Free and forced oscillations, resonance and damping

Physics · Oscillations. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A body that is displaced and then left alone makes free oscillations at its own natural frequency. If an outside periodic force keeps pushing it, the oscillations are forced, and the body vibrates at the frequency of the driving force, not at its own.

Resonance happens when the driving frequency equals the natural frequency. Energy is then passed to the body most effectively and the amplitude becomes very large. Tuning a radio and heating food in a microwave oven use resonance.

In real systems friction takes energy away, so the amplitude falls with time. This is damping. More damping gives a smaller amplitude at resonance and a flatter, broader resonance curve.

Rules to remember

free oscillation: no outside periodic force; frequency = natural frequency
forced oscillation: frequency = frequency of the driving force
resonance: driving frequency = natural frequency, giving the largest amplitude
examples of resonance: pushing a swing in time with its motion, tuning a radio, a microwave oven, soldiers marching in step on a bridge
damping: energy is dissipated by friction, so the amplitude decreases with time
light damping gives a tall, sharp resonance curve; heavy damping gives a low, broad one
shock absorbers in a car use damping to stop the body oscillating for long after a bump

Common mistake

In a forced oscillation the body does not vibrate at its natural frequency or at an average of the two. Once the motion is steady it vibrates at the driving frequency.

Worked example

Four pendulums of lengths 20 cm, 40 cm, 60 cm and 80 cm hang from the same stretched string. A fifth pendulum of length 60 cm, with a heavy bob, hangs from the same string and is set swinging. Which of the four swings with the largest amplitude?

The heavy pendulum is the driver. It forces the others to oscillate at its own frequency.
The natural frequency of a pendulum depends on its length, so only the 60 cm pendulum has the same natural frequency as the driver.
For that pendulum the driving frequency equals the natural frequency, so it is in resonance.

Answer: The 60 cm pendulum; the others swing with much smaller amplitudes

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