Four studies on how motion behaves

01

Period: why a swing takes the time it takes

A simple pendulum is a mass on a rigid arm of length L. For small swings, the time for one full swing does not depend on how far the bob is pulled back. It depends on the arm length and on gravity, through the relation T = 2π√(L/g). Make the arm four times longer and the period doubles. This is why a pendulum clock needs a long arm for a slow beat.

The formula is an approximation. As the swing widens, the period grows, and an exact answer needs an elliptic integral. The approximation is good for gentle swings, which is where most clocks live.

Question for paper. If you lengthen the arm to twice its size, what happens to the period of a small swing? Why does the answer not depend on the mass of the bob?

02

Energy: what stays the same

With no friction and no air resistance, a pendulum trades energy between height and speed. At the top of a swing the bob is momentarily still and high. At the bottom it is fastest and lowest. The sum of the two stays fixed for the whole motion.

A double pendulum follows the same rule, but there are two arms, so energy can pass between them in ways that are hard to follow by eye. The total is still fixed in the idealised model. A physical pivot loses a little energy to friction, which is why a real double pendulum eventually settles.

The instrument on the home page computes the idealised motion. Its energy does not drain while it runs, which makes it a useful reference against which a real device can be compared.

Question for paper. In a single swing, where is the bob's speed greatest, and where is its height greatest? Are those two places the same?

03

Phase: plotting angle against speed

A phase portrait plots an angle on one axis and its rate of change, the angular velocity, on the other. A single pendulum with no friction traces a closed loop each time it swings. The loop is the whole story. Its size is set by the energy, and its shape is set by how far the swing reaches.

A double pendulum needs four numbers: two angles and two speeds. We cannot draw four dimensions at once, so we project. We can plot one angle against its speed, or follow the tip of the lower arm across the plane. The tip path on the home instrument is one such projection, and it is often the easiest picture to read.

Question for paper. Sketch the loop for a pendulum released from rest at a large angle. Why is it not a circle, even though the motion repeats?

04

Sensitivity: two starts that almost match

Start two double pendulums with identical arms and speeds, and an offset of 0.001 radians on the lower arm of one of them. For a while the two look alike. Then they separate, and within a short time they move in quite different ways. Nothing random has been added. The difference is in the starting angle, and the equations magnify it.

This is the practical meaning of sensitive dependence on starting conditions. A prediction is only as good as the measurement of the start. Some systems turn a small uncertainty into a large one within a few swings, and the double pendulum is a clear example.

Open the instrument on the home page, turn on the twin, and watch the gap readout. Then press Reset and compare the two runs again.

Question for paper. If your measurement of the starting angle is off by a tenth of a degree, does the uncertainty stay small or grow? What would you need to know to decide?