Determining Pressure Dimensions with Fundamental Forces

This solution explains how to find the dimensions of pressure using force $[F]$, acceleration $[A]$, and time $[T]$ as the fundamental physical quantities. We will break down the problem step-by-step, expressing pressure in terms of these chosen units.

Defining Pressure

First, let's recall the definition of pressure. Pressure is defined as the force acting perpendicularly on a unit area of a surface. The formula for pressure ($P$) is:

$ P = \frac{F}{Area} $

Here, $F$ represents force and $Area$ represents the area over which the force is applied.

Fundamental Quantities Analysis

We are given that the fundamental physical quantities are force $[F]$, acceleration $[A]$, and time $[T]$. Our goal is to express the dimensions of pressure solely in terms of these quantities.

We need to find the dimensions of Area using $[F]$, $[A]$, and $[T]$. Let's relate these quantities:

• The relationship between force, mass ($M$), and acceleration is $F = ma$. This means the dimension of mass can be expressed as:

  $ [M] = \frac{[F]}{[A]} $

• Acceleration is defined as the rate of change of velocity, and velocity is the rate of change of displacement ($L$). So, $A = \frac{L}{T^2}$. This allows us to express the dimension of length as:

  $ [L] = [A] [T^2] $

Expressing Area using F, A, T

Since Area is the square of length ($Area = L^2$), we can find its dimensions using the expression for $[L]$ derived above:

$ [Area] = [L^2] = ([A] [T^2])^2 $

Simplifying this, we get the dimensions of area in terms of acceleration and time:

$ [Area] = [A^2 T^4] $

Calculating Pressure Dimensions

Now we can substitute the dimensions of Area back into the formula for pressure ($P = \frac{F}{Area}$):

$ [P] = \frac{[F]}{[Area]} $

Substituting the dimensions of Area we found:

$ [P] = \frac{[F]}{[A^2 T^4]} $

To express this using standard dimensional notation with negative exponents, we get:

$ [P] = [F A^{-2} T^{-4}] $

Final Result Verification

By choosing force $[F]$, acceleration $[A]$, and time $[T]$ as fundamental quantities, we have derived the dimensions of pressure. Comparing our result with the given options:

• Option 1: $[F A^{-2} T^{-2}]$
• Option 2: $[F A^{-1} T^{-3}]$
• Option 3: $[F A^{-2} T^{-4}]$
• Option 4: $[F A^{-1} T^{-4}]$

Our calculated dimensions, $[F A^{-2} T^{-4}]$, match one of the provided options.