Question

Give an example of dimensionless quantity but have a unit.

Answer 1

To find the quantity that has no dimension but has a unit, we need to first understand the significance of dimensions and what it means. Also, we need to know what a unit is.

Complete answer:
Dimension of a quantity in physics means the minimum number of powers to which the fundamental units must be raised to obtain one unit of that quantity. If we take that ${{M}^{a}}{{L}^{b}}{{T}^{c}}$ to be the derived dimensional formula of some quantity then $a$, $b$ and $c$ is the exponent and is the dimension of that quantity.

Unit on the other hand is something like a constant magnitude which is used to measure the magnitudes of other quantities of the same nature. If we take $x$ to be a unit of some quantity then we use this unit to measure the same quantity of some other material and compare their properties and many other things that come with it.

So in this solution, we are to list some quantities that do not have the exponent to the dimensional formula or have a dimension of $1$ but do have the unit. Some of them are angular momentum, which has a unit of radian but a dimension of $1$. The next is Atomic mass, which has no dimension but has a unit of $kg$. Solid angle is also an example of such quantity.

Note: The dimensional analysis method cannot be applicable in the trigonometric and logarithmic functions. Similarly, there are many other limitations to both dimensions and units of a quantity.