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