Oscillations: Free and Forced Oscillations

Free Oscillations

Free oscillations occur when an object vibrates at its natural or resonant frequency in the absence of any external forces.
In this type of oscillation, the system oscillates free from external influences after an initial displacement.
Examples of free oscillation include a pendulum swinging back and forth or the vibrations of a guitar string when it is plucked.
The amplitude of free oscillations tends to decrease over time, this phenomenon is known as damping.

Damping in Free Oscillations

Light damping allows the system to continue oscillating with decreasing amplitude.
Critical damping returns the system to its equilibrium position in the shortest time without any oscillations.
Heavy damping returns the system to equilibrium position very slowly, without oscillating.
Oscillatory damping is the case where the system oscillates, but not around the equilibrium position.

Forced Oscillations

In forced oscillations, an external periodic force is applied to a system, causing it to oscillate at the frequency of the applied force.
The frequency of the forced oscillation is known as the driving frequency.
In general, the amplitude of the forced oscillation increases with the amplitude of the forcing function, but is also dependent on the relative frequencies of the natural and driving systems.

Resonance

Resonance can occur in forced oscillations, where the frequency of the external force equals the natural frequency of the system.
At resonance, the amplitude of the oscillation reaches a maximum, and the system absorbs energy from the oscillating force at the fastest rate.
Resonance can be beneficial (as in musical instruments) or potentially destructive (as in collapsing bridges or buildings).

Remember to understand the principles behind these concepts and how to apply them in calculations and solving problems. Keep practicing past questions and examples to strengthen your understanding and application of these topics.