How Pendulum Physics Works

A pendulum is a mass on a rod or string, swinging under gravity. It is the oldest precision instrument in history and one of the tidiest pieces of physics there is, because its behaviour depends on almost nothing.

The result that surprises people is that the mass of the bob does not matter at all. A heavy weight and a light one on identical rods swing at identical rates. This guide explains why, what the period actually depends on, where the famous formula stops being accurate, and how games turn all of it into timing puzzles.

The period and what it depends on

The period of a pendulum is how long one complete swing takes — out and back. For small swings it is given by one of the best-known formulas in physics:

T = 2 * pi * sqrt(L / g)

where L is the length of the rod and g is gravity. Two things are notable about what is not in that equation.

• Mass is absent. The weight of the bob has no effect whatsoever on the period.
• Amplitude is absent. For small swings, a wide swing takes the same time as a narrow one.

Length is the only thing you can change

Since mass and amplitude drop out, the only practical way to change a pendulum's timing is to change its length — and the relationship is a square root, which makes it less dramatic than people expect.

To halve the swing rate you need four times the length. That square-root relationship is why a grandfather clock's pendulum is about a metre long — that happens to give a period close to two seconds.

Phase drift

Two pendulums of different lengths have different periods, so even if they start together they gradually separate and only realign after many cycles. This is the mechanism that makes Pendulum Puzzle a puzzle rather than a single timed step: the safe window through a row of blades moves every cycle.

The game prints each blade's period above its pivot, which turns an opaque timing challenge into something you can genuinely reason about.

The small-angle approximation

That tidy formula is an approximation, and it is worth knowing where it breaks down.

The true equation of motion for a pendulum is:

d2(theta)/dt2 = -(g / L) * sin(theta)

That sin θ has no neat closed-form solution. The standard trick is that for small angles, sin θ is very nearly θ itself — within 1% up to about 14 degrees. Making that substitution turns the equation into simple harmonic motion, which does have a clean solution, and out falls the familiar formula.

What happens at wide angles

Beyond about 20 degrees the approximation starts to drift. The real period is slightly longer than the formula predicts, and it grows with amplitude.

The games on this site integrate the full equation rather than the approximation, so wide swings genuinely slow near the top of the arc. That is why the timing in Pendulum Puzzle never feels quite mechanical — it is correct rather than idealised.

Energy through a swing

A pendulum is a continuous conversion between two forms of energy, and understanding that conversion is what makes swinging games playable.

• At the extremes of the swing, the bob is momentarily stationary. All the energy is potential — stored in height.
• At the bottom of the swing, the bob is at its lowest and moving fastest. All the energy is kinetic.

Between those points, energy trades smoothly back and forth. With no friction the total stays constant, so a pendulum released from a given height returns to exactly that height on the other side.

Why release timing decides everything

If you are riding a pendulum and let go, you inherit its velocity at that instant. That velocity is ω × L, directed along the tangent of the arc.

• Release at the bottom: maximum speed, almost horizontal. Best for distance.
• Release partway up: less speed, steeper angle. Best for height.
• Release at the top: almost no speed at all. You simply drop.

That is the entire skill of Pendulum Dash, and it is why a release that feels 'too early' is usually right.

Pumping a swing

Anyone who has been on a playground swing knows you can increase the amplitude without anyone pushing you. The mechanism is worth understanding because it appears in games.

Standing up at the bottom of the arc raises your centre of mass while the swing is moving fastest. Crouching at the extremes lowers it while the swing is momentarily still. The net effect over a full cycle is that you put in more energy than you take out, and the amplitude grows.

This is parametric amplification — driving a system by changing one of its parameters (here, the effective length) at twice its natural frequency.

The same principle is why reeling in a rope mid-swing in Rope Swing speeds you up. Shortening the radius while under tension does work on the system, and the energy has to go somewhere.

How games use pendulums

As hazards

A swinging blade is the classic timing obstacle. Its value is that it is perfectly predictable — unlike a patrolling enemy, it has no decisions to make, so failure is unambiguously the player's timing.

As transport

Grabbing a swinging bob and releasing it converts a timing challenge into a momentum-management one. Pendulum Dash makes the release point the entire decision.

As counterweights

In Chain Reaction Lab, swinging counterweights add a timing element to a spatial puzzle. The cascade has to arrive when the weight is in the right part of its arc.

Why games model them explicitly

A pendulum could be built from a rope constraint and a rigid body, but integrating the angle directly gives repeatable timing that a constraint chain cannot match. When the gameplay depends on a stable period, the explicit model is worth the extra code — and it is also why the blades in Pendulum Puzzle carry no damping at all.

Playing pendulum games well

1. Watch one full cycle before committing. The period is printed; use it. Waiting two seconds costs a fraction of the time bonus and saves the run.
2. Cross just after the blade passes. The gap behind a passing pendulum is open for the longest, because the blade has to travel to the far extreme and back.
3. Prioritise the short rods. A low period number means a fast blade and a narrow window. Read those first.
4. Release at the low point for distance. That is where the bob is fastest and its velocity is most horizontal.
5. Do not stop between obstacles. The gap between two pendulums of different periods is not a safe zone — the next one may arrive while you are deciding.

• Assuming heavier blades swing slower. Mass has no effect on period. Only rod length and gravity matter, so judge by the rod, not by the bob.
• Timing off the blade rather than off the gap. Watch the space you intend to occupy, not the object you are avoiding.
• Releasing late on a swing. Past the low point you trade speed for height. If you needed distance, you have already lost it.

Why pendulums mattered historically

The properties that make a pendulum a good game obstacle are the same ones that made it the most important precision instrument for three centuries.

Timekeeping

Because the period is almost independent of amplitude for small swings, a pendulum keeps time even as its swing decays. A clock only needs to top up the energy — it does not need to maintain a particular amplitude precisely. Pendulum clocks were the most accurate timekeepers available until the 1930s.

The residual amplitude dependence is exactly the effect described above, and clockmakers worked hard to minimise it. The best pendulum clocks kept time to within a fraction of a second per day.

Measuring gravity

Rearranging the period formula gives gravity in terms of length and period, both of which can be measured very precisely. This was for a long time the standard method of measuring local gravitational acceleration, and it revealed that gravity varies measurably with latitude and altitude.

Temperature compensation

Since period depends on length, and metals expand when heated, a pendulum clock runs slow in summer. The solution was compensating pendulums built from two metals with different expansion rates arranged so their effects cancel — an elegant piece of engineering that follows directly from T = 2π√(L/g).

Frequently asked questions

References