Dimensional analysis

Checking an equation

Narrated video. Transcript:

Can you add 5 metres to 3 seconds? Of course not. You can only add, subtract or equate quantities of the same kind.

This is the principle of homogeneity of dimensions: every term in a correct equation must have the same dimensions.

Let’s test s = ut + ½at². Each term gets its own box. s is a distance, so it is [L].

u times t is [L T⁻¹] times [T], which is [L]. a times t² is [L T⁻²] times [T²], also [L]. The ½ is just a number. Watch every box turn green.

Now try s = ut + at³. The last box comes out as [L T]. It turns orange, so this equation is certainly wrong.

But be careful. Passing the test does not prove an equation is right. K = mv² passes, yet the true formula is ½mv². Dimensions cannot see pure numbers.

One more rule: whatever goes inside sin, cos, log or an exponential must be dimensionless. Sin of 5 metres makes no sense; sin of ωt is fine.

So if an equation fails the dimension test, it is wrong. If it passes, it may be right. Now test some equations yourself.

Try it: is it consistent?

Interactive simulation. Things to try:

• Work out the dimensions of each box in your head, then choose an answer.
• Green boxes match the first term; orange boxes do not.
• One equation passes the test but is still wrong. Can you spot it?

Deducing a formula

Narrated video. Transcript:

Dimensions can do more than check formulas. They can help you find one.

What sets the time period of a simple pendulum? Perhaps the mass of the bob m, the length of the string l, and the acceleration due to gravity g.

Guess T = k mᵃ lᵇ gᶜ, where k is a pure number. Now match the dimensions of both sides.

The left side is [T]. The right side is [M]ᵃ [L]ᵇ [L T⁻²]ᶜ, which is [Mᵃ Lᵇ⁺ᶜ T⁻²ᶜ].

Compare the powers. Mass: a = 0. Time: −2c = 1, so c = −½. Length: b + c = 0, so b = ½.

So T = k√(l/g), and the mass has dropped out! Watch a light bob and a heavy bob on equal strings swing in step.

Dimensions cannot give the number k. Experiment shows that k = 2π, so T = 2π√(l/g).

The method has limits. It cannot find pure numbers, it cannot handle sums of terms, and in mechanics it cannot find more than three unknown powers.

The same idea converts units. 1 N is 1 kg m s⁻². Put 1 kg = 1000 g and 1 m = 100 cm, and you get 10⁵ g cm s⁻², which is 10⁵ dyne.

Try it: find the pendulum formula

Interactive simulation. Things to try:

• Adjust a, b and c until the bars match the dashed outline: one block of T and nothing else.
• Change the mass of the bob. The period does not change.
• Make the string four times longer and watch the period double.

Try it: convert units

Interactive simulation. Things to try:

• n₂ = n₁ [M₁/M₂]ᵃ [L₁/L₂]ᵇ [T₁/T₂]ᶜ, where a, b, c are the dimensions of the quantity.
• Press “CGS” to see 1 N = 10⁵ dyne and 1 J = 10⁷ erg.
• Try minutes or hours as the unit of time.

Deduce it

The speed v of a wave on a stretched string depends on the tension F and the mass per unit length μ. Which relation do dimensions allow?

1. v ∝ √(F/μ)
2. v ∝ √(μ/F)
3. v ∝ F/μ
4. v ∝ Fμ

Convert it

In SI, G = 6.67 × 10⁻¹¹ N m² kg⁻². What is its value in CGS units (g, cm, s)?

1. 6.67 × 10⁻¹¹
2. 6.67 × 10⁻⁸
3. 6.67 × 10⁻¹⁴
4. 6.67 × 10⁻⁵

Summary

• Simple pendulum: T = 2π√(l/g)
• Converting units: n₂ = n₁ [M₁/M₂]ᵃ [L₁/L₂]ᵇ [T₁/T₂]ᶜ
• Example: 1 N = 10⁵ dyne, 1 J = 10⁷ erg

• Homogeneity: every term in a correct equation has the same dimensions.
• Failing the dimension test means the equation is wrong. Passing it means the equation may be right.
• Matching powers of M, L and T can deduce a relation, such as T ∝ √(l/g), but never the pure number in front.
• The method cannot handle sums of terms or trig, log and exponential functions, and in mechanics it finds at most three unknown powers.