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Understanding Pendulum Motion and Arc Distance Calculations in Simple Harmonic Motion

This document explores the relationship between pendulum motion and arc distance, emphasizing the close approximation of arc and straight distances within 1%. It utilizes trigonometric functions and rotational formulas to derive key equations such as ( F = mg sin(q) ) and ( T = 2pi sqrt{m/g} ). With practical examples and calculations for a pendulum at a 5° arc, this guide serves as a valuable resource for those studying dynamics and forces in physics.

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Understanding Pendulum Motion and Arc Distance Calculations in Simple Harmonic Motion

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  1. q L therefore, the arc distance is within 1% of the straight distance (dottted line) (see next page) T x mg Cos q mg Sin q mg T= 2p m/k From SHM T = 2p m mg/L = 2p L/g f = 1 g/L 2p F = mg Sin q for @5o of arc Sin q =.087 F = mg Sinq = mg q Sin q = x/L x = Sin q L Sin q = q x = q L q = x/L F = mg(x/L) or, F = (mg/L)x F = kx k = mg/L

  2. q Pendulum with 5o arc or less L = 1.0 m Trig. Functions for triangles Rotational formulas for circles L q q 5o (1 rad/57.3O) = .0873 rad L L x x Sin 5o = x/1.0 m x = Sin 5o(1.0 m) x = .0872 m q = x/L x = .0873 rad(1.0 m) x = .0873 m

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