Dimensional Analysis

Dimensional analysis, as introduced in NCERT Class 11, Chapter 2 (Units and Measurements), lets us deduce the form of a relation by matching dimensions on both sides. Assume the period T depends on length as L^{a} and on gravitational acceleration as g^{b}, so T = k L^{a} g^{b}. The dimensions are [T^{1}] = [L^{1}]^{a} [L^{1}T^{-2}]^{b} = [L^{a+b} T^{-2b}]. Matching the time power gives -2b = 1, so b = -1/2, and matching the length power gives a + b = 0, so a = 1/2. Hence the period is proportional to L^{1/2}. The option L^{1} is wrong because it ignores the constraint from the gravity term. The option L^{-1/2} is wrong as it reverses the required sign of the exponent. The option independent of L is wrong since experiments and the derivation both show a clear length dependence. A final check confirms the familiar pendulum formula T = 2π√(L/g) indeed contains the square root of length.