angle ω t and its position vector will make an angle of ω t + φ with the +ve x -axis. Next, consider the projection of the position vector OP on the x -axis. This will be OP ′ . The position of P ′ on the x -axis, as the particle P moves on the circle, is given by x ( t ) = A cos ( ω t + φ ) which is the defining equation of SHM.
This shows that if P moves uniformly on a circle, its projection P ′ on a diameter of the circle executes SHM. The particle P and the circle on which it moves are sometimes referred to as the reference particle and the reference circle, respectively. We can take projection of the motion of P on any diameter, say the y -axis. In that case, the displacement y ( t ) of P ′ on the y -axis is given by y = A sin ( ω t + φ ) which is also an SHM of the same amplitude as that of the projection on x -axis, but differing by a phase of π / .
In spite of this connection between circular motion and SHM, the force acting on a particle in linear simple harmonic motion is very different from the centripetal force needed to keep a particle in uniform circular motion. u Example . The figure given below depicts two circular motions. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated in the figures.
Obtain the simple harmonic motions of the x -projection of the radius vector of the rotating particle P in each case. Answer At t = , OP makes an angle of o = π / rad with the (positive direction of) x -axis. After time t , it covers an angle in the anticlockwise sense, and makes an angle of + π t with the x -axis. The projection of OP on the x-axis at time t is given by, x (