Lesson 5: Dimensional Formula

5.1 The idea

Every physical quantity has a definition that is agreed upon worldwide. For example, speed = distance ÷ time. We use letters for the three mechanical fundamental quantities:

• L for length, M for mass, T for time.

(Careful: here T means time dimension, not temperature.)

• When the definition of a quantity is written using these symbols we get its dimensional formula.
• Square brackets [ ] mean "the dimensions of":

[speed] = [distance] ÷ [time] = L ÷ T = LT⁻¹

In general, most derived quantities can be written as

[A] = L^a M^b T^c

where a, b, c are the dimensional exponents (positive, negative or zero).

Dimensional formula of a physical quantity: an expression showing how the quantity depends on the fundamental quantities L, M, T and the power of each.

5.2 Getting the unit from the dimensional formula

Replace L, M, T by their SI units (m, kg, s).

• Speed: LT⁻¹ → m s⁻¹ = m/s.

5.3 Worked example — acceleration

Acceleration is the time rate of change of velocity: [a] = [velocity] ÷ [time] = LT⁻¹ ÷ T = LT⁻²; unit: m/s².

5.4 Table of dimensional formulas

More derived examples (your own practice):

• Momentum p = mv: M × LT⁻¹ = MLT⁻¹, unit kg·m/s.
• Power P = W/t: ML²T⁻² ÷ T = ML²T⁻³, unit watt (W) = kg·m²·s⁻³.
• Weight density / specific gravity relative density = ρ/ρ_water → dimensionless, no unit.
• Frequency f = 1/T: T⁻¹, unit hertz (Hz) = s⁻¹.
• Viscosity coefficient η = Fd/(Av): [MLT⁻² · L] ÷ [L² · LT⁻¹] = ML⁻¹T⁻¹, unit kg/(m·s) = N·s/m² (used in Chapter 4).

5.5 Rules to remember ("Pay attention")

1. You can add or subtract only quantities of the same kind — those having the same dimensional formula. You cannot add a mass (kg) to a distance (m).
2. If they are of the same kind but in different units, convert one first: 1 m + 170 cm = 100 cm + 170 cm = 270 cm.
3. Quantities with different dimensional formulas can be multiplied or divided, and a new quantity results (distance ÷ time → speed; force × distance → work).
4. Pure numbers, fractions and constants like ½, 2, π, are dimensionless.