UNIFORM CIRCULAR MOTION AND SHM

 Consider a particle moving with a constant angular speed ω0 in an anticlockwise direction on a circular path having center O and radius A as shown in the figure. 

At time t = 0, its angular position w.r.t. the reference line OX is ∠ POX = φ. 

At time t = t, having undergone angular displacement w₀t reaching Q from P, its angular position is.

QOX = w₀t + φ

The co-ordinates of point Q are 

x = A cos ( w₀t + φ ) and … … … ( 1 )

y = A sin ( w0t + φ). … … … …   ( 2 )

As the particle moves on the circular path, its feet of perpendiculars on X- and Y- axes move as per the equations ( 1 ) and ( 2 ) and their motion is simple harmonic.

Thus, a given SHM can be described as the projected motion of a particle, known as the reference particle, performing an appropriate uniform circular motion on the diameter of the circle known as the reference circle. The radius of the reference circle is equal to the amplitude of the corresponding SHO and the angular speed of the reference particle is equal to the angular frequency of the SHO. Also, the angular position of the reference particle w.r.t. the reference line at any time is equal to the phase of the SHO at that time.

Combining two SHMs with phase difference of p / 2 and same amplitude results in uniform circular motion and if the amplitudes are different, the motion is on an elliptical path. Combining SHMs in different ways, different types of motion can be obtained.