# Dimensional Analysis

## Introduction

Most physical quantities can be expressed in terms of five basic dimensions, listed below with their SI units:

• Mass, M, kg
• Length, L, m
• Time, T, s
• Electric Current, I, A
• Temperature, θ, K

For example, acceleration is length per unit time per unit time or LT-2.

## Dimensional Analysis

Only quantities with like dimensions may be added(+), subtracted(-) or compared (=,<,>). This rule provides a powerful tool for checking whether or not equations are dimensionally consistent. It is also possible to use dimensional analysis to suggest plausible equations when we know which quantities are involved.

## Example of checking for dimensional consistency

Consider one of the equations of constant acceleration,

s = ut + 1/2 at2. (1)

The equation contains three terms: s, ut and 1/2at2. All three terms must have the same dimensions.

• s: displacement = a unit of length, L
• ut: velocity x time = LT-1 x T = L
• 1/2at2 = acceleration x time = LT-2 x T2 = L

All three terms have units of length and hence this equation is dimensionally valid. Of course this does not tell us if the equation is physically correct, nor does it tell us whether the constant 1/2 is correct or not.

## Example of generating equations

We want to know how the speed of waves, v, on a string depends its mass m, length l, and tension Q? We can solve this problem using dimensional analysis.

First work out the dimensions of all the terms:

• Speed, v : LT-1
• Mass, m : M
• Length, l : L
• Tension, Q : A force = mass x acceleration: MLT-2

Our equation is going to take the form v = malbQc where the power constants a,b and c are unknown.

Re-writing our equation using dimensions: (LT-1) = (Ma) (Lb) (Mc Lc T-2c)

To be dimensionally consistent, each dimension must appear to the same power on each side. Hence:

• For L: 1 = b + c
• For M: 0 = a + c
• For T: -1 = -2c

Solving these equations, we get: a = -1/2, b=1/2 and c = 1/2.

Hence, v = k m-1/2 l1/2 Q1/2 where k is an arbitrary constant.

## Introduction

Most physical quantities can be expressed in terms of five basic dimensions, listed below with their SI units:

• Mass, M, kg
• Length, L, m
• Time, T, s
• Electric Current, I, A
• Temperature, θ, K

For example, acceleration is length per unit time per unit time or LT-2.