OCR M3 Textbook Question (Exercise 6 Question 7)

actually there is a much easier way. The SHM approximation is simply that sin(theta) is approximately theta, or cos(theta) is approximately 1 - 0.5(theta)^2. Therefore you have used the SHM approximation in your argument to get the right answer.

I understand that if you resolve in the transverse direction (or take an energy equation and differentiate) you get theta = -(g/l)*sin(theta), and use sin(theta) is approximately theta for small theta.. However, this in itself has nothing to do with T, which only has a component in the radial direction. For the initial section I applied the usual method which is taking a conservation of energy equation and using this in mv^2/r to find an expression for T. I understand that using 1 - 0.5(theta)^2 will work - however, I don't see why this is an "SHM approximation" since I've not used the SHM equation as part of this approximation(e.g getting it into acceleration = -k*displacement via some approximation). Does this mean the question is just poorly worded or am I missing something?

e.g I understand the link between SHM and a pendulum with a small angular amplitude where you resolve in the transverse direction and use a small angle approximation to get the SHM differential equation. I also understand how to get the given answer using the small angle approximation for cosine. However, I don't see where SHM has actually come in to this particular question...

So it is all to do with potential energy wells. If you imagine the PE plotted as a graph of PE against theta,, you'll obviously have a minimum at theta=0. Anything in any sort of potential energy well will undergo SHM as long as the perturbations from equilibrium are small enough that it can be modelled as a quadratic minimum. You achieve this in this instance by making the amplitude of oscillation small enough that the sine(theta) term is almost equivalent to theta. This means that we are only looking at the small section of the PE well that can be modelled as quadratic, and hence SHM will occur. By making the small angle approximation, you are satisfying the conditions for SHM. I probably haven't explained this very clearly, but do you understand what I'm getting at?