Question

Derive the dimensional formula of pressure.

Answer 1

Hint: Study about how pressure is applied on a body. Obtain the expression for pressure and try to define how pressure is dependent on which quantity. Express pressure in terms of the fundamental quantities. Then find out the dimension of each quantity. Finally, we will get the dimension of pressure.

Complete Step-by-Step solution:
Pressure on an object can be defined as the physical force exerted on the object. Pressure can be defined as the force applied perpendicular to the surface of an object per unit area of that surface. Pressure can be mathematically expressed as,
$\text{pressure = }\dfrac{\text{force applied}}{\text{area of contact}}$
$\text{or, P=}\dfrac{F}{A}$
Where p is the pressure on the object, F is the force applied on the object and A is the area of contact of the force and the surface of the object.
To find the dimension of pressure first we need to find the dimension of force.
So, we can express force as,
$F=m\times a$
Where, F is the force, M is the mass of the object and a is the acceleration of the object.
Now again,
Dimension of acceleration is, $\left[ {{M}^{0}}{{L}^{1}}{{T}^{-2}} \right]$
So, dimension of force will be, $\left[ {{M}^{1}}{{L}^{0}}{{T}^{0}} \right]\times \left[ {{M}^{0}}{{L}^{1}}{{T}^{-2}} \right]=\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right]$
Dimension of area is, $\left[ {{M}^{0}}{{L}^{2}}{{T}^{0}} \right]$
So, dimension of pressure will be, $\dfrac{\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right]}{\left[ {{M}^{0}}{{L}^{2}}{{T}^{0}} \right]}=\left[ {{M}^{1}}{{L}^{-1}}{{T}^{-2}} \right]$
This is the dimension of pressure.

Note: Point forces are independent of area. When we measure pressure on an object we take the perpendicular component of the force to the surface of the body. Pressure is scalar, so it does not have any direction.
Again, to obtain the dimensional formula of any quantity we should not try to directly find the dimension. First, we should break the quantity into its fundamental quantities.