occurring at ½ instead of zero. ⊳

. SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION In this section, we show that the projection of uniform circular motion on a diameter of the circle follows simple harmonic motion. A simple experiment (Fig. .

) helps us visualise this connection. Tie a ball to the end of a string and make it move in a horizontal plane about a fixed point with a constant angular speed. The ball would then perform a uniform circular motion in the horizontal plane. Observe the ball sideways or from the front, fixing your attention in the plane of motion.

The ball will appear to execute to and fro motion along a horizontal line with the point of rotation as the midpoint. You could alternatively observe the shadow of the ball on a wall which is perpendicular to the plane of the circle. In this process what we are observing is the motion of the ball on a diameter of the circle normal to the direction of viewing. Fig.

. Circular motion of a ball in a plane viewed edge-on is SHM. Fig. .

Plots of Eq. ( . ) for φ = for two different periods. u Example .

Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic? Give the period for each case. ( ) sin ω t – cos ω t ( ) sin ω t Answer sin ω t – cos ω t = sin ω t – sin ( π / – ω t ) = cos ( π / ) sin ( ω t – π / ) = √ sin ( ω t – π / ) Fig. .

describes the same situation mathematically. Suppose a particle P is moving uniformly on a circle of radius A with angular speed ω . The sense of rotation is anticlockwise. The initial position vector of the particle, i.e., the vector OP at t = makes an angle of φ with the positive direction of x -axis.

In time t , it will cover a further 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 ( t ) = A cos + T t