Question

A dimensionless quantity:
A) Never has a unit
B) Always has a unit
C) May have a unit
D) Does not exist

Answer 1

Hint: A physical quantity not having dimensions or say a dimensionless quantity is which cannot be expressed in terms of physical quantities. We will explain how units and dimensions are related and how we can determine the units of the physical quantity from its dimension.

Complete answer:
Let’s start with the dimension of a physical quantity for which we will need to address the fundamental quantities. A fundamental quantity belongs to a class of physical quantities of its own kind only, each of them having no dependence with other fundamental quantities. These quantities are used to determine the dimensions of all other physical quantities. The fundamental quantities are, mass $$\left[M\right]$$, length $\left[L\right]$, time $\left[T\right]$, current $\left[I\right]$, luminous intensity $\left[cd\right]$, temperature $\left[K\right]$ and amount of substance $\left[mol\right]$. These quantities are used to determine the dimensions of other physical quantities. How do we calculate these dimensions? Well, dimensions of a physical quantity are the powers to which the fundamental units are raised in order to represent that quantity. Let’s take velocity for an example. We know that velocity is the rate of displacement which is mathematically represented as $v={\displaystyle \frac{s}{t}}$, where $s$ and $t$ are total displacement and total time for that displacement respectively. The dimensions for velocity will be the dimensions of displacement (length) over dimensions of time. So the dimensions of velocity are ${\displaystyle \frac{\left[L\right]}{\left[T\right]}}$ or $\left[L{T}^{-1}\right]$. Does that ring any rings? We know that S.I. unit of velocity is $m{s}^{-1}$ or length per time. This is exactly how we can use dimensions to determine the unit of a physical quantity. Another example, the dimensions for force are $\left[M^1{L}^1{T}^{-2}\right]$ (Calculate this yourself). So the S.I. unit of force will be $kg.m.s^{-1}$ or $Newton$. Dimensionless quantities on the other hand are the quantities which do not possess any dimensions. Example of such a quantity is, specific gravity (ratio of absolute density of an object to absolute density of water). As these quantities are often a pure ratio between similar physical quantities they do not possess dimension and hence no units. But when we look at the dimensionless quantity- angle, it has a unit assigned to it, the radians or degrees. So we cannot say that all dimensionless quantities are unit-less.
Thus the correct answer is option C.

Note: Notice that when we write dimensions, we always put them inside square brackets. We can use dimensional analysis to predict the unit of a physical quantity having dimensions or to check the correctness of a given relation. Mostly dimensionless quantities are rations of two quantities with the same dimensions. The best example of it is strain. Strain is the ratio of change of length to the original length.