Simple harmonic motion

A mass bouncing on a spring. A pendulum clock ticking once a second. A car settling after it thumps over a speed bump. A plucked guitar string. These look like four different things, but underneath they are the same motion, repeating and smooth, and once you can describe one of them you can describe all of them.

Here is a spring hanging from the ceiling with a weight on the end. Pull the weight down and let go. It does not just snap back and stop. It overshoots, rises past where it started, falls again, and keeps bouncing up and down in a steady rhythm. That back-and-forth is the thing we are going to pin down.

First, a guess

Picture a pendulum: a string with a weight tied to the bottom, swinging side to side, like a clock's. Now imagine two of them, exact same length of string, but one has a heavy metal ball and the other a light wooden one. You pull both back the same amount and release them together.

Play with the oscillator

Before any formula, get a feel for it. This is a mass on a spring. Drag the mass heavier or lighter, make the spring stiffer or softer, and change how far you pull it. Watch two numbers: the period (time for one full bounce) and the frequency (bounces per second).

Two things should jump out. Changing the pull - the amplitude - moves the mass further but does not change the period at all. And the smooth curve on the right is a perfect sine wave, the very same shape the spinning dot on the left traces out with its height. Both of those facts are the heart of this lesson.

Why it swings: a force that always points home

Stretch a spring a little and it pulls back a little. Stretch it twice as far and it pulls back twice as hard. Nothing exotic - you can feel this with any rubber band. Whatever the direction you pull, the spring pulls back the other way, toward the resting position. That is a restoring force, and we can write it in one line:

Here $x$ is how far you are from the resting spot and $k$ is the stiffness of the spring - a stiff spring has a big $k$. The minus sign is the whole story: it says the force always points opposite to the displacement, back toward the middle. That is stated from the everyday behavior of springs, not derived.

Now bring in the one law of motion we already have: a force produces an acceleration, and heavier things accelerate less for the same force, $a=F/m$ (from force, mass, acceleration). Put the two together:

Read that out loud: the acceleration is proportional to the displacement and points the opposite way. The further out you are, the harder you get yanked back. That single relationship is what simple harmonic motion means, and it is derived, not assumed.

What motion obeys that rule?

We need a position $x(t)$ whose acceleration - its second rate of change over time - is just itself flipped negative and scaled. If you have met derivatives that is a clean way to see it, but you do not need them: it is enough to know that one familiar shape does exactly this. The cosine wave. Its slope is a sine, and the slope of that is a cosine again, back where it started but negated. So the motion is

$A$ is the amplitude, how far you first pulled it. $\omega$ (omega) is the angular frequency, and it depends only on the stiffness and the mass. Plug the cosine back in and its acceleration comes out to $-{\omega }^{2}x$, which matches our rule the moment ${\omega }^2=k/m$. The cosine works, and nothing simpler does.

Notice what is missing from $\omega$: the amplitude $A$. Pull the mass twice as far and it has twice as far to travel, but the spring also pulls it twice as hard, so it moves twice as fast and arrives in the same time. The period is set by the mechanism, not by how big the swing is. That is why the pull slider in the widget never touched the period.

The period, and the pendulum surprise

One full cycle takes the dot all the way around the circle, an angle of $2\pi$, so the period is

Heavier mass sits under the square root on top, so more mass means a longer, lazier period. A stiffer spring sits underneath, so more stiffness means a quicker one. Both match what you felt in the widget.

Now back to the pendulum. For small swings a pendulum has a restoring force proportional to displacement too, so it is also simple harmonic motion, and its period turns out to be $T=2\pi \sqrt{L/g}$, set by the length $L$ and gravity $g$. The mass is nowhere in it. Why? Gravity pulls harder on a heavier bob, which would speed it up, but a heavier bob is also harder to get moving in exactly the same proportion, and the two effects cancel perfectly. So the heavy and light pendulums keep the same time, just as the probe claimed.

The sine is the shadow of a circle

The spinning dot in the widget is not decoration. Take a point moving around a circle at a steady rate and look at just its height. As the point goes round, its height rises, peaks, falls, bottoms out, and returns - tracing an exact cosine curve. Simple harmonic motion is literally the shadow of uniform circular motion cast onto one line.

The same idea, in code

A programmer's way to convince yourself: never write the cosine at all. Just encode the rule $a=-(k/m)x$ and step it forward in tiny time slices. The cosine emerges on its own. This is exactly the numerical integration idea from simulating motion.

k, m = 8.0, 0.5      # stiffness, mass
x, v = 1.0, 0.0      # start pulled to x = 1, released from rest
dt = 0.001

for step in range(20000):
    a = -(k / m) * x     # the only physics: force pulls back, F = -k x
    v = v + a * dt       # update velocity first (semi-implicit, stays stable)
    x = x + v * dt       # then position
    # x now traces A*cos(omega*t) with omega = sqrt(k/m) = 4 rad/s

Check yourself