Dimensional Analysis: Definition, Uses and Limitations

If you’ve ever looked at a physics equation and wondered, Is this even correct?, You’re already thinking in the direction of dimensional analysis.

It is a quick checking method used to see if an equation or formula could be correct or undoubtedly incorrect. Instead of performing complex steps and heavy calculations to verify a formula, we use dimensional analysis to do this task in an easier and simpler way.

However, despite its simplicity, it also has some limitations that must be understood.

You can also use it to convert units. For example, force into dyne, meter into centimeter, and many others.

In this article, we will discuss what dimensional analysis is, how this technique is used to perform different tasks with examples, and also its limitations.

What is Dimensional Analysis in Physics?

Dimensional analysis is the analysis that uses the idea of physical dimensions to check the validity of equation and to derive relationships between different physical quantities. It is also used to change the units of a quantity from one system to another.

In simple terms, it is a method used to study the relationship between physical quantities by comparing their dimensions and in the conversion of units.

For example, it helps check equations like v = u+at to find out whether the relation is very likely correct or frmly incorrect.

It is also useful for finding relations between physical quantities, for example, determining how the time period of a simple pendulum is related to its length and gravitational acceleration. It can also be used for unit conversion, such as expressing a force of 3 N in dyne.

What are the Uses of Dimensional Analysis?

The main uses of dimensional analysis are:

1. Principle of Homogeneity of Dimensions
2. Derive a Relation Between Physical Quantities
3. Conversion of Units

1) Principle of Homogeneity of Dimensions

This principle states that

The dimensions on both sides of an equation must be the same, regardless of the form of the formula.

Or

The dimensions of each term in an equation must be the same.

Anything separated by a plus (+), minus (−), or equal (=) sign in an equation is called a term. For example, x = 3+4s, here x, 3, and 4s are terms of the equation where x and s are pure numbers.

We can only add things if they are of the same kind, like apples with apples or mangoes with mangoes. Similarly, we can only add terms if they have the same dimensions or units. For example, speed can only be added to speed, and mass can only be added to mass.

Also, we can equate an equation only if the terms on both sides have the same units. Therefore, each term in an equation must have the same dimensions.

Based on these rules, we check whether an equation is dimensionally consistent or not.

An equation is said to be dimensionally consistent or dimensionally homogeneous if every term in it has the same dimensions.

If an equation is dimensionally consistent, it may be correct. However, if it is not consistent, it violates the principle of homogeneity and is definitely incorrect.

Note: A check of dimensional consistency doesn’t tell us anything more than checking the consistency of units, but it has an advantage. With dimensions, we don’t have to commit to a specific set of units or worry about conversions between different unit systems.

Principle of Homogeneity Examples

Example 1

The most common example used to explain the principle of homogeneity is the second equation of motion:

where s is displacement, u is initial velocity, t is time, and a is acceleration.

Left-hand side (L.H.S):

[s] = [L]

Right-hand side (R.H.S):

[ut] = [LT^-1][T] = [L]

[1/2 at²] = [LT^-2][T²] = [L]

If,

L.H.S = R.H.S

[L] = [L] + [L]

[L] = [L]

As all the terms have the same dimensions and L.H.S = R.H.S., this equation holds the principle of homogeneity and is dimensionally consistent.

Note that the number ½ doesn’t contribute in this check. So, dimensional analysis cannot verify whether it is ½ at² or 2at².

In simple words, dimensional analysis can show when an equation is wrong, but it cannot prove that it is fully correct.

Example 2

In this example, we would like to check if the famous physics equation E=mc² is dimensionally consistent or not. Let’s see

L.H.S (Left-Hand Side):

[E] = [ML²T^-2]

R.H.S. (Right-Hand Side):

[mc²] = [M][LT^-1]²

= [M][L²T^-2]

[mc²] = [ML²T^-2]

If,

L.H.S. = R.H.S.

[ML²T^-2] = [ML²T^-2]

It shows that L.H.S. = R.H.S., which means this equation is dimensionally consistent.

If there is a case when L.H.S ≠ R.H.S or all the terms don’t have the same dimensions, the equation doesn’t hold the principle of homogeneity and is incorrect. For example, v_i = v_f + ½ at², where v_i and v_f are initial and final velocities, respectively, a is the acceleration, and t is the time.

Also note that dimensional analysis does not check numerical constants. So even if the dimensions match, the equation may still not be completely correct.

The principle of homogeneity is also helpful when you’re confused about the relation, like wondering whether it should be T = 2π√(l/g) or T = 2π√(g/l).

Instead of guessing, you can check if the dimensions on both sides are the same. If they match, the equation might be correct. If they don’t match, the equation is definitely wrong.

2) Derive Relation Between Physical Quantities

Dimensional analysis can also be used to derive relations if we know which quantities a certain physical quantity depends on.

To find relations using dimensional analysis, we need to know the dimensions of physical quantities. If it is hard to remember them, we should at least know the basic formulas of the involved quantities, because they can help us find the dimensions.

Example: Derive the formula of the Planck length using dimensional analysis. It depends on Planck’s constant ħ, the gravitational constant G, and the speed of light c.

1st step: As the Planck length depends on ħ, G, and c, we have

l_p ∝ ħ^x G^y c^z

2nd step: Remove the proportionality sign, and a proportionality constant k would appear. So, we have

l_p = k ħ^x G^y c^z

From experiments, it is found that the proportionality constant, which is Planck length, is 1, so it is dimensionless.

One important limitation is that dimensional analysis only works properly when the proportionality constant has no dimensions. For example, in the law of gravitation, the proportionality constant G has dimensions, so we cannot fully derive this formula using dimensional analysis.

3rd step: Write the dimensional formula of each quantity in the calculation formula, i.e.,

[l_p] = [ħ_x] [G_y] [c_z]

[L] = [ML²T^-1]^x [M^-1L³T^-2]^y [LT^-1]^z

[M⁰L¹T⁰]= [M^x-yL^2x+3y+zT^-x-2y-z]

4th step: Equate powers of the same dimensions on both sides

M: 0 = x-y ⇒ x = y

L: 1 = 2x+3y+z

T: 0 = -x-2y-z

As x = y from M, solving equations of L and T gives

Put the value of z in 1=5y+z, we have

1=5y-3y

1=2y

y=1/2

It gives us the value of x and z as follows

From M,

x=y ⇒ x=1/2

From T, we have

z=-3y = -3 (1/2)

z = -3/2

Putting these values of variables in the Planck length formula, we have

l_p = ħ^1/2 G^1/2 c^-3/2

= √ħG/c³

3) Conversion of Units Using Dimensional Analysis

Here we’ll use the concept of dimensions to convert units from one system to another. Like from the SI system to the CGS system.

We already know that a physical quantity depends on a number and units, i.e.,

Physical quantity = n x u

If we convert a quantity from one unit system to another, the numerical value and unit change. But the physical quantity remains unchanged. In other words, the two expressions are equal in representing the same quantity. Like 5 m=500 cm. In general form, we can write it as

n₁u₁=n₂u₂

It is important to note that even though the numerical value and units change, the dimensions of the quantity remain unchanged. We’ll use this concept to convert units.

Example: Suppose we want to find the value of 1 kg/m³ into CGS system. We need to convert units from kg/m³ into g/cm³.

Dimensional formula of density = [ML^-3]

As, n₁u₁=n₂u₂ and we need to find the value of n₂, so we have

n₂ = n₁(u₁/u₂)

=1(M₁L₁^-3/M₂L₂^-3)

=(M₁/M₂)(L₁/L₂)^-3

= (kg/g) (m/cm)^-3

=(1000 g/g)(m/100 m)^-3

= (1000)(10^-2)^-3 = (10³)(10^-6)

= (10^3-6)

n₂ = 10^-3

So, 1 kg/m³ density in the CGS system would be 10^-3 g/cm³ or 0.001 g/cm³.

What are the Limitations of Dimensional Analysis?

In the previous section, we explored how useful dimensional analysis is. But it’s just as important to recognize its limitations, so you can judge when it will actually help and when it won’t.

1) While checking the correctness of the equation using the homogeneity principle, it won’t be able to differentiate between ½ and 2 in the formula, as these are constant numbers.

For example, the 2nd equation of motion is s = ut+½ at², even if it is written like s = ut+2at², the homogeneity principle would show it correct. So, a dimensionally consistent equation doesn’t need to be correct, but if it is not consistent, it would definitely be wrong.

2) Deriving relations using dimensional analysis cannot find if there are any proportionality constants like 2, cosθ, 2π, e^-m on which the formula depends.

For example, the time period of a simple pendulum, T = 2π √l/g. If you derive this formula using dimensional analysis, you would only get an expression like T = k √l/g, where k is a proportionality constant. It won’t tell anything about k; you have to find it experimentally.

3) Dimensional analysis cannot be used to derive a formula if the proportionality constant has dimensions, like the law of gravity.

This is because the method works by matching the powers of the same dimensions on both sides. A dimensional constant adds extra variables that we don’t know.

However, if physical constants are already included explicitly in the equation, dimensional analysis can still be used to check consistency. Like the mentioned example of finding the formula for the Planck length.

4) Deriving a relation using dimensional analysis doesn’t work if the relation contains addition or subtraction, like in 2nd equation of motion. It would only derive relations correctly if all the quantities are in multiplication or division form.

5) This method of deriving relations works only when the number of unknown exponents is less than or equal to the number of independent exponent equations. 

In mechanics, there are only three fundamental dimensions (M, L, T), so equating exponents can provide at most three independent equations. If a physical relation involves more than three independent variables (or quantities), the number of unknowns exceeds the number of equations, and the system becomes underdetermined. 

In such cases, dimensional analysis cannot uniquely determine the relation, although it may still suggest a possible functional form.

6) While deriving a relation, dimensional analysis cannot determine the exact form of dimensionless quantities like ​m₁/m₂. It may tell you a ratio exists, but not how it appears in the formula. It could be m₁/m₂, (m₁/m₂)², sin(m₁/m₂) or could be anyother form.

Dimensional Analysis Practice Questions (Try Yourself)

1) Check if this equation is correct: v = u + at²

2) Derive the formula of the Planck time using dimensional analysis. It depends on Planck’s constant ħ, the gravitational constant G, and the speed of light c.

3) Convert

• 10 g into mg
• 72 km/h into m/s
• 2500 cm into meters
• 1 N into dyne
• 1 Pascal into dyne/cm²

using dimensional analysis.