Question #b03f7

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Question #b03f7

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Answer 1 · 2016-02-28T09:47:21

For the SHM approximation to be more accurate.

Explanation:

I think you meant $sin θ \approx θ$ $sintheta ~~ theta$ instead of $m g sin θ \approx sin θ$ $mgsintheta~~sintheta$ .

The simple harmonic motion (SHM) is only an approximation for swinging pendulums, unlike the case for spring mass system where it is exactly SHM.

The differential equation for SHM is

$\frac{d^{2} x}{d t^{2}} + ω^{2} x = constant$ $frac{"d"^2x}{"d"t^2} + omega^2x = "constant"$

For pendulums, the equation of motion is

$\frac{d^{2} θ}{d t^{2}} + \frac{g}{l} sin θ = 0$ $frac{"d"^2 theta}{"d"t^2} + g/l sintheta = 0$

which is not exactly SHM. Only when $θ \to 0$ $theta -> 0$ , does $sin θ \to θ$ $sin theta -> theta$ , do we begin to observe SHM-like behaviors, such as having a period independent of the amplitude.

For beginners, we do not concern ourselves with frictional losses, although you do have a point that increasing the amplitude increases the rate of energy loss from the system in real life.

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