Oscillations: Springs, Pendulums and Harmonic Motion

Pull on a spring and it pulls back. Stretch it twice as far and it pulls back twice as hard. Robert Hooke described this in 1678:

F = −kx

Here x is how far the spring is stretched or squeezed from its natural length, k is the spring constant and F is the force the spring exerts. The minus sign means the force points opposite the displacement: stretch it down and it pulls up, squeeze it and it pushes out.

Hooke's law holds only up to a point. Stretch a spring too far and it bends out of shape for good and stops following the rule.

Any repeating motion is described by three numbers. The amplitude A says how big the swings are. The period T says how long one full cycle takes: out, back, out the other way and back again. The frequency f says how many cycles happen each second:

f = 1 / T

A heart beating 72 times a minute has a frequency of 1.2 Hz and a period of about 0.83 s. When the restoring force follows Hooke's law, the motion is simple harmonic motion, and the period doesn't depend on the amplitude. Big swings and small swings take the same time.

A mass m bouncing on a spring of stiffness k has a period of

T = 2π√(m/k)

A heavier mass is harder to speed up and slow down, so it oscillates more slowly. A stiffer spring pulls back harder, so it oscillates faster. Quadrupling the mass doubles the period. The formula holds for a mass hanging vertically too: gravity only shifts where equilibrium is, not how fast it bounces.

A simple pendulum is a small heavy bob on a light string of length L. Gravity pulls it back toward the bottom of its swing. For small swings, up to about 15°, its period is

T = 2π√(L/g)

The mass of the bob isn't in the formula. A heavier bob feels a bigger pull, but it's also harder to move, and the two effects cancel. Only the length and the strength of gravity matter. A pendulum 0.99 m long has a period of almost exactly 2 s on Earth, and on the Moon, where g is about 1.62 N/kg, the same pendulum would take about 4.9 s.

For bigger swings the period grows a little, which is why pendulum clocks use small arcs.

During each cycle, energy changes form back and forth. At the ends of the motion the object stops for an instant, so all the energy is stored: as elastic potential energy in a spring or gravitational potential energy in a pendulum. At equilibrium the object moves fastest, so the energy is all kinetic.

For a spring, the stored energy at displacement x is ½kx², so the total energy is

E = ½kA² = ½mv_max²

Doubling the amplitude gives four times the energy. Without friction the total never changes, so the swing would go on forever.

Real oscillations lose energy to friction and air resistance, so their amplitude shrinks. This is damping. Sometimes it's what we want: a car's shock absorbers damp the springs so the car doesn't keep bouncing after a bump.

Every oscillator has a natural frequency, the one it settles into on its own. If you push it with a repeating force at that same frequency, each push adds energy at the right moment and the amplitude grows. This is resonance. It's how you pump a swing higher with small, well-timed pushes, and how a singer can make a wine glass ring.

Resonance can also cause trouble. When London's Millennium Bridge opened in June 2000, the bridge's slight sideways sway made walkers fall into step with it, which pushed it further, so it was closed after two days. It reopened in 2002 with dampers fitted.

The steady beat of a pendulum made it the first really accurate way to keep time, and the study of springs grew alongside it. Together they set up the mathematics of oscillation used across physics today.