Question

Write the dimensional formula of pressure.

Answer 1

Hint : To solve this question we need to use the basic formula of the pressure. Then we need to write the dimensions of each of the quantities present in the formula to get the final answer.

Formula used: The formulae used in solving this question are given by
$P = \dfrac{F}{A}$ , here $P$ is the pressure on a surface due to a force of $F$ perpendicular to the surface of area $A$ .
$F = ma$ , here $F$ is the force acting on a particle of mass $m$ which is moving with acceleration $a$ .

Complete step by step answer
We know that the pressure is defined as the force acting normal to a surface per unit its surface area, that is,
$P = \dfrac{F}{A}$
Now, for obtaining the dimensional formula of pressure, we have to write the dimensional formulae of the force and the area. So we have
$\left[ P \right] = \dfrac{{\left[ F \right]}}{{\left[ A \right]}}$ ………………..(1)
We know that force on a particle is related with the acceleration of the particle by the Newton’s second law of motion as
$F = ma$
So the dimensional formula of the force can be written as
$\left[ F \right] = \left[ m \right]\left[ a \right]$
$\Rightarrow \left[ F \right] = \left[ {{M^1}{L^0}{T^0}} \right]\left[ {{M^0}{L^1}{T^{ - 2}}} \right]$
On simplifying we get
$\Rightarrow \left[ F \right] = \left[ {{M^1}{L^1}{T^{ - 2}}} \right]$ ………………..(2)
Now, we know that the dimensional formula of area is given by
$\left[ a \right] = \left[ {{M^0}{L^2}{T^0}} \right]$ ………………..(3)
Substituting (2) and (3) in (1) we get
$\left[ P \right] = \dfrac{{\left[ {{M^1}{L^1}{T^{ - 2}}} \right]}}{{\left[ {{M^0}{L^2}{T^0}} \right]}}$
$\Rightarrow \left[ P \right] = \left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]$
Hence, the dimensional formula of the pressure is $\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]$ .

Note
For solving such types of questions, it is required to obtain each physical quantity given in the problem in terms of the fundamental or base quantities. So we must know the physical formula which can relate the quantity with the fundamental quantities. It must be noted that there can be multiple such formulas and any one of them can be used for this purpose.