Suppose you want to estimate the period (time for one swing) of a simple pendulum of length L swinging under gravity g. 1. Identify the quantities that could affect the period T: length L (meters) and acceleration due to gravity g (meters/second^2). 2. Write a general form: T = k * L^a * g^b, where k is a dimensionless constant and a, b are exponents to be found. 3. Use dimensional analysis: - Dimensions of T: [T] = seconds (s). - Dimensions of L: [L] = meters (m). - Dimensions of g: [g] = m/s^2. So, [T] = [L]^a * [g]^b = m^a * (m/s^2)^b = m^{a+b} * s^{-2b}. 4. Equate exponents for meters and seconds: - For meters: a + b = 0. - For seconds: -2b = 1 → b = -1/2. Then a = -b = 1/2. 5. The formula becomes T = k * L^{1/2} * g^{-1/2} = k * sqrt(L/g). 6. Dimensional analysis tells us the functional form but not k. Experiments show k ≈ 2π. So the real formula is T = 2π * sqrt(L/g). Dimensional analysis got the right dependence on L and g but missed the 2π constant.