Question

Derive the dimensional formula for pressure.

Answer 1

Hint: Dimensional formula of a given physical quantity is defined as the expression which shows how and which of the fundamental quantities represent the dimensions of a physical quantity.

Complete answer:
We know that pressure is defined as the force which is applied perpendicular to the surface of an object or a body per unit area over which that force is distributed. In other words, it can also be defined as the ratio of the force to the area over which the force is acting. Pressure is expressed in the unit pascal (Pa).
Pressure = Force $ \times Are{a^{ - 1}}$ - (1)
As we know that, force is given by the formula,
Force = Mass $ \times $ Acceleration
Where ‘F’ is the force, ‘m’ is the mass of the object or the particle and ‘a’ is the acceleration.
And mathematically acceleration is given as,
Acceleration = Velocity $ \times tim{e^{ - 1}} = \left[ {L{T^{ - 1}}} \right]{\left[ T \right]^{ - 1}} = \left[ {L{T^{ - 2}}} \right]$
So, dimensional formula of force is given as,
Force = $\left[ M \right]\left[ {L{T^{ - 2}}} \right] = ML{T^{ - 2}}$ - (2)
Dimensional formula of area is given as,
Area = ${M^0}{L^2}{T^0}$ - (3)
Substitute equation (2) and (3) in equation (1) we get,
Pressure (P) = Force $ \times Are{a^{ - 1}}$
P = $[ML{T^{ - 2}}]{[{M^0}{L^2}{T^0}]^{ - 1}}$
$\therefore P = M{L^{ - 1}}{T^{ - 2}}$

Hence, the dimensional formula of pressure (P) is $M{L^{ - 1}}{T^{ - 2}}$.

Additional Information:
There are various kinds of pressure such as, atmospheric pressure, absolute pressure, gauge pressure and differential pressure. The simplest difference between the absolute pressure and gauge pressure is that absolute pressure uses absolute zero as its zero point, whereas gauge pressure uses atmospheric pressure as its zero point.

Note:
The equation which is obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the given physical quantity. The physical constant quantities which have no dimensions are called dimensionless constants.