Question Detail
Question
The period T of a simple pendulum depends on its length l and the acceleration due to gravity g. Using dimensional analysis, which of the following is the correct relationship? (Assume k is a dimensionless constant)
Correct Answer
Option B is the correct answer.
Detailed Explanation
Let T ∝ lˣgʸ. Dimensions: [T] = T, [l] = L, [g] = LT⁻². So T = M⁰L⁰T = Lˣ (LT⁻²)ʸ = Lˣ⁺ʸ T⁻²ʸ. Equating powers: for T: 1 = -2y ⇒ y = -1/2; for L: 0 = x + y ⇒ x = -y = 1/2. Therefore, T ∝ l¹/² g⁻¹/² = √(l/g), so T = k √(l/g).
Hint
Set up a proportionality T ∝ lˣ gʸ, insert dimensions, and solve for the exponents x and y.