understand and use the terms light, critical and heavy damping and sketch displacement–time graphs illustrating these types of damping

Damped & Forced Oscillations – A‑Level Physics 9702

1. What is Damping?

When a system that can oscillate (like a pendulum or a spring) loses energy over time, its motion gradually slows. This loss of energy is called damping and is caused by friction, air resistance, or any other “drag” force that opposes motion.

Mathematically, the equation of motion for a damped harmonic oscillator is:

\$m\ddot{x} + c\dot{x} + kx = 0\$

where m is mass, c is the damping coefficient, k is the spring constant, and x is displacement.

2. Types of Damping

Depending on the value of the damping coefficient c, we classify damping into three categories:

  1. Light (Under‑damped) – c is small. The system oscillates many times, but the amplitude gradually decreases. Think of a tuned guitar string that keeps ringing for a while.

  2. Critical – c is just right. The system returns to equilibrium as quickly as possible without oscillating. Imagine a door that stops instantly when you push it but doesn’t bounce back.

  3. Heavy (Over‑damped) – c is large. The system returns to equilibrium slowly and never oscillates. Picture a car’s shock absorber that takes a long time to settle after hitting a bump.

3. Sketching Displacement–Time Graphs

Below are simple, emoji‑style sketches of the three damping types. The red line represents displacement x(t), the blue line is time t.

  • Light Damping: 🔴⚡⚡⚡⚡ (many oscillations, decreasing amplitude)

  • Critical: 🔴⬇️ (single sharp drop to zero)

  • Heavy: 🔴⬇️⬇️⬇️ (slow, no oscillations)

In a real graph, the light‑damped curve would look like a decaying sine wave, the critical curve a single exponential decay, and the heavy curve a slow exponential approach to zero.

4. Forced Oscillations & Resonance

When an external periodic force F(t) = F_0 \sin(\omega t) drives the system, the equation becomes:

\$m\ddot{x} + c\dot{x} + kx = F_0 \sin(\omega t)\$

Key points:

  • The system can reach a steady‑state oscillation at the driving frequency \omega.

  • If \omega matches the system’s natural frequency \omega_0 = \sqrt{k/m}, the amplitude can become very large – this is resonance (🎯).

  • With damping, the maximum amplitude occurs slightly below \omega_0, and the peak is finite.

Analogy: Think of a child on a swing. Pushing at the right rhythm (matching the swing’s natural frequency) makes the swing go higher. Pushing too early or too late (different \omega) makes it harder to increase height.

5. Summary Table

Damping TypeCharacteristicGraph Shape
Light (Under‑damped)Oscillates, amplitude ↓Decaying sine wave 🔴⚡⚡⚡
CriticalFastest return, no oscillationSingle exponential drop 🔴⬇️
Heavy (Over‑damped)Slow return, no oscillationSlow exponential approach 🔴⬇️⬇️⬇️

6. Quick Quiz

  1. What happens to the amplitude of a lightly damped oscillator over time? (Answer: It decreases gradually.)

  2. Which damping type returns to equilibrium fastest without oscillating? (Answer: Critical damping.)

  3. When does resonance occur in a forced oscillator? (Answer: When the driving frequency matches the natural frequency.)

Great job! Keep practicing by drawing your own displacement‑time graphs and experimenting with different damping coefficients in a simulation or a simple spring‑mass setup. 🎓🧪