Question

Which of the following physical quantities has a unit but no dimensions?
A. Relative velocity
B. Relative density
C. Angle
D. Energy

Answer 1

Hint: This question is a theoretical one, based on the concept of the units and the dimensional formulae of the physical quantities. We will define all the terms given in the options, their units, and their dimensions, in order to find the physical quantity that has a unit but no dimensions.

Complete step by step solution:
We will define all the terms given in the options, their units, and their dimensions.
The relative velocity – the velocity of an object with respect to the other. The expression for the same is given as \[V=|{{V}_{1}}-{{V}_{2}}|\]. The unit of the relative velocity is the same as that of the velocity. Thus, the unit of the relative velocity is\[m{{s}^{-1}}\]. The dimensional formula of the relative velocity is\[[{{\text{M}}^{0}}{{\text{L}}^{1}}{{\text{T}}^{-1}}]\].
The relative density – the ratio of the density of the substance to the density of the water in case of the solid or liquid AND the ratio of the density of the substance to the density of the air in case of the gas. As this term is a ratio, thus, the unit gets canceled out. Therefore, the relative density will even be dimensionless.
Angle – An angle is formed by the two rays, called the sides of the angle, sharing a common endpoint. The units of the angle are steradian and radian. But, the angle has no dimensional formula.
Energy – the ability to do work is called energy. The unit of the energy is Joule and the dimensional formula of the energy is\[[{{\text{M}}^{1}}{{\text{L}}^{2}}{{\text{T}}^{-2}}]\].
$\therefore$ The physical quantity, the angle has the units radian and steradian, but no dimensions.
As the angle has units but no dimension, thus, option (C) is correct.

Note: To compute the dimensional formula using the units of the parameters should be known to solve these types of problems. The ratios will have no unit, as the units of the numerator and the denominator gets canceled out. Some parameters will have no unit, even though they will have dimensions, such as the angle.