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To compensate the energy loss in the system in a damped system due to the retarding forces, an external force is applied. This force acts in the direction of motion of the oscillator and does a positive work on the system. As a result of which, the amplitude of motion remains constant when the energy input per cycle exactly equals the energy lost due to damping. The system which oscillates in this manner experiences forced oscillation.
When the system executing forced oscillation behaves in such a way that its natural frequency becomes equal to the frequency of oscillation, the system is said to be in resonance. At resonance, the applied force remains in phase with the velocity so that the power transferred to the system is of maximum value.
The soldiers marching on a bridge are asked to break the steps to avoid the resonance condition.
Two Body Oscillations:
When two masses are connected with each other by strings and oscillating together then the system is called two-body oscillator. It is as shown in the figure:
The masses are placed on a frictionless track and joined up by ideal strings as shown in the figure. There are two kinds of motion which distinguish themselves by being very simple. In one kind of motion x1 and x2 remain equal and the whole system oscillates back and forth without the stretching of the middle string. In this kind of motion the frequency of oscillation is given by,
In the other kind of motion, x1 and x2 remain exactly opposite and the motion is like “in and out” type:
In this case, the net restoring force on the body is three times as compared to the previous case and thus the frequency of oscillation of the system is given by,
This simple motion of the two-body oscillator is called normal modes. They have the property that when the system starts motion in one of these modes then it will continue in that mode.
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Waves and Optics