Oscillations

Oscillatory motion is periodic motion that repeats itself in equal intervals of time, and simple harmonic motion is the special case the whole chapter is built on. Its defining feature is a restoring force that is proportional to the displacement and always directed back towards the mean position. In symbols the same statement is written $a \propto -x$, or with the constant named, $a = -\omega^{2}x$, where $\omega$ is the angular frequency measured in radians per second. The negative sign is not decoration: it is what turns an outward push into a back-and-forth motion, and it is the detail the examiner most often builds a question on.

From that single condition the shape of the motion follows. The particle moves fastest as it passes through the mean position, where the displacement $x$ is zero, and it comes momentarily to rest at the two extremes, where the displacement equals the amplitude. Acceleration behaves the opposite way round, because $a = -\omega^{2}x$ makes it zero at the mean position and largest at the extremes, where the restoring force is strongest. So the restoring force is minimum at the mean position and maximum at the extreme positions, and these two readings mirror each other, which is why a question can ask about either end of the swing and expect the same idea.

The amplitude is the maximum displacement of the body on either side of its equilibrium position, and it is what a displacement-time graph is read for. Time period and frequency are locked together by $T = 1/f$, and both connect to the angular frequency through $\omega = 2\pi f = 2\pi/T$. One consequence is easy to miss: the period of a vibrating body does not depend on the amplitude, so a wider swing and a narrow swing of the same oscillator take the same time. A body performing simple harmonic motion also does not settle. In the absence of resistive forces the oscillations never stop, because the total energy is conserved and is only handed back and forth between its kinetic and potential forms.