Angular acceleration of a pendulum is given by $$\frac{d^2\theta}{dt^2}= -\frac{mGR}{I}\sin(\theta)$$ How can we make a Taylor series approximation 5 terms when $\theta=0$?
This is for my computing class. Usually for taylor series, a function is sum of differenial and factorials..
what about a taylor series for a second order diffenrtial equation? what if the value of sin is not zero or near to zero? I have tried searching internet to understand this
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When $\theta$ approaches $0$, (small angle approximation) $\sin\theta$ can be expanded with the Taylor power series $$\sin\theta=\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}+o(\theta^6)$$ Thus $\sin\theta$ can be replaced like this: $$\frac{d^2\theta}{dt^2}= -\frac{mGR}{I}\sin\theta=-\frac{mGR}{I}\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}\right)$$
$$\sin\theta=\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}+\frac{\theta^9}{9!}+...$$