Stiffness and Mass Set the Natural Frequency

Drag the weight to one side and let go. As soon as you release it, the weight starts vibrating on its own. The graph below shows how its displacement changes over time: a smooth, rolling wave. This is simple harmonic motion. Now try one experiment. Pull it far and release, then pull it just a little and release. The height of the wave (the amplitude) changes, but the spacing between one swing and the next stays the same. In other words, how hard you shake it cannot change the beat. The system itself sets the beat, which is why we call it the natural frequency, the one it is born with.

Now keep the mass the same and change only the spring. Here k is the stiffness of the spring, how stiff it is. Raise k and the vibration speeds up noticeably. A stiffer spring pulls the weight back toward its resting position more forcefully. For the same weight, a stronger pull back means a faster round trip. Think of a car's suspension. A hard sports-car spring bounces quick and tight, while a soft spring wallows slowly. The stronger the restoring pull, the faster the beat. That is the first rule.

This time the other way around: keep the spring the same and increase only the mass. The bigger the block, the heavier the weight. Raise m and the vibration grows slower and slower. A heavy thing has a lot of inertia. Once it moves it resists stopping, and once it is still it resists starting. So even with the same spring pulling, a heavy weight responds sluggishly and takes longer for one round trip. It is like a big bell ringing deeper and slower than a small one. The greater the mass, the slower the beat. That is the second rule, and it works in exactly the opposite direction from stiffness.

Tie the two rules into one expression and you get the natural angular frequency ωₙ = √(km). Stiffness k sits on top (the numerator), mass m sits on the bottom (the denominator). So when k grows, ωₙ grows; when m grows, ωₙ shrinks, exactly what we just saw. One more thing matters: there is a square root over it all. Move the k and m sliders to change the ratio km and the curve has a √ shape, steep at first and gentler as it goes. That means quadrupling the stiffness makes the frequency only twice as fast. To double the beat, you must make the spring four times stiffer. Once you know the frequency, the period T = 2πωₙ (the time for one round trip) and the frequency f = ωₙ2π (cycles per second) follow at once.

The same principle works just as well beyond springs. A pendulum is a little special. Its natural frequency is ωₙ = √(gL), with only the length L in it and the weight of the bob nowhere to be found. So a heavy bob and a light bob swing at the same beat as long as the string is the same length. This is exactly what Galileo noticed watching a cathedral chandelier. A tall building is like a giant pendulum: the taller it is, the softer and heavier it is, so it sways slowly. That is why the top of a skyscraper drifts back and forth in the wind once every few seconds. A tuning fork is the opposite. Its short, stiff, light prongs vibrate very fast, hundreds of times a second, so our ears hear a steady pitch. Switch between the three with the buttons and you will see the same story: the balance of stiffness and mass sets the beat.