities are enclosed in square brackets [ ] to represent its dimensions.

• Thus, length has the dimension [L], mass [M], time [T], electric current [A], thermodynamic temperature [K], luminous intensity [cd], and amount of substance [mol].

• In mechanics, all the physical quantities can be written in terms of the dimensions [L], [M] and [T]. For example, the volume occupied by an object is `[L] × [L] × [L] = [L]^3 = [L^3]`.

Dimensional Formulae and Dimensional Equations

`"Dimensional Formula"`
The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity.
`"Example-"` The dimensional formula of the volume is `[M° L^3 T°]`, and that of speed or velocity is `[M° L T^(-1)]`.

`"Dimensional Equation"`
• An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity.

• Thus, the dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities.

`"Example-"` The dimensional equations of volume `[V] = [M^0 L^3 T^0]`

Dimensional Analysis and Its Applications

• The dimensional analysis helps us in deducing certain relations among different physical quantities and checking the derivation, accuracy and dimensional consistency or homogeneity of various mathematical expressions.

• When magnitudes of two or more physical quantities are multiplied, their units should be treated in the same manner as ordinary algebraic symbols.

• The physical quantities represented by symbols on both sides of a mathematical equation must have the same dimensions.

Checking the Dimensional Consistency of Equations

`"Principle of Homogeneity"`
The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. This simple principle called the principle of homogeneity of dimensions.

• If the dimensions of all the terms are not same, the equation is wrong.

• Dimensions are customarily used as a preliminary test of the consistency of an equation, when there is some doubt about the correctness of the equation. However, the dimensional consistency does not guarantee correct equations.

• A pure number, ratio of similar physical quantities, such as angle as the ratio (length/length), refractive index as the ratio (speed of light in vacuum/speed of light in medium) etc., has no dimensions.

• Now we can test the dimensional consistency or homogeneity of the equation.
For the distance `x` travelled by a particle or body in time `t`
`x=x_0 + v_0 t + 1/2 at^2`

The dimensions of each term may be written as
`[x] = [L]`
`[x_0 ] = [L]`
`[v_0 t] = [L T^–1] [T] = [L]`
`[(1/2) a t^2] = [L T^–2] [T^2] = [L]`

As each term on the right hand side of this equation has the same dimension, namely that of length, which is same as the dimension of left hand side of the equation, hence this equation is a dimensionally correct equation.

• If an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be

Deducing Relation among the Physical Quantities

Consider a simple pendulum, having a bob attached to a string, that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length (`l`), mass of the bob (`m`) and acceleration due to gravity (`g`).

We can derive the expression for its time period using method of dimensions.

The dependence of time period T on the quantities `l, g` and `m` as a product may be written as :
`T=kl^x g^y m^z`
where k is dimensionless constant and x, y and z are the exponents.

By considering dimensions on both sides, we have

`[L^oM^oT^1]=[L^1 ][L^1 T^–2 ]^y [M^1 ]^z`
`= L^(x+y) T^(–2y) M^z`
On equating the dimensions on both sides, we have
`x + y = 0; –2y = 1;` and `z = 0`

So that `x=1//2, y=-1//2, z=0`

Then, `T=k l^(1//2) g^(-1//2)`

or, `T=k sqrt(l/g)`
here `k=2pi`
`T=2pi sqrt(l/g)`