natural frequency mass-spring

Drag the mass, pull it aside, and let go, and the moment you release it the mass starts to vibrate on its own. The graph below shows how that displacement changes with time, a smooth wave. That is simple harmonic motion. Pull it far and let go, then pull it a little and let go, and the height of the wave changes but the spacing of one swell stays the same. How hard you shake it cannot change the beat. The system itself sets the beat, so we call it the inborn frequency. The height of the wave and the spacing of the wave are different stories. One is the hand and the other is the system. Release it far or near and the clock of the wave does not care.

Now leave the mass as it is and change only the spring. k is the stiffness of the spring, how stiff it is. Raise k and the vibration gets noticeably faster. A stiff spring pulls the mass back toward equilibrium harder, so the same weight is returned more strongly and makes the round trip faster. A hard sports-car spring bounces quick, and a soft spring sways slow. The stronger the restoring force, the faster the beat. That is the first rule. The stronger the force that returns it, the shorter the round trip. The soft side stretches the time and the stiff side shortens it. A sports car and a soft seat are the two ends of this rule.

Now the other way. Leave the spring and raise only the mass. The bigger the block the heavier the mass, and when you raise m the vibration gets slower. A heavy thing has large inertia. Once it moves it hates to stop, and once it is stopped it hates to start. So even when the same spring pulls, a heavy mass answers sluggishly and one round trip takes longer. A large bell rings heavier and slower than a small bell. The larger the mass, the slower the beat. That is the second rule, and it acts in the opposite direction from stiffness. A mass with large inertia answers late to the same force. Starting and stopping both get heavier. A large bell takes its time because it hates to turn around.

Tie the two rules into one formula and the natural angular frequency is ωₙ = √(km). The stiffness k sits in the numerator and the mass m in the denominator. So when k grows the frequency grows, and when m grows the frequency shrinks, just as we saw. The square root matters. Change the ratio km of k and m and the curve has the shape of a square root, steep at first and then gentler. Raise the stiffness four times and the frequency only gets twice as fast, so to make the beat twice as fast you have to make the spring four times stiffer. Once you know the frequency, the period T = 2πωₙ and the frequency f = ωₙ2π follow at once. The square root grows the beat more slowly than the stiffness. So a small change of the spring moves the beat by less than that change. Four times the stiffness buys only twice the beat.

The same principle works outside a spring. A pendulum is a little special. Its natural frequency is ωₙ = √(gL), and only the length L of the string enters. The weight of the bob does not enter at all. So a heavy bob and a light bob swing at the same beat if the string length is the same. That is what Galileo noticed in a cathedral chandelier. A tall building is like a huge pendulum. The taller it is the softer and heavier it is, and it sways slowly, which is why the top of a skyscraper sways lazily in the wind. A tuning fork is the opposite. Short, stiff, light tines tremble hundreds of times in 1 second, so fast that the ear hears a steady tone. Switch the three with a button and you see the same story, that the balance of stiffness and mass sets the beat. Only the length of the string sets the beat, and the weight of the bob drops out. A building and a tuning fork are the two ends of that balance. The same formula, with a different pair in the fraction, still sets the clock.