8.2.2 Natural Frequency

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

$\displaystyle \begin{aligned}({{F}_{{net}}}&=ma)\\mg-(ke+kx)&=ma\end{aligned}$

But $\displaystyle mg=ke$ , so

$\displaystyle \begin{aligned}-kx&=ma\\a&=-\frac{k}{m}x\end{aligned}$

Comparing this with the SHM equation $\displaystyle a=-{{\omega }^{2}}x$ , we can deduce that

$\displaystyle {{\omega }^{2}}=\frac{k}{m}$

Hence, the natural frequency is $\displaystyle {{f}_{n}}=\frac{1}{{2\pi }}\sqrt{{\frac{k}{m}}}$ and the natural period is $\displaystyle {{T}_{n}}=2\pi \sqrt{{\frac{m}{k}}}$ .

One way to interpret the $\displaystyle {{\omega }^{2}}=\frac{k}{m}$ 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