What if a swing had much longer ropes?

On a swing, you go forward and back, forward and back.

Some swings have short ropes. Some have long ropes. What if the ropes were much longer? Would you swing faster or slower?

A playground swing is a pendulum: a weight hanging from a fixed point that swings back and forth. OpenStax, a free physics textbook, even names a child's swing as an everyday example of a pendulum.

The time a pendulum takes to swing over and back once is called its period. In this lab you can change the length of a swing's ropes in a model and watch what happens to the period. What do you think a swing with ropes four times longer would do?

OpenStax lists pendulums among common oscillators, from clocks to a child's swing, and shows that for small displacements a pendulum is a simple harmonic oscillator. Its period then depends only on its length and the acceleration due to gravity.

This lab tests the length dependence in the pendulum model. The bob starts at 20° from the vertical, gravity is 9.81 m/s², there is no damping, and the length changes from 1 m to 4 m. The question is not only whether the period grows, but by how much: in proportion to L, or more gently?

First, a short swing, one meter long.

Over and back takes about two seconds. Whoosh, whoosh! It swings about 30 times every minute.

In the model, a swing 1 meter long, pulled back 20° and let go, takes about 2.0 seconds to swing over and back. That is about 30 swings every minute.

At the bottom of each swing it is moving at about 1.1 meters per second. Watch the graph of the angle: it rises and falls in smooth, even waves, because nothing in the model slows the swing down.

With L = 1 m, g = 9.81 m/s² and a 20° release, the model's exact period is 2.021 s; the small-swing formula 2π√(L/g) gives 2.006 s, the difference being the slight lengthening at larger amplitudes. That is 29.7 swings per minute, with a speed of 1.09 m/s at the bottom.

The model integrates the full equation θ″ = −(g/L) sin θ and also computes the exact period from it, so both numbers are available: the readout shows the exact one. OpenStax University Physics adds that the period is completely independent of other factors such as the bob's mass, which this lab tests separately.

Now the ropes are four times longer.

The swing goes slowly, back and forth. Each swing takes about four seconds. That is twice as long!

Make the ropes 4 meters long, four times as long. Now each swing over and back takes about 4.0 seconds, twice as long as before, and there are only about 15 swings a minute.

It is not four times as slow, only twice. Here is another surprise: at the bottom, the long swing is moving faster, about 2.2 meters per second, because it has a much bigger arc to travel. It goes faster but still takes longer, because its path is so much longer.

At L = 4 m the model gives T = 4.043 s, exactly 2 × 2.021 s, and 14.8 swings per minute. The bottom speed √(2gL(1 − cos θ₀)) doubles too, to 2.18 m/s, because the drop height grows with L.

The pattern continues: 9 m gives 6.064 s (×3) and 0.25 m gives 1.011 s (×½). Since T ∝ √L, a pendulum's length must be quadrupled to double its period. At 10 m, the model's longest, T is 6.39 s. On the angle graph, the 4 m swing draws one wave for every two drawn by the 1 m swing.

Gravity pulls the swing back to the middle.

A long swing has a long way to go each time. So it takes longer to get there and back.

When a swing is pulled back, gravity pulls it toward the bottom. It rushes past, climbs the other side, and gravity pulls it back again. A longer swing travels a much longer curved path for the same angle, so even though it moves faster at the bottom, it takes longer to get there and back.

The two effects don't cancel: the path grows faster than the speed, so the period grows, but only by the square root of the length. Our model is a simple pendulum: a small weight on a light, stiff rope, with no air or friction unless you add them.

Along the arc s = Lθ, the restoring force is −mg sin θ ≈ −(mg/L)s for small angles, the form of a spring with effective constant mg/L. That gives ω = √(g/L) and T = 2π√(L/g). Doubling the length halves the effective stiffness per meter of arc, so the oscillation slows by √2.

Model limits, from its documentation: a point bob on a weightless, rigid rod in one plane, uniform gravity, and linear damping if switched on. A real swing's ropes, seat and rider have size and mass, so a real swing is only approximately a simple pendulum.

You can time a swing with a watch. Count how long it takes to go over and back.

Scientists time long pendulums to measure how strong gravity is!

Because a pendulum's swing time depends only on its length and on gravity, OpenStax explains that timing a pendulum of known length lets you work out how strong gravity is where you are. It even suggests trying it at home with a string and a weight.

That is useful for more than homework. Maps of gravity over large regions help geologists study how Earth's plates move and search for oil and minerals. OpenStax also notes that the swing time hardly depends on how far you pull the pendulum, as long as the swing is small, under about 15°.

OpenStax College Physics derives T = 2π√(L/g) and stresses that the only things affecting a simple pendulum's period are its length and g; the period is nearly independent of amplitude, especially below about 15°. Inverting it, g = 4π²L/T², so a pendulum of precisely known length measures local gravity; OpenStax's take-home experiment does exactly that with a string and a stopwatch.

OpenStax adds that maps of g over large regions aid the study of plate tectonics and the search for oil fields and mineral deposits. Its worked example uses a pendulum of length 75.000 cm to determine g to five digits, with the angle kept small for accuracy.