Simple Harmonic Motion (S.H.M.) and Its Equation

SIMPLE HARMONIC MOTION is defined as an oscillatory motion in which the restoring force -- the force that always acts to pull the system back towards its mean (equilibrium) position -- is directly proportional to the displacement of the system from that mean position, and is always directed opposite to the displacement. Written as an equation,\n\n$F = -kx$\n\nwhere $x$ is the displacement measured from the mean position and $k$ is a positive constant (for a spring, $k$ is simply the spring's own force constant, taken up in detail in Section 13.7; for any other S.H.M.-producing system, $k$ plays exactly the same mathematical role). The negative sign is essential: it shows that whichever way the system has been displaced, the force always points back the other way, towards the mean position.\n\nApplying Newton's second law, $F = m\,d^2x/dt^2$, to this restoring force gives the DEFINING DIFFERENTIAL EQUATION of S.H.M.:\n\n$m\frac{d^2x}{dt^2} = -kx \quad\Longrightarrow\quad \frac{d^2x}{dt^2} + \omega^2 x = 0, \qquad \text{where } \omega^2 = \frac{k}{m}$\n\nThe general solution of this second-order differential equation -- which can be verified directly by substitution, differentiating twice and checking it satisfies the equation -- is\n\n$x(t) = A\cos(\omega t + \phi)$\n\nHere: $A$, the AMPLITUDE, is the maximum magnitude of displacement, reached whenever $\cos(\omega t+\phi) = \pm 1$; $\omega$, the ANGULAR FREQUENCY, is fixed entirely by the physical properties of the oscillating system itself, through $\omega=\sqrt{k/m}$, and does NOT depend at all on how the motion happened to be started; and $(\omega t+\phi)$, the PHASE of the motion at time $t$, together with $\phi$, the PHASE CONSTANT (or initial phase), which is fixed by the particle's own displacement and velocity at the instant $t=0$, and is discussed further in Section 13.5.\n\nBecause $\omega$ depends only on the system's own fixed properties ($k$ and $m$), the resulting period\n\n$T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$\n\ndoes NOT depend on the amplitude $A$ at all -- a small oscillation and a much larger oscillation of the very same …