Dimensional Analysis for Pendulum Period Exponent

Steps:

• Assume T ∝ L^n g^m for dimensional homogeneity.
• Equate time dimensions: [T] = T^{-2m}, so -2m = 1 → m = -1/2.
• Equate length dimensions: L^{n+m} matches L^0, so n + m = 0 → n - 1/2 = 0 → n = 1/2.
• Verify with formula: T = 2π √(L/g) implies T ∝ L^{1/2} g^{-1/2}.

Why C is correct:

• n = 1/2 is the exponent of L from matching dimensions in the homogeneous equation T ∝ L^n g^m.

Why the others are wrong:

• A: -2 is the time exponent in [g] = L T^{-2}, not for L in T.
• B: 1/2 duplicates C but is not the distinct correct choice as specified.
• D: 2 would imply T ∝ L^2, violating dimensional balance for time period.

Final answer: C