Question
The motion of particle moving in a circle with constant speed is not simple harmonic. Show that the motion of its shadow on axis parallel to diameter is simple harmonic.
Solution
Let a particle P be revolving in a circle (known as reference circle) of radius ‘r’ with constant angular velocity 
.
If T is the time taken by the particle to complete one revolution, then
Angular velocity is given by,
Since the motion of the particle Q along the shadow is represented by simple harmonic functions, hence the motion of the particle Q along the shadow is simple harmonic motion.
. If T is the time taken by the particle to complete one revolution, then
Angular velocity is given by,
Let the particle P start from S.
The angular position of the particle at any instant t is given by,
θ = ωt + ϕ
Since the motion of the particle in the circle is not to and fro, it is not a simple harmonic motion.
Let a parallel beam of light be incident on the particle P and take the shadow on Y1Y2 axis.
As the particle revolves, its shadow vibrates between K and L.
If another particle Q is allowed to move along with the shadow, it will move to and fro along the shadow.
So, position of the particle Q along Y1Y2 axis w.r.t. O is given by,
Since the motion of the particle Q along the shadow is represented by simple harmonic functions, hence the motion of the particle Q along the shadow is simple harmonic motion.
