Two simple pendulums have lengths ℓ₁ and ℓ₂ and undergo small angular displacements θ₁ and θ₂ from their mean positions. If their angular accelerations have equal magnitude at an instant, which relation is correct?
Options

Question

Two simple pendulums have lengths ℓ₁ and ℓ₂ and undergo small angular displacements θ₁ and θ₂ from their mean positions. If their angular accelerations have equal magnitude at an instant, which relation is correct?

Options

G1 Physics · Accepted Answer

For a simple pendulum, α = -(g/ℓ)sinθ. For small angular displacement, sinθ ≈ θ. Hence, α ≈ -gθ/ℓ. If the angular accelerations have equal magnitude, gθ₁/ℓ₁ = gθ₂/ℓ₂. Cancel g: θ₁/ℓ₁ = θ₂/ℓ₂. Cross-multiplying: θ₁ℓ₂ = θ₂ℓ₁. Correct option: D.