Dimensions of Physical Quantities

What are Dimensions?

Dimensions describe the nature of a physical quantity in terms of base quantities. Every physical quantity can be expressed as a combination of the seven fundamental quantities:

• Length $[L]$
• Mass $[M]$
• Time $[T]$
• Electric current $[A]$
• Temperature $[K]$
• Luminous intensity $[cd]$
• Amount of substance $[mol]$

Dimensional Representation

The representation of a physical quantity using powers of fundamental quantities is called its dimensional formula.

Example:

• Speed = length/time → $[M^0 L^1 T^{-1}]$
• Acceleration = velocity/time → $[M^0 L^1 T^{-2}]$
• Force = mass × acceleration → $[M^1 L^1 T^{-2}]$

Interpreting Dimensions

If a quantity has:

• No mass involved → exponent of $M$ is 0
• No time involved → exponent of $T$ is 0

Example:

• Volume = $[L^3]$
• Pressure = force/area → $[M^1 L^{-1} T^{-2}]$

Key Points:

• Dimensions do not include numerical constants (like $\pi$, 1/2).
• Dimensionless quantities like angle, strain, refractive index are written as $[M^0 L^0 T^0]$.
• Only quantities with the same dimensions can be added/subtracted.

Dimensions help analyze, derive, and check the validity of physical equations.

Example:

Find the dimensions of: (a) Kinetic energy $= \frac{1}{2}mv^2$ (b) Work $= F \times d$

Solution: (a) $E = \frac{1}{2}mv^2$ → $[M][L^2 T^{-2}] = [M L^2 T^{-2}]$ (b) $F = [M L T^{-2}],\ d = [L]$ → Work = $[M L T^{-2}][L] = [M L^2 T^{-2}]$