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.

Let a particle P be revolving in a circle (known as reference circle) of radius ‘r’ with constant angular velocity co. If T is the time taken by the particle to complete one revolution, then

Let the particle P start from S. The angular position of the particle at any instant t is given by

θ = ωt + ϕ

 

The motion of the particle P in a circle is not to and fro, therefore not simple harmonic motion. Let incident a parallel beam of light on the particle Pand 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 and the 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.

The position of the particle executing simple harmonic motion is given by

Let the position of particle be +8 cm at t₁ and -8cm at t_2. 

Therefore the time taken by particle to move between two points 8cm on either side of mean position is,

The minimum time is 0.4s.

Position of particle is given by, 

y = A = cos + B sin               ...(1)

The velocity of particle is given by, 

  

Acceleration of particle is given by,



Since the acceleration of particle is directly proportional to displacement and directed towards mean position, therefore the motion is simple harmonic motion.

Now, 



Squaring (2) and (3) and adding, we have

  

Dividing (2) (3), we have

 



Epoch = 

Let at any instant, the particle be at P at a distance y from mean position and v be the velocity of particle at P.

Kinetic energy: The kinetic energy of particle executing simple harmonic oscillation at any instant is given by



The velocity of particle at a distance y from the mean position is,



Potential energy: The potential energy stored in the particle is equal to the work done in displacing the particle from mean position to y. Let the particle be displaced through a distance x from mean position. The restoring force F acting on particle is,



Therefore, work done against this restoring force in moving the particle through a small distance dx is,



Therefore, total work done by restoring force in displacing the particle from mean position to P is,


The potential energy stored in the body is equal in magnitude and opposite in sign of the work done by restoring force. Thus


Total energy is,

E = K + U



It is clear from the above equation that total energy is independent of the position of particle during its motion. Thus, total energy is constant.