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The damped and forced oscillations can be best understood through a RLC circuit. A typical RLC series circuit is shown below:
Let the capacitor has initial charge Qmax before the switch is closed. When the switch is closed, the total energy stored in the capacitor and inductor at any time will be governed by the differential equation:
Here the –ve sign implies that the energy is decreasing with time. The above equation can further be modified by considering the energy stored due to the inductor and capacitor both so that the equation will be:
The solution of this equation will be as follows:
(i) When R=0, the circuit becomes LC oscillator circuit
(ii) When R is very small then the solution of the equation is  , where  is the angular frequency with which the circuit undergoes damped oscillation.
(iii) When R is large oscillation damp out more rapidly and when R ≥ Rc, no oscillation takes place, where Rc is the critical value of resistance and is given by,  . In the situation when R = Rc the system is critically damped and when R > Rc, the system is over damped.
The damped oscillation is as shown above for a RLC circuit. When an external force is applied to the system to maintain the oscillation in the system, the type of oscillation is then called as forced oscillation.
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