Lesson 7 of 10 · 7 min
Dimensions and dimensional formulae
NCERT §1.4, §1.5
Riya's g, the bob's weight, its density and the energy of its swing look like unrelated numbers. Strip away the numbers and each one is just mass, length and time raised to some powers.
The lesson in notes
In short
The dimensions of a quantity are the powers to which the base quantities are raised to express it; they describe its nature, not its size.
Each base quantity is one dimension, written in square brackets: [M] for mass, [L] for length, [T] for time, [A] for electric current, [K] for thermodynamic temperature, [mol] for amount of substance and [cd] for luminous intensity.
In mechanics every quantity can be written with [M], [L] and [T] alone. Volume is [L³]; force = mass × acceleration is [M L T⁻²].
Magnitudes do not enter: speed, initial velocity, final velocity and change in velocity all have the same dimensions, [L T⁻¹].
The dimensional formula shows which base quantities appear and with what powers, zero powers included: volume [M⁰ L³ T⁰], velocity [M⁰ L T⁻¹], acceleration [M⁰ L T⁻²], density [M L⁻³ T⁰].
A dimensional equation sets a quantity equal to its dimensional formula, for example [F] = [M L T⁻²] or [ρ] = [M L⁻³ T⁰].
A dimensional formula can be read off from any defining equation: momentum = mass × velocity gives [M L T⁻¹]; work = force × distance gives [M L² T⁻²].