A point P of mass m moves in a vertical plane along a circle of radius A with uniform angular speed ω. The point Q is the projection of P on the vertical diameter passing through the centre O.
a. (i) Taking downward displacement of Q from O as positive, derive an expression for the acceleration of Q in terms of its displacement y.
b. (ii) Hence prove that Q performs simple harmonic motion and write the expression for the restoring force acting on it.
c. (iii) Deduce an expression for the total mechanical energy of Q in terms of m, A, ω and y. State clearly which forms of energy are included.