ch15-p103 - 103. For simple harmonic motion, Eq. 15-24 must...

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Unformatted text preview: 103. For simple harmonic motion, Eq. 15-24 must reduce to c ch h τ = − L Fgsinθ → − L Fgθ where θ is in radians. We take the percent difference (in absolute value) d− LF sinθ i − d− LF θ i = 1 − g g − LFg sin θ θ sin θ and set this equal to 0.010 (corresponding to 1.0%). In order to solve for θ (since this is not possible “in closed form”), several approaches are available. Some calculators have built-in numerical routines to facilitate this, and most math software packages have this capability. Alternatively, we could expand sinθ ≈ θ – θ 3/6 (valid for small θ) and thereby find an approximate solution (which, in turn, might provide a seed value for a numerical search). Here we show the latter approach: 1− θ ≈ 0.010 θ −θ3 / 6 1 1−θ 2 6 ≈ 1010 . which leads to θ ≈ 6(0.01/1.01) = 0.24 rad = 14.0° . A more accurate value (found numerically) for the θ value which results in a 1.0% deviation is 13.986°. ...
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