Natural frequency and free vibration
Pull a swing back and let it go. It swings by itself, at a fixed number of swings every second. That number is its natural frequency. It depends on the object (for a pendulum: its length; for a spring: its mass and stiffness), not on how hard you pushed.
When an object vibrates by itself after one push, it is a free vibration. Friction and air slowly steal energy, so the swings get smaller. This slow loss is called damping.
New word: frequency means how many swings happen in one second. Its unit is the hertz (Hz).
Forced vibration
Now keep pushing the object with a repeating force from outside. The pusher is the driver. The object cannot ignore it. After a short time it vibrates at the driver's frequency, not at its own natural one. This is forced vibration (forced oscillation).
If the driver's frequency is far from the natural frequency, the swing stays small. A tuning fork pressed on a table makes the table vibrate at the fork's frequency: the table is forced to follow.
Resonance
Resonance happens when the driver's frequency is equal (or very close) to the natural frequency of the object. Each push then arrives at exactly the right moment and adds energy to the swing. The size of the swing (amplitude) becomes the largest.
Why so big? Each push helps the motion instead of fighting it. Small pushes add up, like pushing a swing in time.
The graph of amplitude against driver frequency has a sharp peak at the natural frequency. Less damping gives a taller, narrower peak. More damping gives a lower, wider peak. With no damping the amplitude would grow without limit; in real life friction always limits it.
Where resonance helps and where it hurts
Useful: a radio or TV tuner is an electric circuit tuned to resonate with one station's frequency. Musical instruments use resonance in their body or air column to make sound louder. An MRI scanner uses the resonance of atoms in a magnetic field.
Harmful: a bridge can sway dangerously if people march in step at its natural frequency, so soldiers break step. A washing machine shakes violently when its drum speed matches the frame's natural frequency. A wine glass can crack when a loud note matches its natural frequency. Engineers add damping or change the natural frequency to stay safe.
Try it: resonance with a bottle and a swing
Do it: blow gently across the top of an empty bottle. You hear a note: the air inside resonates. Pour in some water and blow again. The note gets higher because the air column is shorter. Next time you are on a swing, try pushing at different rhythms and see which one lifts you the most.
Predict, then check: in the 3D, guess which pendulum will swing big at 2.03 Hz. Then drag the slider and see.
Key formulas and definitions
- Frequency f = number of swings per second (unit: hertz, Hz); period T = 1/f
- Pendulum natural frequency f0 = (1/2π)√(g/L): shorter L gives larger f0
- Spring natural frequency f0 = (1/2π)√(k/m)
- Resonance when driver frequency f = natural frequency f0
- Key terms: free vibration, forced vibration, driver, damping, amplitude
Worked examples
1. A pendulum swings 20 times in 10 s. Find its natural frequency and period.
f = 20 / 10 = 2 Hz. T = 1/f = 0.5 s.
2. A bridge has a natural frequency of 1.8 Hz. Soldiers march at 2 steps per second. Is there resonance? What should they do?
Their rhythm is 2 Hz, close to 1.8 Hz, so a large sway is possible. They should break step, so no repeating push builds up.
3. A swing has a natural frequency of 0.4 Hz. How often should you push it for the biggest swing?
Push at the same frequency: 0.4 times per second, that is once every T = 1/0.4 = 2.5 s.
4. Two pendulums have lengths 1 m and 4 m. Which has the higher natural frequency? What is the ratio?
f0 is proportional to 1/√L. The 1 m pendulum has the higher frequency. Ratio f(1 m) : f(4 m) = 1/1 : 1/2 = 2 : 1.
5. A mass on a spring has f0 = 3 Hz. The mass is made 4 times larger. What is the new natural frequency?
f0 is proportional to 1/√m. Four times the mass halves the frequency: 1.5 Hz.
6. A washing machine drum spins at 10 Hz and shakes badly. The frame's natural frequency is 10 Hz. Name two ways to reduce the shake.
Change the drum speed away from 10 Hz, or change the frame's natural frequency (stiffer or heavier). Adding damping (rubber feet) also reduces the peak.
Common mistakes
- Thinking resonance means the loudest or fastest vibration. It means the driver frequency equals the natural frequency; the amplitude is large.
- Believing a forced vibration happens at the natural frequency. It happens at the driver's frequency.
- Saying a stronger push is needed for resonance. Small pushes at the right rhythm are enough.
- Forgetting damping. Without friction the amplitude would grow forever; real objects always have some damping.