Being negative for the sake of it
The other day in class I was deriving the equation for the simple pendulum and started wondering how to make the solution negative. I knew it had to be to fit in with the result a= -ω2x but why should the component of mg be -ve? I looked the derivation up in several places but wasn't satisfied. Most sources used a similar approach to hyperphysics. That the force is -ve because it is but to my mind -mgsinθ is simply in the opposite direction to mgsinθ so that doesn't make sense.
I asked one of the maths teachers and he agreed that you can't simply say that the Force is -ve because of the direction so at least I realised that I wasn't missing something obvious. The only sensible way I could think of introducing a -ve was after substituting sinθ=x/L x is the displacement so if this is to the right then F is to the left so must be opposite sign.
Eventually I found an explanation for the early introduction of the -ve sign that I am happy with. It comes from considering the angular motion of the pendulum. τ = Iα Now the torque is in the opposite direction to increasing θ so τ = -LFsinθ which makes sense. Cancel out a few things and you arrive at F=-mgsinθ. No good for our students though who don't study rotational dynamics. Roll on the new syllabus.
Steve Sque, University of Exeter
And if it's any consolation that animated pendulum made me feel sick too.