Two simple pendulums of lengths l₁ and l₂ are oscillating with small angular amplitudes ₁ and ₂ , respectively. If their maximum linear speeds at their lowest points are exactly equal, the ratio of their angular amplitudes ( ₁ ₂ ) is:

Step-by-step solution

For a simple pendulum of length l oscillating with a small angular amplitude ₀ , the linear amplitude is A = l ₀ . The angular frequency of the simple pendulum is = g l . The maximum linear speed of the pendulum bob is given by v_ = A . Substituting the expressions for A and : v_ = (l ₀) g l = ₀ gl Given that the maximum linear speeds of the two pendulums are equal: ₁ gl₁ = ₂ gl₂ Rearranging to find the ratio of their angular amplitudes: ₁ ₂ = l₂ l₁ = l₂ l₁ Answer: l₂ l₁