Question

The dimensional formula for relative density is :
$A. [ML^{-3}]$
$B. [M^0L^{-3}]$
$C. [M^0L^0T^{-1}]$
$D. [M^0L^0T^0]$

Answer 1

Hint: Write the definition of relative density. Then we will form a mathematical expression for the relative density. Therefore, we will determine its dimensional formula. We will also have to know about dimensionless quantities.

Complete step by step solution:
Relative density is defined by the ratio of the density of a substance at any temperature to the density of pure water at 4° C. Hence, relative density is the ratio of two similar quantities. Let’s take the density of a substance at temperature t°C to be $\rho$ and also let $\rho_w$ be the density of water at the temperature 4°C. Therefore, the relative density of that particular substance at the temperature t°C is given by,
$D=\dfrac{\rho}{\rho_w}$
Now, since both the denominator and the numerator are density, they will have the same dimensional formulas. For both of $\rho$ and $\rho_w$, the dimensional formula is $ML^{-3}$.
Hence, the dimensional formula for relative density is given by
$[D]=\dfrac{M{{L}^{-3}}}{M{{L}^{-3}}}={{M}^{0}}{{L}^{0}}{{T}^{0}}$
Hence, option D is the correct answer.
Additional information: Apart from the main three dimensions M, L and T, there are also four fundamental dimensions. They are,
[Temperature]=$\Theta$ , [Amount of matter]=N, [Current]=A, [Luminous intensity]=J .
The physical quantities that are the ratios of two similar types of quantities, are called dimensionless quantities. Their dimension is basically unity. A few examples of such dimensionless quantities are Angle, Dielectric constant, relative permeability, refractive index of a medium etc.
It’s important to mention 4°C in the definition of relative density, because water has its density maximum at 4°C.

Note: Keep in mind the following things,
1. If the dimensional formula is $[M^0L^0T^0]$ , it means that actually the dimension is unity. The answer could be $[M^0L^0T^0A^0]$ as well.
2. When a physical quantity is written inside [], it generally means the dimensional formula of that particular quantity.
3. Being dimensionless does not imply that the quantity has to be unitless. For example, Angle is a dimensionless quantity but it has units like degrees, radians etc.