Before simple harmonic motion gets mathematical, you need its vocabulary. An oscillation is any motion that repeats back and forth about a central point — and five key quantities (displacement, amplitude, period, frequency and angular frequency) describe it completely.
Pull a pendulum to one side and release it, and it swings through the middle, out to the far side, and back again — over and over. That repeating to-and-fro is the defining feature: the object’s displacement x keeps varying with time about one special point.
That special point is the equilibrium position, labelled x = 0. It’s the place where the object would happily sit still, because the resultant force there is zero. Displace the object either side of it and the motion always carries it back through this fixed central point.
Plotted against time, the displacement of an oscillating object traces out a wave shape — and everything you need to describe the motion can be read straight off that graph.
Displacement is the distance of the object from its equilibrium position at a given instant. It’s a vector: a positive value means the object is on one side of equilibrium, a negative value means the other. It’s measured in metres (m).
The amplitude is the largest value the displacement reaches — the distance from equilibrium out to an extreme position. The oscillation reaches this maximum on both sides, at +x0 and −x0. Amplitude is also measured in metres (m).
The period is the time taken to complete one full oscillation — for example, from one extreme, across to the other extreme, and all the way back. It’s measured in seconds (s). If every cycle takes the same time, the oscillations are described as isochronous.
The frequency counts how many complete oscillations happen each second, measured in hertz (Hz). Since one oscillation takes a time T, frequency and period are reciprocals of each other — which is also why Hz is equivalent to s⁻¹:
Angular frequency is the rate of change of angular displacement, measured in radians per second (rad s⁻¹). One complete oscillation corresponds to a full circle of 2π radians, so:
Why does a back-and-forth motion get an angular frequency? Because an oscillation can be matched, moment for moment, to an object going round a circle. Each stage of the oscillation corresponds to a fraction of one full revolution:
Adding those angles up: π/2 + π/2 + π = 2π radians for one complete oscillation. That’s exactly where the equation ω = 2π ÷ T comes from — the motion sweeps through 2π radians’ worth of cycle in a time T. It also means fractions of an oscillation translate directly into angles: half an oscillation is π radians, a quarter is π/2.
A child on a playground swing completes 0.4 oscillations every second.
(a) Calculate the time period of the motion.
(b) Calculate the angular frequency of the motion.
The arm of a metronome swings from one extreme position across to the opposite extreme in 0.60 s.
(a) State the angular displacement covered in this movement.
(b) Determine the angular frequency of the oscillation.
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