Question

The physical quantity which has the dimensional formula $\left[ M{{T}^{-3}} \right]$ is:
A. surface tension
B. Density
C. Solar constant
D. Compressibility

Answer 1

Hint: Find using the formulas of each of the options, find their dimensional formula and compare with $\left[ M{{T}^{-3}} \right]$. Use the formulas, $\sigma =\dfrac{F}{L}$ for surface, $density=\dfrac{mass}{volume}$, $\text{solar constant = }\dfrac{\text{energy}}{\text{time }\!\!\times\!\!\text{ area}}$ and $\beta =-\dfrac{1}{V}\dfrac{dV}{dP}$ for compressibility.

Formula used:
$\sigma =\dfrac{F}{L}$
$density=\dfrac{mass}{volume}$
$\text{solar constant = }\dfrac{\text{energy}}{\text{time }\!\!\times\!\!\text{ area}}$
$\beta =-\dfrac{1}{V}\dfrac{dV}{dP}$

Complete step by step answer:
Let us discuss each of the options to find the dimensional formula one by one and check which one of them is equal to $\left[ M{{T}^{-3}} \right]$.
Option A: In fluids the fluid molecules are together due to the forces of attraction between them. Therefore, the inner molecules exert a force of attraction on the molecules that are on the surface of the fluid. The force experienced by the surface molecules per unit length is called the surface tension of the fluid.
Therefore surface tension $\sigma =\dfrac{F}{L}$. Let the dimensional formula of surface tension be $\left[ \sigma \right]$.
Therefore, $\left[ \sigma \right]=\left[ \dfrac{F}{L} \right]$
Dimensional formula of force is $\left[ F \right]=\left[ ML{{T}^{-2}} \right]$ and the dimensional formula of length is [L].
Hence, $\left[ \sigma \right]=\left[ \dfrac{F}{L} \right]=\dfrac{\left[ ML{{T}^{-2}} \right]}{\left[ L \right]}=\left[ M{{T}^{-2}} \right]$
And $\left[ M{{T}^{-2}} \right]\ne \left[ M{{T}^{-3}} \right]$.
Option B : Density is defined as the mass of the substance in one unit of volume.
Therefore, $density=\dfrac{mass}{volume}$.
Dimensional formula of mass is [M].
Dimensional formula of volume is $\left[ {{L}^{3}} \right]$.
Therefore, dimensional formula of density is $\dfrac{\left[ M \right]}{\left[ {{L}^{3}} \right]}=\left[ M{{L}^{-3}} \right]$.
This gives that $\left[ M{{L}^{-3}} \right]\ne \left[ M{{T}^{-3}} \right]$.
Option C : Solar constant is the amount of radiations entering earth’s atmosphere per unit area in one unit time. This means that $\text{solar constant = }\dfrac{\text{energy}}{\text{time }\!\!\times\!\!\text{ area}}$.
The dimensional formula of energy is $\left[ M{{L}^{2}}{{T}^{-2}} \right]$
The dimensional formulas of area and time are $\left[ {{L}^{2}} \right]$ and [T] respectively.
Therefore, the dimensional formula of the solar constant is $\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ {{L}^{2}} \right]\left[ T \right]}=\left[ M{{T}^{-3}} \right]$
Hence, the correct option is C.

Note: Compressibility is defined as the relative change in volume of a fluid or rigid body per unit pressure applied on it. It tells us how much a body can compress when a pressure is applied on it. It is given as $\beta =-\dfrac{1}{V}\dfrac{dV}{dP}$.
The dimensional formula of compressibility is equal to the inverse of the dimensional formula of pressure. i.e. $\left[ {{M}^{-1}}{{L}^{1}}{{T}^{2}} \right]$.