Why if a mass tied to an ideal string is given a small displacement, its motion is SHM. However, for a large displacement it is not SHM but oscillatory?
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If you are talking about the motion of a pendulum, its equation is given by $\frac{d^2 \theta}{d t^2} = - \frac{g}{L} \sin \theta$. However, the simple harmonic motion is described by $\frac{d^2 \theta}{d t^2} = - \frac{g}{L}\theta$ - note the absence of the sine here. For small displacements we can approximate $\sin \theta \simeq \theta$, using a Taylor series expansion, which means that for small angular displacements the motion is indeed simple harmonic to a reasonable degree of accuracy. As $\theta$ becomes larger, the non-linearity of the sine function become more pronounced, and the motion departs from the simple harmonic form. This has a solution in terms of elliptic integrals.
Clara Díaz Sanchez
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