Two simple pendulums having lengths l₁ and l₂ with negligible string mass undergo angular displacements θ₁ and θ₂, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?

Solution

Physics sir · 3 steps

Write α for a simple pendulum: it goes as θ/l, so equal α means equal θ/l.

1. Torque about pivot. Gravity gives a restoring torque τ = -mgl sin θ about the point of suspension. The bob's moment of inertia about that point is I = ml².

   α = τ/I = -(mgl sin θ)/ml² = -g/l sin θ ≈ -g/l θ

2. Set them equal. Equal angular accelerations give g/l₁θ₁ = g/l₂θ₂. The g cancels, leaving θ₁/l₁ = θ₂/l₂.
3. So. Cross-multiplying gives θ₁ l₂ = θ₂ l₁, which is option (a).

The idea to keep. For a simple pendulum, α = -g/lθ. One l from the torque cancels one l from ml², so a longer pendulum needs a bigger angle for the same α.

Answer from the NTA final answer key. Solution by PrepWiser.