Every pendulum is “born” with a natural frequency. This frequency (and period) is not affected by the amplitude of oscillation.

The natural frequency of a spring-mass system depend on the stiffness (the spring constant of the spring) and the inertia (the mass of the mass) of the system.

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The formula for the spring-mass system can be derived in just a few lines from N2L.

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The natural frequency of a pendulum is dependent only on its length. (The acceleration of free fall is the other factor, but we usually cannot choose the value of g.)

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Very intricate patterns can be traced out by a “double pendulum”. See the pendulum showing off its artistic streak below.

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The CRO (cathode ray oscilloscope) goes artistic too.

The restoring force keeps the mass accelerating towards the equilibrium position. The inertia (of the mass) causes the mass to always overshoot the equilibrium position. Hence the eternal karma of oscillations.