# 1. Introduction

It is important to realise that it only makes sense to add the same sort of quantities, e.g. area may be added to area but area may not be added to temperature! These considerations lead to a powerful method to analyse scientific equations called dimensional analysis. One should note that while units are arbitrarily chosen (an alien civilisation will not use seconds or weeks), dimensions represent fundamental quantities such as time.

Basic dimensions are written as follows:

Dimension Symbol
Length L
Time T
Mass M
Temperature K
Electrical current I

See the module on Units for a review of SI units.

Example 1: An area can be expressed as a length times a length. Therefore the dimensions of area are L × L = L2, which we can express as: [area] = L2.

A given area could be expressed in the SI units of square metres, or indeed in any appropriate units.

In some equations symbols appear which do not have any associated dimension, e.g., in the formula for the area of a circle, πr2, π is just a number and does not have a dimension.

Exercise 1: Calculate the dimensions of the following quantities:

(a)Volume

Solution:

A volume is given by multiplying three lengths together:
 Dimension of volume = L × L × L L3

So [volume] = L3

(b)Speed

Solution:

Speed is the rate of change of distance with respect to time:
 Dimension of speed = L/T LT−1

So [speed] = LT−1 (The SI units of speed are metres per second.)

(c)Acceleration

Solution:

Acceleration is the rate of change of speed with respect to time:
 Dimension of acceleration = LT−1/T LT−2

So [acceleration] = LT−2 (The SI units of acceleration are metres per second per second)

(d)Density

Solution:

Density is the mass per unit volume, so using the dimension of volume we get:
 Dimension of density = ML−3

So [density] = M−3 (The SI units of density are kg m−3)

Click on questions to reveal solutions

Quiz 1: Pick out the units that have a different dimension to the other three:

(a)kg m2 s−2 Incorrect - please try again!
(b)g mm2 s−2 Incorrect - please try again!
(c)kg2 m s−2 Correct - well done!
(d)mg cm2 s−2 Incorrect - please try again!

Explanation: kg2 m s−2 has dimensions = M2LT−2

It can be checked that all the other answers have dimension ML2T−2

# 2. Checking Equations

Example 2: Consider the equation:

y = x + ½kx3

Since any terms which are added together or subtracted MUST have the same dimensions, in this case y, x and ½kx3 have to have the same dimensions. We say that such a scientific equation is dimensionally correct. However, if it is not true, then the equation must be wrong.

If in the above equation x and y are both lengths (dimension L) and ½ is a dimensionless number, then for the kx3 term to have the same dimension as the other two, we would need:

dimension of k × L3 ∴ dimension of k = L/L3 = L = L−2

So k would have dimensions of one over area, i.e., [k] = L−2

Quiz 2: Hooke’s law states that the force, F, in a spring extended by a length x is given by F = − kx. From Newton’s second law F = ma, where m is the mass and a is the acceleration, choose the dimension of the spring constant k:

(a)MT−2 Correct - well done!
(b)MT2 Incorrect - please try again!
(c)ML−2T−2 Incorrect - please try again!
(d)ML2T2 Incorrect - please try again!

Explanation: From Hooke’s law, F = −kx, we see that we can write:

k = F/x

Now F = ma, so the dimensions of force are given by:

 [F] = M × (LT−2) = MLT−2

Therefore the spring constant has dimensions:

 [k] = MLT−2/L = MT−2

Example 3: The expressions for kinetic energy E = ½mv2 (where m is the mass of the body and v is its speed) and potential energy E = mgh (where g is the acceleration due to gravity and h is the height of the body) look very different, although both describe energy.

One way to see this is to note that they have the same dimension:

 ½ = mv2 Dimension of kinetic energy ⇒ M (LT−1)2 = ML2T−2
 mgh Dimension of potential energy ⇒ M (LT−2)2 L = ML2T−2

Both expressions have the same dimensions, so they can therefore be added and subtracted from each other.

Exercise 2: Check that the dimensions of each side of the equations below agree:

(a)The volume V of a cylinder of radius r and height h: V = πr2h.

Solution: A volume is given by multiplying three lengths together:

 Dimension of volume = L × L × L = L3

So [volume] = L3

(b)v = u + at for an object with initial speed u, (constant) acceleration a and final speed v after a time t.

Solution: We want to check the dimensions of the equation v = u + at. Since v and u are both speeds, they have dimensions LT−1. Therefore we only need to verify that at has this dimension. To see this consider:

[at] = (LT−2 × T = LT−2 × T = LT−1

So the equation is dimensionally correct because all the terms have dimensions of speed.

(c)E = mc2, where E is energy, m is mass and c is the speed of light.

Solution: We want to check the dimensions of the equation E = mc2. Since E is an energy it has dimensions ML2T−2. The right hand side of the equation can also be seen to have this dimension, if we recall that m is a mass and c is the speed of light with dimension LT−1. Therefore:

 [mc2] = M(LT−1)2 = ML2T−2

So the equation is dimensionally correct as all terms have dimensions of energy.

(d)c = λν, where c is the speed of light, λ is the wavelength and ν is the frequency.

Solution: We want to check the dimensions of the equation c = λν. Since c is the speed of light it has dimensions LT−1. The right hand side of the equation involves wavelength [λ] = L and frequency [λ] = T−1. We thus have:

[λν] = L × T−1

- which indeed also has dimensions of speed, so the equation is dimensionally correct.

Click on questions to reveal solutions

Note: dimensional analysis is a way of checking that equations might be true: it does not prove that they are definitely correct. Dimensional analysis would suggest that both Einstein’s equation E = mc2 and the (incorrect) equation E = ½mc2 might be true. On the other hand, dimensional analysis shows that E = mc3 makes no sense.

# 2. Dimensionless Quantities

Some quantities are said to be dimensionless. These are pure numbers which would be the same no matter what units were used (e.g., the mass of a proton is roughly 1850 times the mass of an electron, regardless of how you measure mass).

Example 4: The ratio of one mass m1 to another mass m2 is dimensionless:

dimension of the fraction m1/ m2 = M/ M = 1

The dimensions have canceled and the result is a number (which is independent of the units, i.e., it would be the same whether the masses were measured in kilograms or tonnes).

Note: angles are defined in terms of ratios of lengths. They are therefore dimensionless! Functions of dimensionless variables are themselves dimensionless (????).

Very many functions are dimensionless. The following quantities are important cases of dimensionless quantities:

Trigonometric Functions Logarithms
Exponentials Numbers, e.g. π

Note the following properties:

• Functions of dimensionless variables are dimensionless. (????)
• Dimensionless functions must have dimensionless arguments. (????)

Quiz 3: If the number of radioactive atoms is found to be given as a function of time t by:

N(t) = N0 exp (−kt)

- where N0 is the number of atoms at time t = 0, what is the dimension of k?

(a)LT Incorrect - please try again!
(b)log (T) Incorrect - please try again!
(c)T Incorrect - please try again!
(d)T−1 Correct - well done!

Explanation: In N(t) = N0 exp (−kt) , the exponential and its argument must be dimensionless. Therefore kt has to be dimensionless. Thus:

dimensions of k × T = 1

So the dimension of k must be inverse time, i.e., [k] = T−1.

Exercise 3: Determine the dimensions of the expressions below:

(a)In a Young’s slits experiment the angle θ of constructive interference is related to the wavelength λ of the light, the spacing of the slits d and the order number n by d sin (θ) = . Show that this is dimensionally correct.

Solution: We want to check the dimensions of d sin(θ) = . Both d and λ have dimensions of length. The angle θ and sin (θ), as well as the number n, must all be dimensionless. Therefore we have:

L × 1 ∴ [d sin (θ)] = 1 × L = [nλ]
(b)The Boltzmann distribution in thermodynamics involves the factor exp (−E/(kT)) where E represents energy, T is the temperature and k is Boltzmann’s constant. Find the dimensions of k.

Solution: The factor exp (−E ⁄ (kT)) is an exponential and so must be dimensionless. Therefore its argument E ⁄ (kT) must also be dimensionless.

The minus sign simply corresponds to multiplying by minus one and is dimensionless. Energy, E has dimensions [E] = ML2T−2 and temperature has dimensions K, so the dimensions of Boltzmann’s constant are:

[k] = ML2T−2/ K = ML2T−2K−1

The SI units of Boltzmann’s constant are therefore kg m2 s−2 K−1 or J K−1.

Click on questions to reveal solutions

Quiz 4: Use dimensional analysis to see which of the following expressions is allowed if P is a pressure, t is a time, m is a mass, r is a distance, v is a velocity and T is a temperature:

(a) log ( Pt / mr ) Incorrect - please try again!
(b) log ( Prt2 / m ) Correct - well done!
(c) log (Pr2/ mt2) Incorrect - please try again!
(d) log (Pr/ mtT) Incorrect - please try again!

Explanation: First note that the argument of a logarithm must be dimensionless. Now pressure is force over area, so it has dimensions:

 [P] = MLT−2/ L2 = ML−1T−2

Therefore the combination Prt2 ⁄ m is dimensionless since:

 [Prt2/ m] = (ML−1T−2) × L × T2/ M = M/ M = 1

None of the other combinations are dimensionless and so it would be completely meaningless to take their logarithms.

# 4. Quiz on Dimensional Analysis

Choose the solutions from the options given

1.Newton’s law of gravity states that the gravitational force between two masses, m1 and m2, separated by a distance r is given by F = Gm1m2 ⁄ r2. What are the dimensions of G?
(a)M−1L3T−2
(b)M2L−2
(c)MLT−2
(d)M−1 L−3 T2
2.The coefficient of thermal expansion, α, of a metal bar of length ℓ whose length expands by Δℓ when its temperature increases by ΔT is given by Δℓ = α ℓ ΔT. What are the dimensions of α?
(a)K−1
(b)L2T−1
(c)L2T−1
(d)L−2K−1
3.The position of a mass at the end of a spring is found as a function of time to be Asin(ωt). What are the dimensions of A and ω?
(a)L & T
(b)L & Dimensionless
(c)sin (L) & T−1
(d)L & T−1