Swings

Swings and the flying trapeze are examples of one of the classic problems of physics: the pendulum. Hanging a weight from a string and making it swing may be very simple, but the pendulum is a classic example of a nonlinear oscillator, and it is actually impossible to solve its exact motion analytically (but quite easy on a computer) . However, application of conservation of energy allows us to get some important answers without having to know the details. (Image courtesy of the San Francisco School of Circus Arts.)

Gravitational Potential Energy

Potential energy is an extremely important form of energy in physics. I've already talked about kinetic energy in the skidmarks chapter. Potential energy is a way of keeping track of energy that is stored and which can be put back into kinetic energy at a later time. Examples of potential energy include: gravitational (discussed here), electrical (eg. in a charged capacitor), magnetic (eg. in an inductor with current) and nuclear.

Gravitational potential energy keeps track of work you do against gravity. For instance, when you bike up a hill, you are probably going quite slow at the top after working very hard to get there. What did all that work do for you? Was it wasted? Some of it was, the work you did against friction that went into heating your gears, bearings and even the tires. But a lot of the work didn't go into friction. It went into getting you up the hill, and we say that it got stored in gravitational potential energy. How do you convert it into kinetic energy? By coasting downhill again, of course.

The concept of work was introduced in the skidmarks chapter: if you move a distance d against a force F you do work W = Fd. Gravitational potential energy just keeps track of the work you do against gravity in going up a hill. Since the force of gravity is mg downward (where m is the mass and g = 9.8 m/s²), the work you do against gravity by rising a height h is mgh. So, labeling vertical position with y, we get the formula for gravitational potential energy:

U = mgy

It doesn't matter where you set y = 0, because it's only changes in U and therefore y that count.

An important note: we don't talk about potential energy associated with work done against friction. Friction is a nonconservative force: it's very hard to recover kinetic energy from work that you do against friction. Gravity, on the other hand, is a conservative force: you can recover ALL of the work done against gravity back into kinetic energy. Potential energy is used to keep track of work done against conservative forces.

Conservation of Energy: The Swing

Armed with the concept of gravitational potential energy, we can answer a simple question about a swing (or pendulum or flying trapeze). If you release it from rest a height h above the bottom, how fast will it be going at the bottom?

At the top, the mass is at rest, so its kinetic energy K is zero. Let's set y = 0 to be at the bottom of the swing. The mass is released at a height y = h, so it has gravitational potential energy U = mgh. The basic idea is that total energy is conserved:

E = U + K = constant = mgh

As the mass swings down, it picks up as much kinetic energy as the potential energy it loses. One can therefore calculate K at any height. At the bottom of the swing, U = 0, so K = mgh, which can be solved for v using K = ½mv²:

v = 2gh

Note that the speed doesn't depend on mass. A big kid has just the same speed as a small kid if released from the same height.

swing GIF animation

This GIF animation shows how energy sloshes back and forth between potential and kinetic energy while the total energy remains constant. Note that the potential energy is proportional to the height of the mass. The animation plays at the correct speed for a rope length of 3 m. (Hit RELOAD to show the animation 10 more times.)

Oscillation Period

Another important aspect of pendula is the swing period, the time it takes to go through a complete swing. This is used in "grandfather" clocks, for example, as a timing source. Just as the speed of the bob at the bottom is independent of mass, the oscillation period is also independent of mass, because mass cancels out of the equations. The oscillation period does depend on starting height, but weakly; in fact, for small oscillations, the oscillation period

is independent of starting height (called amplitude):

= 2g/l

But, because the pendulum is a nonlinear oscillator, its period changes as the swing amplitude gets large. It is therefore very important for clock pendula to have small swings. The plot below shows how the period increases with the angle you start the swing from. The period is normalized to the period for very small swings. As you can see, a pendulum started at an angle of 90 degrees has a period 18% longer than a small swing.

plot of tau vs amplitude
Normalized swing period vs starting angle for a pendulum.

Equations

gravitational potential energy: U = mgy
conservation of total energy: E = U + K = constant
small amplitude swing period: = 2g/l

Summary

Potential energy keeps track of work done against conservative forces.
Potential and kinetic energy transfer back and forth in swing motion while total energy is conserved.
Since a pendulum is a nonlinear oscillator, it's oscillation period depends on swing amplitude.

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Sam Hokin / samh@fusion.kth.se