What are the conditions for resonance in a system?

Resonance in a system occurs when the frequency of an external force matches the natural frequency of the system.

Resonance is a phenomenon that occurs in many different physical systems, from musical instruments to electrical circuits, to mechanical systems. It is a condition where an external force or a vibrating system forces another system to oscillate with greater amplitude at a specific frequency. This specific frequency is known as the resonant frequency or natural frequency of the system.

The first condition for resonance to occur is that the system must have the ability to oscillate. This means it must have some kind of restoring force that pushes it back towards its equilibrium position when it is displaced. For example, in a pendulum, this restoring force is provided by gravity. In an electrical circuit, it could be provided by a capacitor or inductor.

The second condition is that there must be some form of damping or resistance in the system. This is because, in a perfectly undamped system, the amplitude of the oscillations would increase indefinitely, which is not physically possible. The damping force can come from various sources, such as friction in a mechanical system or resistance in an electrical circuit.

The third condition is that an external force must be applied to the system at the same frequency as the system's natural frequency. When this happens, the system absorbs energy from the external force and the amplitude of its oscillations increases. This is the essence of resonance - the system is 'in tune' with the external force, leading to large oscillations.

In summary, for resonance to occur in a system, the system must have a natural frequency, there must be some form of damping, and an external force must be applied at the same frequency as the natural frequency. When these conditions are met, the system will oscillate with a large amplitude, which is the hallmark of resonance.