 # Simple Harmonic Motion

 From: Michael Ellis Date: 18 April 1999 Subject: Simple Harmonic Motion In A-level Mechanics we have the following question: A particle of mass m is attached to one end of a light elastic string of natural length a and modulus mg. The other end of the string is fixed to a point A. The particle is held at point O which is a vertically below A, and projected vertically downwards. Find the equation of motion of the particle when it is a distance x below O, and hence prove that the particle subsequently performs S.H.M. about a point B which is a below O for as long as the string remains taut. Please help - this question has really thrown me!

### Maths Help suggests:

The general method for this type of question (proving SHM for particles dangling on the end of an elastic string) requires you to show that the acceleration is a negative multiple of the distance of the particle from the centre of oscillation, namely: This is normally done by applying Newton's 2nd Law ("F=ma") when the particle is a general distance x from the centre of oscillation. This question is slightly different in that x is measured from the point O, but that will sort itself out as we work through the question. Also, the tension in the string will be given by Hooke's Law.

Firstly, a good diagram is essential: Apply Newton's 2nd Law ("F=ma") with the upward direction positive: Use Hooke's Law for the tension in the string: Simplify: Thus we see that the acceleration is a negative multiple of the distance of the particle from the centre of oscillation (here a-x rather than x). This proves that the particle performs S.H.M. for as long as the string remains taut (ie when T exists as shown).