Photo courtesy of the San Francisco School of Circus Arts.
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.
Gravitational Potential Energy
Potential energy is an extremely important form of energy in
physics. I've already talked about kinetic energy on the
skidmarks page. 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 on the
skidmarks page: 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.
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):
=
2 l/g
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.
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:
= 2l/g
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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