17.3 Damped and Forced Oscillations, Resonance

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1. What is Damping?

In an ideal world, if you set a pendulum swinging, it would swing forever. But in real life, every oscillating system eventually slows down and stops. This happens because of resistive forces — forces that act against the motion of the oscillator.

Common examples of resistive forces include:

• Air resistance — the air pushes back against a moving object
• Friction — surfaces rubbing against each other
• Fluid drag — resistance from liquids like water or oil

These resistive forces always act in the opposite direction to the motion of the oscillating object. Every time the object moves, the resistive force takes away a little energy. Over time, the object has less and less energy, so its oscillations get smaller and smaller — until it eventually comes to rest at the equilibrium position (the centre point of the oscillation).

This process — the reduction in energy and amplitude of oscillations due to resistive forces — is called damping.

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2.1 Light Damping

Light damping occurs when the resistive force is small. The oscillator still completes many oscillations back and forth, but the amplitude (the maximum displacement from equilibrium) gradually gets smaller each time.

Key features of light damping:

• The oscillator does oscillate — it keeps going back and forth many times.
• The amplitude decreases exponentially (this means it shrinks by a fixed proportion each cycle, not a fixed amount — so the decrease gets smaller and smaller over time).
• The period and frequency stay almost the same throughout, even as the amplitude drops.
• Eventually, the oscillator comes to rest at equilibrium.

Real-life example: A pendulum swinging in air. It swings back and forth many times, but each swing is slightly smaller than the last.

Displacement–time graph for light damping:

x
|  /\      /\    /\   /\
| /  \    /  \  /  \ /  \
|/    \  /    \/    X    ---
|      \/
|_________________________ t

The graph looks like a sine or cosine wave whose peaks and troughs get smaller and smaller over time. The wave shrinks within an exponential envelope — an imaginary curved boundary that follows the decreasing peaks.