Period, Frequency, and Amplitude

Period (T) is how long one full cycle takes, measured in seconds. Frequency (f) is how many cycles happen per second, measured in hertz. Because one is "time per cycle" and the other is "cycles per time," they're reciprocals: f = 1/T and T = 1/f. A pendulum that takes 2 seconds per swing-cycle has a frequency of 0.5 Hz. Double the period and you halve the frequency, every time, no exceptions.

Here's the part that feels wrong at first: pull a pendulum back further, or stretch a spring more, and it does NOT take longer to complete a cycle. That's because simple harmonic motion comes from a restoring force proportional to displacement — pull back twice as far, and the restoring force (and acceleration) is twice as strong too. The extra distance the mass has to cover is exactly compensated by the extra speed and force it gets. Those two effects cancel perfectly, so the round-trip time stays fixed. Amplitude only sets how far the oscillator travels and how fast it moves at top speed — not how long the trip takes.

This independence from amplitude is what makes a force "simple harmonic" rather than just oscillating. It connects directly to the energy picture next: a bigger amplitude means more total mechanical energy sloshing between kinetic and potential, but that energy swap still completes on the same clock. Whether you're looking at a mass on a spring or a pendulum at small angles, the timing is set by mass and stiffness (or length and gravity), never by how far you pulled it.