Question

A physical quantity is measured and the result is expressed as nu where u is the unit used and n is the numerical value. If the result is expressed in various units then,
A) $n \propto {\text{size of }}u$
B) $n \propto {u^2}$
C) $n \propto \sqrt u $
D) $n \propto \dfrac{1}{u}$

Answer 1

Hint: Any physical quantity (P) can be expressed as numerical value(n) along with units(u) as $P = {\text{nu}}$. For example, length can be written as L=10m, here L is physical quantity i.e. length and 10 is the numerical value along with meters m as its unit.

Complete step by step solution:
We know that any physical quantity can be a representation as $P = {\text{nu}}$, where $n$ is the numerical value and $u$ is its unit.
Let us take an example of mass, suppose if a body has a mass of \[{\mathbf{10}}{\text{ }}{\mathbf{kg}}\]. In this \[{\mathbf{10}}\] is the numerical value (n) and $kg$ is the unit (u).

Case 1:
Converting these 10 kgs to gms (unit value is decreasing ) which gives 10000 gms (because we know that 1 kg = 1000 gms). So here we get n=10000 and u is gms.
If you observe here, as we decrease the value of the unit, the numerical value is increasing

Case 2:
Converting 10 kgs to quintal, the unit value is increasing from kgs to quintal which gives 0.1 quintals (because we know that 1 kg = 0.01 quintal). Thus in this case n = 0.1 and u is quintal. So if we observe here, as we increase the value of the unit, the numerical value is decreasing
From these two cases we can say that n is inversely proportional to u which can be shown as:
\[ \Rightarrow {\mathbf{n}}\; \propto \;\dfrac{1}{u}\]

Thus option D is correct.

Note:
When we are converting the physical quantity from one unit to another unit, only numerical value changes but the actual dimension of the physical quantity remains the same. For example in 1 minute = 60 seconds, we can say that the value of time here is either 1 min or 60 seconds as both are the same, but only the representation changes. The dimension of time remains the same and it is $\left[ T \right]$.