Oscillations

Our world is full of motion that repeats: a heartbeat, a plucked guitar string, the swing of a clock pendulum, the quartz crystal that times your watch, the atoms vibrating in a crystal lattice, even the gentle sway of a tall building in the wind. Any motion that repeats itself at regular intervals is called periodic motion or oscillation. The remarkable thing physics uncovers here is unification: an astonishing range of these oscillators, mechanical and electrical alike, obey the same small set of equations. Master one — the block on a spring — and you have, in effect, mastered them all.

The frequency \(f\) of an oscillation is the number of complete cycles per second, measured in hertz (1 Hz = 1 cycle/s). The time for one full cycle is the period \(T = 1/f\). The simplest, and by far the most important, oscillation is simple harmonic motion (SHM), in which the displacement is a sinusoidal function of time:

Because the cosine repeats every \(2\pi\) radians, the motion repeats when \(\omega t\) advances by \(2\pi\). That ties the angular frequency \(\omega\) (in rad/s) to the period and frequency:

Differentiate the displacement once for velocity and again for acceleration. Each derivative of a cosine brings down a factor of \(\omega\) and shifts the phase:

Apply Newton's second law to \(a = -\omega^{2}x\). The force needed is \(F = ma = -m\omega^{2}x\) — a restoring force proportional to displacement and pointing back toward equilibrium. That is exactly Hooke's law for a spring, \(F = -kx\), provided we identify the spring constant with \(k = m\omega^{2}\):

As the block oscillates, energy shuttles back and forth between the spring's potential store and the block's kinetic store. Using \(x(t)\) and \(v(t)\) from above:

The "springiness" need not be a spring. A disk hung from a wire and twisted through an angle \(\theta\) feels a restoring torque proportional to the twist — the angular form of Hooke's law, \(\tau = -\kappa\theta\), where \(\kappa\) (kappa) is the torsion constant of the wire. This is a torsion pendulum. Replacing stiffness \(k \to \kappa\) and inertia \(m \to I\) in the spring result gives the period directly:

In a pendulum it is gravity, not a spring, that supplies the restoring torque. For a simple pendulum — a bob of mass \(m\) on a massless string of length \(L\) — the restoring torque for small angles is proportional to the angular displacement, so the motion is approximately SHM:

A physical pendulum is any rigid body swinging about a fixed pivot. With rotational inertia \(I\) about the pivot and the pivot a distance \(h\) from the centre of mass, the small-angle period is:

There is a beautiful geometric picture that ties SHM to circular motion. Imagine a reference particle moving steadily around a circle of radius \(x_{m}\) at angular speed \(\omega\). Its shadow — the projection of its position onto a diameter — moves back and forth in exactly simple harmonic motion:

Real oscillators lose energy to friction or air drag, so their amplitude decays — this is damped motion. For a block on a spring also subject to a drag force \(-bv\) proportional to speed (with damping constant \(b\)), the solution is a cosine whose amplitude shrinks exponentially:

Push an oscillator with an external periodic force at driving angular frequency \(\omega_{d}\) and it settles into oscillation at \(\omega_{d}\), not at its own natural frequency \(\omega\). The amplitude of the response depends on how close the two are:

Most problems reduce to identifying the effective stiffness and inertia, then using \(T = 2\pi\sqrt{\text{inertia}/\text{stiffness}}\), or to reading amplitude and frequency off \(x = x_{m}\cos(\omega t + \phi)\) and differentiating. Useful relations: \(\omega = 2\pi f = 2\pi/T\), \(v_{m} = \omega x_{m}\), \(a_{m} = \omega^{2}x_{m}\), \(E = \tfrac{1}{2}kx_{m}^{2}\), \(g = 9.8\,\mathrm{m/s^{2}}\).