Question

What is the dimensional formula of pressure difference?

Answer 1

Step 1: Definition

The stress at such a position within such a confined fluid, or rather the perpendicular force per unit area is called pressure.

Step 2: Formula:

$\mathrm{P}=\frac{\mathrm{F}}{\mathrm{A}}$

Here,

F is the force.

A is the acceleration.

Step 3: Derivation of dimensional formula

Since,

$\mathrm{Pressure}=\mathrm{Force}\times {\mathrm{Area}}^{-1}...\left(1\right)$

$\mathrm{Force}=\mathrm{Mass}\times \mathrm{Acceleration}$

And,

$$\begin{gathered} \mathrm{Acceleration}=\mathrm{Velocity}\times {\mathrm{time}}^{-1} \\ =[\mathrm{L}{\mathrm{T}}^{-1}]\left[{\mathrm{T}}^{-1}\right] \\ =[\mathrm{L}{\mathrm{T}}^{-2}] \end{gathered}$$

∴ The dimensional formula of force $$\begin{gathered} =\left[\mathrm{M}\right]\times [\mathrm{L}{\mathrm{T}}^{-2}] \\ ={\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-2}...\left(2\right) \end{gathered}$$

The dimensional formula of area $={\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^0...\left(3\right)$

By substituting equations (2) and (3) in equation (1) we get,

Pressure, $\mathrm{P}=[{\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-2}]\times {\left[{\mathrm{L}}^2\right]}^{-1}$

$=\left[{\mathrm{M}}^1{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\right]$

Here,

M is the mass, L is the length, and T is the time.

Therefore, the dimensional formula of pressure difference is $\left[{\mathrm{M}}^1{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\right]$.