Both sides of a correct equation, and every term in it, have the same dimensions. A mismatch proves an equation wrong. A match does not prove it right, because pure numbers have no dimensions.
“5 metres = 5 seconds” has no meaning. A length and a time are different kinds of quantity, and they cannot be equal. The same applies to every equation in physics.
The principle of homogeneity of dimensions states: in a correct equation, both sides have the same dimensions, and every term that is added or subtracted has those same dimensions.
This gives a check that needs no numbers. Find the dimensions of each term. If any term differs, the equation is wrong.
Check s = ut + ½at², where s is displacement, u is initial velocity, a is acceleration and t is time.
Every term is [L], so the equation is dimensionally consistent.
A student writes v = at². Use dimensions to show it is wrong.
Left side: v is a velocity, [L T⁻¹].
Right side: at² = [L T⁻²] × [T²] = [L].
[L T⁻¹] ≠ [L], so the equation is wrong. The correct relation v = at gives [L T⁻²] × [T] = [L T⁻¹], which matches.
A quantity’s dimensions show how its unit is built from base units. To convert, replace each base unit by its size in the new units.
Density has dimensions [M L⁻³]. To convert 2 kg/L to kg/m³: 1 L = 10⁻³ m³. So 2 kg/L = 2 kg / (10⁻³ m³) = 2000 kg/m³.
One student writes the period of a pendulum as T = 2π√(l/g). Another writes T = √(l/g). Can dimensional analysis decide which is correct?
Inside the root, l/g = [L] ÷ [L T⁻²] = [T²]. The square root is [T], which matches the period T. Both formulas pass.
They differ only by 2π, a pure number with no dimensions. Dimensional analysis cannot detect it.
The factor 2π comes from solving the equation of motion of the pendulum, or from measurement. For a pendulum with l = 1.00 m and g = 9.81 m/s², the measured period is about 2.01 s, which agrees with 2π√(l/g) and not with √(l/g) ≈ 0.32 s.
A measurement is a number and a unit, and SI fixes the units worldwide. Significant figures record how precise a measurement is, and the rounding rules carry that precision through a calculation. Dimensions describe what kind of quantity something is, and the principle of homogeneity uses them to check equations and convert units.
In each case, a result is kept to what the measurements and the physics support: significant figures limit the precision claimed, and dimensions limit which equations can be correct.
1The principle of homogeneity of dimensions states that…
2The two sides of an equation have different dimensions. The equation is…
3In s = ut + ½at², every term has the dimensions…
4Why can dimensional analysis not detect a missing ½ or 2π?
5An equation passes the dimensional check. It is…
6What does a dimensional check of v = at² show?
7Which statement describes dimensional analysis best?
That completes the unit on units and measurement.
Noescio