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Unformatted text preview: 103. For simple harmonic motion, Eq. 1524 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
builtin 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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 Spring '08
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 Physics, Simple Harmonic Motion

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