Question

Give a reason as to why relative density has no unit.
A) Because it is imaginary
B) Because it is a ratio of two-dimensionally same quantities
C) Because it is a ratio of two-dimensionally different quantities
D) None of these

Answer 1

Hint: When the term relative attached to a quantity it simply means it is compared to some other quantity for the sake of universalization. The density of a body is the amount present per unit volume of the body.

Complete solution:
Here it is asked why relative density doesn’t have a unit?
When the word relative is attached to a quantity it simply means that it is measured against the standard quantity of its dimension.
We know that the density of a body is the amount of mass present per unit volume of that body
$\rho = \dfrac{{mass}}{{volume}}$
The unit of density is $kg{m^{ - 3}}$
Now we have got two fluids to say one is oil and one is the water we know that both of them have some densities
Say density of the oil is $800kg{m^{ - 3}}$ and density of water is universally $1000kg{m^{ - 3}}$
Now if it is asked what is the relative density of oil with respect to water
This statement simply means treating the density of water as a reference to what will be the density of oil
So, to find this we divide the density of oil buy water
\[\Rightarrow {\rho _r} = \dfrac{{{\rho _{oil}}}}{{{\rho _{water}}}}\]
$\Rightarrow {\rho _r} = \dfrac{{800kg{m^{ - 3}}}}{{1000kg{m^{ - 3}}}}$
$\Rightarrow {\rho _r} = 0.8$
Here as we can see that the relative density $\left( {{\rho _r}} \right)$ has no units because when we divide the density of oil with the density of water their units got cancelled because they are the quantity of the same dimension i.e $kg{m^{ - 3}}$

Final answer is (B), the relative density does not have a unit because it is a ratio of two-dimensionally same quantities.

Note: When we divide two-dimensionally the same quantity they get cancelled and their ratio is dimensionless. Specific density is the density of a fluid with respect to some standard fluid (water). The relative density concept is highly used in the pressure measurement of the fluids.