Large initial displacements/angles on a pendulum

Question

Why does large initial displacements/angles on a pendulum affect the period and the accuracy of oscillation? I know at small angles it does not effect the period however large angles (maybe above 30$^o$) do..

Thank you

Answer 1

If you follow the derivation of the equation of motion, there is an approximation made that $\sin \theta = \theta$. Before the approximation the equation is $\ddot \theta + \frac gl \sin \theta =0$. Once you make the approximation you have a simple harmonic oscillator. As the Taylor series of $\sin \theta$ around zero is $\sin \theta \approx \theta - \frac 16 \theta^2 $ we are ignoring the cubic (and higher) term(s). This is valid when the angle is small.