Can you prove a formula wrong without an experiment?

What "dimensions" means

Think of a shoebox. Its volume is length × breadth × height, three lengths multiplied together. In dimension terms that's [L³]. Volume doesn't depend on mass or time, so the powers of [M] and [T] are zero.

NCERT puts it like this: the dimensions of a physical quantity are the powers, or exponents, to which the base quantities are raised to represent it. The book writes the base quantities in square brackets: [L] for length, [M] for mass, [T] for time. Section 1.4 also uses brackets for the other base quantities: electric current is [A], thermodynamic temperature is [K], luminous intensity is [cd] and amount of substance is [mol].

Force is mass × acceleration, and acceleration is length ÷ time². So force has dimensions [M L T⁻²]. Size doesn't matter here, only the kind of quantity. Initial velocity, final velocity and speed are all [L T⁻¹].

Two terms that often get mixed up:

• Dimensional formula: the expression itself, such as [M L T⁻²] for force or [M L⁻³ T⁰] for density.
• Dimensional equation: a quantity set equal to its formula, such as [F] = [M L T⁻²].

The one rule

Can you add your age to your height? Sixteen years plus 170 centimetres gives a number, but it means nothing. Physics works the same way. The principle of homogeneity of dimensions says you can add or subtract only quantities with the same dimensions. Every term in a correct equation must have the same dimensions.

Test it on x = x₀ + v₀t + ½at².

• x and x₀ are both [L].
• v₀t is [L T⁻¹] × [T] = [L].
• at² is [L T⁻²] × [T²] = [L].
• ½ is a pure number, so it has no dimensions.

Every term is [L], so the equation is dimensionally consistent.

Example 1.4: five formulas for kinetic energy

The book gives five candidate formulas for kinetic energy K. The correct dimensions are [M L² T⁻²]. Check each one:

That rules out three formulas without an experiment. But two survive, and they differ only by a number. Dimensions can't choose between ½ and 3/16. Only the actual definition of kinetic energy tells you it's ½mv².

So the rule runs one way. A dimensionally inconsistent equation is wrong. A consistent one isn't necessarily right.

A slip people make with v²: velocity is [L T⁻¹], so v² is [L² T⁻²]. Square the whole bracket, not just the L.

Example 1.5: building the pendulum formula

Dimensions can sometimes build a formula. Suppose the period T of a simple pendulum depends on its length l, the mass of the bob m and the acceleration due to gravity g. Write

T = k l^x g^y m^z

where k is a dimensionless constant. The left side is [T]. The right side is [L]^x [L T⁻²]^y [M]^z = [L^(x+y) T^(−2y) M^z]. Match the powers:

• L: x + y = 0
• T: −2y = 1, so y = −½
• M: z = 0

That gives x = ½, so T = k √(l/g). The mass drops out completely. On this result, a heavy bob and a light one on the same length of string have the same period.

But what is k? The method can't say. Dimensions find the shape of the formula, not the number in front. The value k = 2π comes from a full theory and from experiment.

The limits, in plain words

1. Dimensionless constants such as 2π stay invisible.
2. To build a formula, you must already know what the quantity depends on.
3. The check can't tell apart quantities with the same dimensions. Swap a height for a radius, both [L], and it won't notice.

Check yourself

A student writes distance as s = vt². Does it pass the dimension check? (Answer at the bottom.)

Questions students ask

Answer: No. vt² is [L T⁻¹] × [T²] = [L T], not [L], so the formula is wrong.