Starting from the tangential component of the weight of a pendulum bob, analyse the motion of a simple pendulum executing small–angle oscillations.
(i) Show that the displacement of the bob along the arc satisfies the differential equation of simple harmonic motion and hence derive an expression for its time period T.
(ii) The initial angular amplitude of the pendulum is increased from 5∘ to 15∘ while keeping L and g unchanged. State qualitatively whether the actual time period will be greater than, equal to, or less than the value obtained in part (i). Give a reason.
(iii) Using the first two terms of the series expansion sinθ≈θ−θ3/6 (with θ in radians), obtain an approximate percentage increase in the time period when the amplitude is 15∘.
(iv) Suggest one experimental precaution connected with amplitude that should be followed while using a simple pendulum to determine g accurately.