Why Soldiers Break Step When Marching Across a Bridge

A swing, a vibrating guitar string, a vehicle's shock absorber and even the atoms in a crystal all move back and forth about a fixed point. The simplest such motion is simple harmonic motion, which also explains why pushing a swing at just the right moment sends it higher.

This lesson covers simple harmonic motion and its link with circular motion, the time periods of a pendulum and a mass-spring system, and free, damped and forced oscillations with resonance.

**Simple harmonic motion is periodic motion in which the restoring force, and so the acceleration, is proportional to the displacement from the mean position and directed towards it, $a=-{\omega }^{2}x$; it is exactly the projection of uniform circular motion onto a diameter.

Describing SHM:**

$$F=-kxa=-{\omega }^{2}xx=A\sin (\omega t+\varphi )$$

- A is the amplitude, $\omega$ the angular frequency and $\varphi$ the phase constant; the period is $T=\frac{2\pi }{\omega }$
- Velocity $v=\omega \sqrt{A^2-x^2}$ is greatest, $\omega A$, at the mean position and zero at the extremes
- Acceleration is zero at the mean position and greatest, ${\omega }^{2}A$, at the extremes

Link with circular motion. A particle P moves round a circle of radius A at constant angular speed $\omega$. Its projection N on a diameter has displacement $x=A\sin \omega t$, so N moves back and forth in SHM with the same period, and N's acceleration is the component of P's centripetal acceleration along the diameter, $-{\omega }^{2}x$.

Worked example. A particle in SHM has amplitude 0.10 m and period 2.0 s:

$$\omega =\frac{2\pi }{2.0}=3.14{\text{ rad s}}^{-1}{v}_{\max }=\omega A=0.314{\text{ m s}}^{-1}{a}_{\max }={\omega }^{2}A=0.987{\text{ m s}}^{-2}$$

At $x=0.060$ m, $v=3.14\sqrt{0.10^2-0.060^2}=0.251$ m s${}^{-1}$.

An everyday example. The piston in a scooter engine, driven by a rotating crankshaft, moves very nearly in simple harmonic motion — the projection of circular motion onto a line.

The substance. Not every periodic motion is simple harmonic — the Earth's orbit repeats, but it is not a back-and-forth motion with acceleration proportional to minus the displacement.

**For a mass-spring system the restoring force $-kx$ gives $T=2\pi \sqrt{\frac{m}{k}}$, and for a simple pendulum with small swings the restoring force $-\frac{mg}{l}x$ gives $T=2\pi \sqrt{\frac{l}{g}}$.

Mass-spring system.** A mass m on a spring of constant k, displaced by x, feels $F=-kx$, so $a=-\frac{k}{m}x$ — SHM with ${\omega }^2=\frac{k}{m}$:

$$T=2\pi \sqrt{\frac{m}{k}}$$

Simple pendulum. A bob on a string of length l, displaced by a small angle $\theta$, feels a restoring force $-mg\sin \theta$. For small angles, $\sin \theta \approx \frac{x}{l}$, so $F=-\frac{mg}{l}x$ and ${\omega }^2=\frac{g}{l}$:

$$T=2\pi \sqrt{\frac{l}{g}}$$

Worked example 1 — a spring. A 0.50 kg mass on a spring with $k=200$ N m${}^{-1}$:

$$T=2\pi \sqrt{\frac{0.50}{200}}=2\pi \times 0.050=0.314\text{ s}$$

Worked example 2 — a pendulum with a 2.0 s period. With $g=9.8$ m s${}^{-2}$, its length must be

$$l=\frac{g{T}^2}{4{\pi }^2}=\frac{9.8\times 2.0^2}{4{\pi }^2}=0.993\text{ m}$$

Springs in combination. Two identical springs in parallel act like one of constant 2k, shortening the period by a factor of $\sqrt{2}$; in series they act like $\frac{k}{2}$, lengthening it by $\sqrt{2}$.

An everyday example. The shock absorbers of an autorickshaw act as springs, so a heavily loaded rickshaw bobs up and down more slowly than an empty one.

The substance. A pendulum's period does not depend on the bob's mass, but a spring's period does — gravity's pull grows with mass, while a spring's force does not.

Free oscillations happen at a body's natural frequency with no outside force, damped oscillations lose amplitude as energy is dissipated, and forced oscillations follow an external periodic force, reaching resonance — a very large amplitude — when the driving frequency matches the natural frequency.

Free oscillations. A body displaced and released vibrates at its natural frequency, $f_0=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$, with constant amplitude if nothing removes energy.

Damped oscillations:

- Friction or air resistance removes energy, so the amplitude shrinks with time
- Light damping allows many slowly fading swings; heavy damping stops the motion quickly

Forced oscillations and resonance:

- An external periodic force makes the body oscillate at the driving frequency, not its natural one
- The amplitude grows as the driving frequency approaches the natural frequency and peaks at resonance, when they are equal
- The less the damping, the taller and sharper the resonance peak

Worked example. A 2.0 kg mass on a spring with $k=800$ N m${}^{-1}$ has natural frequency

$$f_0=\frac{1}{2\pi }\sqrt{\frac{800}{2.0}}=\frac{20}{2\pi }=3.2\text{ Hz}$$

A periodic push at about 3.2 Hz drives it into resonance, while a push at 1.0 Hz produces only a small amplitude.

An everyday example. A child on a park swing goes higher and higher when pushed in time with the swing's natural rhythm — resonance at work.

The substance. Resonance can be useful or dangerous — radio tuning relies on it, while soldiers break step on bridges so their marching never drives the structure at its natural frequency.