Engineers often simplify the motion of a simple pendulum of length by replacing with…

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

Engineers often simplify the motion of a simple pendulum of length LL by replacing sin⁡θ\sin\theta with θ\theta (in radians) when the angular displacement θ\theta is small. (a) Compute lim⁡θ→0sin⁡θθ\displaystyle\lim_{\theta\to 0}\dfrac{\sin\theta}{\theta} to justify this small-angle substitution. (b) For 0<θ<π20<\theta<\dfrac{\pi}{2}, the inequality 0≤θ−sin⁡θ≤θ360\le \theta-\sin\theta\le \dfrac{\theta^{3}}{6} holds. Using this bound, find an upper estimate for the relative error ∣sin⁡θ−θθ∣\displaystyle\left|\dfrac{\sin\theta-\theta}{\theta}\right| when θ=0.1 rad\theta=0.1\text{ rad}. Express your answer as a decimal. (c) Convert the bound from part (b) to a percentage and comment on the practical appropriateness of the small-angle approximation at θ=0.1 rad\theta=0.1\text{ rad}.

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