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 fn. Some people call it the “stiffness-to-inertia ratio” ratio.

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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