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Simple harmonic motion along a straight line can be represented by the projection of uniform circular motion along a diameter of a reference circle.
The projection of the motion of a particle P along y-axis implies that the particle P also exhibits simple harmonic motion. Therefore, the uniform circular motion can be considered as a combination of two simple harmonic motions, one along the x-axis and the other along y-axis, with the two differing in phase 90o.
Damped Harmonic Motion:
In real systems, the dissipative forces retard the oscillatory motion. Consequently, the mechanical energy of the system diminishes with time. Thus, the oscillatory motion of the system is damped. In a damped oscillatory motion following points must be remembered:
1- When the retarding force is much smaller than the restoring force, the oscillatory character of the motion is preserved but the amplitude decreases in time, with the result that the motion ultimately ceases. Any system that behaves in this manner is called a damped oscillator.
2- During the damped oscillatory motion, the amplitude decays exponentially with time.
3- In the absence of retarding force, the system oscillates with its natural frequency.
4- If the system is so viscous that the retarding force is greater than the restoring force then the system is over damped.
5- Irrespective of the case whether the system is over damped or under damped, the friction is present and the energy of the oscillator eventually falls to zero. The lost mechanical energy dissipates into internal energy in the retarding medium.
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Waves and Optics