Question

Explain in detail the dimension, dimensional formula; dimensional equation and principle of homogeneity.

Answer 1

When the unit of any physical quantity is represented in terms of funda mental units then the obtained expression is called the “Dimensional Formula” of the quantity and the raised powers on fundamental units, are called ‘Dimensions’ of the quantity. For example:

Speed = \(\frac{\text{Distance }}{\text{Time }}\)

∴ Unit of Speed = \(\frac{\text{Unit of distance}}{\text{Unit of time}}\)

= (Unit of distance)¹ x (Unit of time)^-1

Thus it is clear that to obtain the unit of speed we have to raise power of 1 over the unit of distance and power of -1 over the unit of time. Thus the units of speed can be written in different systems as ms^-1, cms^-1, foot.s^-1. Hence, it is clear that the power i.e. dimensions remains unchanged even on changing the system of units i.e, “Dimensions of any physical quantity do not depend upon unit.”

Now, the dimensional formula of speed

= \(\frac{\text{Dimensional formula of distance}}{\text {Dimensional formula of time }}\)

= \(\frac{L^1}{T^1}\) = L¹T^-1

= [M⁰L¹T^-1]

Thus [M⁰L¹T^-1] is the dimensional formula of speed and (0,1,-1) are the dimensions of speed.

Dimensional equation: The equation obtained on equating the symbol of physical quantity with its dimensional formula is called dimesnional equation.

For example

Force = [M¹L¹T^-2]

or [F] = [M¹L¹T^-2] Dimensional equation

Principle of homogeneity of dimensions:

According to this principle, for a physical quantity, the dimensions of fundamental quantities on both the sides should be equal i.e., for equation

[M^aL^bT^c] = [M^xL^yT^z]. According to principle of homogeneity

x = a; y = b and z = c