Period of S.H.M.

The period T of an S.H.M. is defined as the time taken by the particle to complete exactly one full oscillation. We can derive an expression for T directly from the general displacement formula. Suppose at time t the displacement is $x = A\sin(\omega t+\phi)$ (Eq. 5.12). Now consider the displacement at a LATER time $t' = t + 2\pi/\omega$:

Since the sine function repeats identically every $2\pi$ radians (i.e. $\sin(\theta+2\pi)=\sin\theta$ for any angle $\theta$), this equals $A\sin(\omega t+\phi) = x$ exactly -- the SAME displacement as at the earlier time t. The identical argument applied to the velocity and acceleration expressions (section 5.5) shows they too return to their original values after the same time interval $2\pi/\omega$. So the particle's ENTIRE state of motion -- position, speed, and direction -- repeats after a time $2\pi/\omega$, which is therefore (by definition) the period:

Since, from section 5.4, $\omega^2 = k/m$ for a spring-block oscillator, substituting $\omega = \sqrt{k/m}$ into Eq. (5.15) gives the practically useful, directly computable formula

More generally -- and this is the form worth memorising, because it will be re-derived for every other kind of S.H.M. system in this chapter (the simple pendulum in 5.12, the torsional/angular oscillator in 5.13, the vibrating magnet in 5.13.1) simply by substituting that system's own force/torque law -- the period can always be written as …