One of the amazing things about a SHM is that its period is independent of the amplitude of oscillation. Instead, it is dependent on the system’s restoring force and inertia.
Take for example the vertical spring-mass system.
If we denote the extension of the spring at the equilibrium position by e, then we can encapsulate the dynamics in one N2L equation
But , so
Comparing this with the SHM equation , we can deduce that
Hence, the natural frequency is and the natural period is .
One way to interpret the relationship is that a higher k value results in a stronger restoring force (per unit displacement), and a smaller m value results in a larger acceleration (per unit displacement), and hence fn. Some people call it the “stiffness-to-inertia ratio” ratio.