Question

a particle is in its state of uniform circular motion. prove that its projection along one of its diameter executes simple harmonic motion.

Please give me answer of this question urgently.

Answer 1

The displacement of the projection along the diameter from the Center of the circle is given by :
        x = R Cos ωt ,               if  x = R at  t = 0.
 Uniform circular motion means that  ω is a constant.

   Then  v = dx/dt = - Rω Sinωt
             a = d²x/dt² = - Rω²  Cos ωt = - ω² x
      Hence its motion is a SHM.