# Why for a large displacement of a mass tied to a string is not SHM

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?

• Hi. Could you elaborate your experimental setup? Are you referring to a mass suspended using a string like a pendulum? Thanks! – Guru Vishnu Jan 26 at 9:07
• This might interest you: physics.stackexchange.com/questions/57623/… – Semoi Jan 26 at 9:47

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.