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 *f*_{n}. Some people call it the “stiffness-to-inertia ratio” ratio.

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**Demonstrations**

Spring-Mass and Pendulum

**Video Explanation**

Derivation of Natural Frequency Formula for Spring-Mass System

**Interesting**

Lissajous Figures in the Sand

Lissajous Figures on CRO

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