A simple pendulum of length executes small-angle oscillations with amplitude

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

Problem

A simple pendulum of length L=0.90 mL=0.90\,\text{m} executes small-angle oscillations with amplitude θmax=15∘\theta_{\text{max}}=15^{\circ}. Treat the motion as simple harmonic about θ=0\theta=0. The mass of the bob is 0.25 kg0.25\,\text{kg} and air resistance is negligible. (a) Express the gravitational potential energy UU of the bob relative to its lowest point in terms of the angular displacement θ\theta and identify the total mechanical energy EtotalE_{\text{total}} for this oscillation. (b) Derive an expression for the fraction of the total mechanical energy that is potential when the bob is at an angular displacement θ\theta. Your final answer should be written solely in terms of θ\theta and θmax\theta_{\text{max}}. Do not substitute numerical values yet. (c) Using your result from part (b), determine the exact fraction of the energy that is kinetic when the bob passes through θ=5∘\theta=5^{\circ}. (d) Calculate the linear speed of the bob at θ=5∘\theta=5^{\circ}. (e) Discuss briefly how the energy fractions would change, if at all, were the amplitude doubled while keeping the small-angle approximation valid.

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