Question: The dimensional formula of relative density is: [PMT/NEET-2003]
a. $ML^{-3}$
b. $LT^{-1}$
c. $MLT^{-2}$
d. Dimensionless

The correct answer is (d) Dimensionless.

Relative density (also known as specific gravity) is a ratio quantity, making it dimensionless.

Definition and Analysis

Relative density = $\frac{\text{Density of substance}}{\text{Density of reference substance (water)}}$

Dimensional Analysis

• Density = $\frac{\text{Mass}}{\text{Volume}} = \frac{[M]}{[L^3]} = [ML^{-3}]$
• Relative density = $\frac{[ML^{-3}]}{[ML^{-3}]} = [M^0L^0T^0]$

Result: The dimensions cancel out completely, making it dimensionless.

Physical Significance

Relative density tells us how many times denser (or lighter) a substance is compared to water:

• Relative density > 1: Substance is denser than water (sinks)
• Relative density < 1: Substance is less dense than water (floats)
• Relative density = 1: Same density as water

Examples for Better Understanding

• Mercury: Relative density ≈ 13.6 (no units)
• Ice: Relative density ≈ 0.92 (no units)
• Gold: Relative density ≈ 19.3 (no units)

Key Point for NEET

Remember that whenever you have a ratio of two quantities with identical dimensions, the result is always dimensionless. This concept appears frequently in physics with quantities like:

• Relative density
• Refractive index
• Coefficient of friction
• Strain

The absence of units makes these quantities universal – they have the same numerical value regardless of the unit system used.