Question

Define dimensions and dimensional formula of physical quantities. Give few examples of dimensional formula.

Answer 1

1. Dimensions:

The dimensions of a physical quantity are the powers to which the fundamental units must be raised in order to obtain the unit of a given physical quantity.

2. Dimensional formula:

When any derived quantity is represented with appropriate powers of symbols of the fundamental quantities, SUCH an expression is called dimensional formula. It is expressed by square bracket with no comma in between the symbols.

3. Examples of dimensional formula:

a. Speed = \(\frac{Distance}{time}\)

∴ Dimensions of speed = \(\frac{[L]}{[T]}\) = [L¹M⁰T^-1]

[Note: As power of M is zero, it can be omitted from dimensional formula. Therefore, dimensions of speed can be written as [L¹T¹]

b. Force = Mass x acceleration = Mass x \(\frac{Distance}{(time)^2}\)

∴ Dimensions of force = \([M]\times\frac{[L]}{[T^2]}\) = [L¹ M¹ T²]

c. Kinetic energy = \(\frac{1}{2}\)mv² = \(\frac{1}{2}\) m\((\frac{Distance}{Time})^2\)

∴ Dimensions of kinetic energy = \([M^1]\times\frac{[L^2]}{T^2}\) = [L¹ M¹ T²]

d. Temperature gradient = \(\frac{Temperature}{Distance}\)

∴ Dimension of temperature gradient = \(\frac{[K]}{[L]}\) = [L^-1 M⁰ T⁰ K¹]