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.Please help - this question has really thrown me!
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:
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).