Physics and Measurement

Dimensions are the powers of base quantities (M, L, T, A, K, mol, cd) in a derived quantity.

What dimensional analysis CAN do:

1. Check correctness of an equation — both sides must have the same dimensions.
2. Convert units between systems (e.g. SI to CGS).
3. Derive relationships between physical quantities up to a dimensionless constant.

What it CANNOT do:

1. Find the dimensionless constants (like 2π in T = 2π√(L/g)).
2. Distinguish between quantities that share dimensions (work and torque are both [ML²T⁻²]).
3. Handle equations involving trigonometric, exponential or log functions of dimensional quantities (their arguments must be dimensionless).

Worked example. Estimate the time period T of a simple pendulum, given it depends on length L, mass m, and g. Assume T = k · L^a · m^b · g^c. Equating dimensions: [T] = [L]^a [M]^b [LT⁻²]^c. So 1 = -2c → c = -1/2. 0 = a + c → a = 1/2. 0 = b → b = 0. Result: T ∝ √(L/g). The constant 2π comes from full Newtonian mechanics, not from dimensions.