A particle P moves with uniform angular speed ω along a circle of radius A. From every position of P a perpendicular is dropped on a fixed diameter to meet the diameter at the point Q.
a. (i) Show that the displacement x of Q from the centre O of the circle can be expressed as x=Acosθ, where θ is the angular position of P measured from the diameter taken positive in the anticlockwise sense.
b. (ii) Using the result of part (i), explain why the point Q executes simple harmonic motion and state its amplitude, angular frequency and initial phase in terms of A, ω and the initial value of θ.
c. (iii) If P is at an angle 30∘ from the positive side of the diameter at t=0, write the equation of motion of Q as a function of time.