Two simple pendulums having lengths ℓ 1 and ℓ 2 with negligible string mass undergo angular displacement θ 1 and θ 2 from their mean positions respectively. If the angular acceleration of both

Question

Two simple pendulums having lengths ℓ 1 ​ and ℓ 2 ​ with negligible string mass undergo angular displacement θ 1 ​ and θ 2 ​ from their mean positions respectively. If the angular acceleration of both pendulums are same, then which equation is correct ?

TestHub · Accepted Answer

For a simple pendulum, the restoring torque is $τ = - m g ℓ sin θ$ .

For small angular displacements, $sin θ \approx θ$ , so $τ = - m g ℓ θ$ .

The moment of inertia of the pendulum about the point of suspension is $I = m ℓ^{2}$ .

The angular acceleration $α$ is given by $α = \frac{τ}{I} = \frac{- m g ℓ θ}{m ℓ ^{2}} = - \frac{g}{ℓ} θ$ .

The magnitude of the angular acceleration is $∣ α ∣ = \frac{g}{ℓ} θ$ .

Given that the angular accelerations of both pendulums are the same, we have $∣ α_{1} ∣ = ∣ α_{2} ∣$ .

$\frac{g}{ℓ _{1}} θ_{1} = \frac{g}{ℓ _{2}} θ_{2}$

$θ_{1} ℓ_{2} = θ_{2} ℓ_{1}$

Answer: $θ_{1} ℓ_{2} = θ_{2} ℓ_{1}$

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