Push a swing at the right moment of each cycle and small pushes build a big swing. Damped Oscillations gives the losses that stop this growth. You can push a swing in rhythm in the Pendulums and Resonance lab.
Why it works
Add a driving force: $m\ddot{x} + b\dot{x} + kx = F_0\cos\omega t$. After the starting transient dies away (it decays like the damped motion), the system settles into motion at the driving frequency: $x = A\cos(\omega t - \delta)$. Substituting and matching the cosine and sine terms gives $$A(\omega) = \dfrac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (b\omega/m)^2}}, \qquad \omega_0 = \sqrt{k/m}.$$ When $\omega$ is near $\omega_0$, the first term in the root nearly vanishes, and only damping keeps the amplitude finite.
Key idea
Driven near its natural frequency, a lightly damped oscillator responds strongly: at $\omega = \omega_0$, $A = \dfrac{F_0}{b\omega_0}$. Far below, $A \approx F_0/k$ (just the static stretch); far above, the response is small. The ratio of the resonant amplitude to the static one is $Q = \dfrac{m\omega_0}{b}$: less damping, sharper and taller peak.
Worked example: At resonance
A $1\ \text{kg}$ mass on a spring ($k = 100\ \text{N/m}$, $b = 1\ \text{N·s/m}$) is driven by $F_0\cos\omega t$ with $F_0 = 2\ \text{N}$ at $\omega = \omega_0$. Find the steady amplitude and compare it with the static stretch.
- $\omega_0 = 10\ \text{rad/s}$, so $A = \dfrac{F_0}{b\omega_0} = \dfrac{2}{1(10)} = 0.20\ \text{m}$.
- Static stretch: $F_0/k = 0.02\ \text{m}$. Resonance multiplies it by $Q = m\omega_0/b = 10$.
Answer: $0.20\ \text{m}$, ten times the static stretch.
Worked example: Off resonance
Same system, driven at $\omega = 8\ \text{rad/s}$. Find the amplitude.
- $\omega_0^2 - \omega^2 = 100 - 64 = 36$ and $b\omega/m = 8$.
- $A = \dfrac{2}{\sqrt{36^2 + 8^2}} = \dfrac{2}{36.9} = 0.054\ \text{m}$, about a quarter of the resonant value.
Answer: About $0.054\ \text{m}$.
Common mistake
Thinking any repeated push in rhythm makes the amplitude grow without limit. Each push adds energy, but a bigger swing also loses more energy per cycle to friction and air; the amplitude levels off where the two balance. Check the formula: at $\omega = \omega_0$ the amplitude is $F_0/(b\omega_0)$, finite for any real damping $b > 0$.
The timing matters as much as the size. Driven slowly, the mass moves in step with the force; driven fast, it moves opposite to it; at resonance the displacement lags the force by a quarter cycle, so the force is in step with the velocity. That is exactly when the force does the most positive work each cycle, as when you push a swing while it moves away from you.
Resonance is useful (tuning a radio, a child on a swing) and sometimes a hazard, which is why engineers add damping to buildings and bridges and avoid natural frequencies near common driving frequencies, such as footsteps or machinery.