What if a swing had a heavier rider?

A little rider swings on a swing. Then a much heavier rider gets on.

Will the heavy rider swing faster? Slower? Or just the same?

Heavy things seem harder to get moving, but gravity pulls on them harder too. So what happens to a swing when a much heavier rider sits on it? Does each swing over and back take longer, or less time?

In this lab the ropes, the starting angle and the pull of gravity stay the same. Only the rider's mass changes, from 1 kilogram to 20, twenty times as heavy. Make your guess, then watch the swing-time readout.

Two effects of mass compete in a pendulum. Gravity's pull on the bob, and so the restoring force, grows in proportion to its mass; but so does its inertia, its resistance to being accelerated. Which wins?

OpenStax poses the question with two simple pendulums hanging from the same ceiling, one with a 10 kg bob and one with a 100 kg bob. This lab runs the same test in the pendulum model: L = 2 m, a 20° release, g = 9.81 m/s², no damping, and a bob mass raised from 1 kg to 20 kg.

First a light rider swings. Over and back takes about three seconds.

The swing goes smoothly, again and again.

In the model, a swing with 2-meter ropes and a 1-kilogram rider, pulled back 20° and let go, takes about 2.9 seconds to swing over and back. That is about 21 swings a minute, and at the bottom the rider is moving at about 1.5 meters per second.

Watch the graph of the angle and remember the swing time. Next, we will change only one thing: how heavy the rider is.

With m = 1 kg, L = 2 m, g = 9.81 m/s² and θ₀ = 20°, the model's exact period is 2.859 s (the small-swing formula gives 2.837 s), 21.0 swings per minute, and the bottom speed is √(2gL(1 − cos θ₀)) = 1.54 m/s.

The model also measures the period from the simulation itself, by timing successive passes through the bottom: 2.859 s, agreeing with the exact value. That measured period is what we will compare when the mass changes.

Now a heavy rider sits on the swing. The rider is twenty times as heavy!

But look: the swing takes the same time, about three seconds. Heavy or light, it is the same!

Now change the rider to 20 kilograms, twenty times as heavy. Pull the swing back to the same 20° and let go. Each swing still takes about 2.9 seconds, and the rider still moves at about 1.5 meters per second at the bottom.

Try the lightest rider the model allows, 0.1 kilograms: the same again. The swing time does not budge. To change it, you have to change the rope length or the strength of gravity.

At m = 20 kg the model gives T = 2.859 s and a bottom speed of 1.54 m/s, identical to the 1 kg case; the simulated period, timed from bottom crossings, is 2.8588 s for 0.1 kg and 20 kg alike.

Only L, g and the amplitude change the period: at L = 0.5 m it falls to 1.429 s, and at the Moon's 1.62 m/s² it rises to 7.03 s, whatever the mass. In the model, the bob's mass changes only how big the weight is drawn.

Gravity pulls harder on a heavy rider.

But a heavy rider is also harder to get moving. The two cancel out, so the swing goes the same.

Gravity pulls a heavier rider back toward the bottom with a bigger force. But a heavier rider is also harder to speed up. The extra pull and the extra heaviness balance exactly, so the swing speeds up and slows down in just the same way.

It's the same reason that, without air, a heavy ball and a light ball fall together. Our model is a simple pendulum: the rider is a small weight on the end of light, stiff ropes, with no air or friction.

The tangential restoring force is −mg sin θ, and the bob's tangential acceleration is Lθ″. Newton's second law gives mLθ″ = −mg sin θ, and m divides out: θ″ = −(g/L) sin θ. OpenStax notes the period is completely independent of factors such as mass, the same cancellation that makes free-fall acceleration independent of mass.

Model limits: a point bob on a weightless, rigid rod, uniform gravity, no damping here. OpenStax describes a physical pendulum, an object that cannot be modeled as a point mass on a string, whose mass distribution must be included in the equation of motion. A real rider has size, so a real swing is only approximately a simple pendulum.

A science book compares a heavy pendulum and a light one. They swing in the same way!

What matters is how long the ropes are.

OpenStax asks readers to picture two pendulums hanging from the ceiling, one with a 10-kilogram weight and one with a 100-kilogram weight, ten times heavier. Pulled back the same amount, they swing exactly alike, because the mass of the weight has no effect on the motion of a simple pendulum.

Real swings are not quite simple pendulums. A rider has a size and shape, and OpenStax explains that for objects like that, how the mass is spread out also matters. So how a rider sits might change the swing time a little, but simply being heavier does not.

OpenStax's check-your-understanding problem has two simple pendulums suspended 2 cm above the floor, with 10 kg and 100 kg bobs, both displaced by 12°; the answer is that their motion will not differ at all, because the bob's mass has no effect on a simple pendulum.

For extended bodies OpenStax introduces the physical pendulum, with period T = 2π√(I/mgL), where I is the moment of inertia and L the distance from the pivot to the center of mass. For a simple pendulum, I = mL² and this reduces to 2π√(L/g). So in principle a rider who sits, leans or stands could shift the center of mass and I a little, changing a real swing's period through how the mass is arranged, not through the total mass.