How Do You Go Up in a Swing?
| How do you
like to go up in a swing,
Up in the air so blue?
Oh, I do think it the pleasantest thing
Ever a child can do!
Up in the air and over the wall,
Till I can see so wide,
River and trees and cattle and all
Over the countryside--
Till I look down on the garden green,
Down on the roof so brown--
Up in the air I go flying again,
Up in the air and down!
-- Robert Louis Stevenson
Very likely everyone who ever reads this page will
already know how
to operate a
swing (and most will already have seen the poem quoted above). But
does it work? The
answer may seem obvious ... until one thinks about it.
On this page I take a look at a qualitative
explanation for the operation of a swing suspended by a single rope
or chain on each side. A swing suspended by rigid rods, such as a
typical lawn swing, is a little different in its principle of
operation, though exactly the same motions will work to get it going.
Throughout this page, I will assume we’re
watching someone on a swing from their right side
. When they’re
swinging forward, they’re following a circular path
around the bar going counter-clockwise. When they’re moving back,
they’re revolving along the same circular path going clockwise.
Riding a swing is a lot
like riding a pendulum. As the pendulum rises, its kinetic energy
changes to potential energy; as it falls, the potential energy
changes back to kinetic energy. The only way to increase the height
of the swing is to increase the total energy of the pendulum.
Where’s the energy
coming from to “pump up” a swing? You can’t give yourself any
kind of direct “push”; there is nothing to push against
Furthermore, you normally only change position at the ends of the
path, when you’re at the top of the arc and momentarily stationary;
energy is force times distance, and an impulse at zero velocity
transfers no energy.
The only place we can
reasonably expect to find the energy coming from is gravity. Somehow,
at the ends of the arc, when you change position, you must
yourself slightly, and thus increasing your
potential energy at the moment when your kinetic energy is least.
But how is this done? Where’s the “lift” coming from?
The first thing we
observe is that the motions of a person on a swing are normally
symmetric. At the peak of the back-swing, the head and body go back,
the legs go forward. At the front, the head and body go forward, the
legs go back. One might speculate that this motion represents a
motion by itself is adequate to drive the swing
. The symmetric
motions aren’t necessary. If you sit in a (slightly moving) swing
and just tilt your head back and forth in the usual rhythm, while
keeping your legs and body rigid, the swing will respond. (I've
tried it.) Similarly,
if you keep your head and body rigid and just swing your lower legs
back and forth in the usual rhythm, the swing will respond.
Note well: Your head
and shoulders are above
your center of mass. Your feet and
lower legs are below
your center of mass. When you’re on
the swing, your center of mass is between your shoulders and your
knees; your head and shoulders are closer to the bar (or "axle"), and
legs are farther from the bar, than your center of mass.
At the back of the arc,
you throw your head back
, but you throw your feet forward
What do these have in common? Both gestures apply a torque to your
torso. More on this a little later.
The Line of Pull
The next thing to
notice is that the line of pull is not
through your center of
mass, save at the very bottom of the arc. This is probably not
going on: At the top of the arc, you are momentarily stationary; in
particular, your body is not rotating. Your angular momentum is
At the bottom of the
arc, when you’re traveling forward, your body is also rotating.
Viewed from the right side, you are rotating counter-clockwise
as well as moving forward (else your feet couldn’t wind up out
there in front of
the rest of you at the top of the arc
in front). So, if we take your center of mass as the origin, you’ve
got "positive" angular momentum (the counterclockwise direction is
typically designated as "positive"). If you’re traveling backward,
you’re rotating clockwise,
and you’ve got negative angular
momentum. Angular momentum is conserved; something must have applied
a torque to you, about your center of mass. What?
In figure 1
, we see a figure
on a stationary swing, hanging straight down. The line of pull of the
chain is through the center of mass. Gravity, of course, always
pulls straight down on the center of mass. So, the figure feels
no net torque at that moment.
There are two forces
acting on you. Gravity can only pull your center of mass straight
down; with your center of mass as the origin, gravity can’t apply a
torque to you. So the torque must come from the other force,
is the pull of the chain. It must not
be pulling through
center of mass. In figures 2
, we see the same figure, not
moving (not "pumping" the swing), but swinging passively back and forth
like a pendulum. None the less, the body of the "pendulum
swinger" rotates counter clockwise as the swing goes forward, and
clockwise as the swing goes backwards. So, the swinger must be
feeling a torque, and that must be coming from the chain. Hence,
the line of pull must
be as shown in figures 2 and 3.
Note that the chain does not go straight from the bar to the
swing: At the "pivot point", where the swinger is holding it, it
bends. From experience, holding onto the chain is necessary to
keep from falling off the swing; the swinger puts a "downward" force on
the chain at that point; the line of pull goes straight from the
swinger's hand to the bar.
To summarize, on the back-swing,
you’re rotating clockwise (the “negative” direction). After
you pass the low point, you’re slowing down, and so is your
rotation. After you pass the peak of the arc in back, you’re
rotating counter-clockwise, and your rotation is accelerating. So,
throughout the back half of your swing, your angular momentum is
increasingly positive, and the pull of the chain must be in front
your center of mass. By a similar argument, during the
“front” half of your swing, the pull of the chain must be behind
your center of mass.
To put that
differently, everywhere except the bottom of the arc, the pull of the
chain is along a line which passes below
your center of mass.
Where the Lift Comes From
At the peak of the arc
going back, you throw your head back and your feet forward. With
your center of mass as origin, you’re applying a positive
(counterclockwise) torque to your head and feet; your torso, and the
swing itself, experience a negative
(clockwise) torque as
a result. But the swing
doesn’t twist (much!) in response to this torque, so something must be
applying an opposite torque to it. That “something” can only be
one thing: The chain. It’s pulling below your center of mass; to
apply a positive torque to the swing and your torso (to keep them
from twisting as you lean back and move your feet forward), the chain
must be pulling harder. So, when you throw
your head back and feet forward, and hence increase their angular
momentum (as they move), you actually give a tug
on the chain,
and your center of mass lifts slightly (figure 4
Let's go over that again a little more carefully. The chain is
toward the bar, which is normally above you at all points in your
arc. So, that extra tug from the chain has a vertical component: It
lifts your center of mass, very slightly. Look carefully at
figure 4. As the swinger's head goes back and feet go forward,
the swinger's body feels a "reaction torque" in the opposite
direction. The result is increased pressure exerted by the
swinger's hand on the pivot point, which consequently moves clockwise
around the swinger's center of mass. By simple trigonometry, the
swinger's center of mass must consequently move toward the bar.
To view it differently,
by throwing your head back and feet forward at the back of the arc,
you’re “rolling yourself up” in the chain a little bit. In
effect, you’re pulling your center of mass up the chain.
The net consequence is that
you gain a little potential energy, which comes back as kinetic
energy when you swing down. At the bottom of the arc, you’re going
faster than you would have been had you not changed position.
At the front peak of
the arc, you do the same thing, in reverse, and because the chain is
now pulling behind
your center of mass, you again end up being
lifted slightly (figure 5
). So, you gain
energy at both ends of the arc.
You may object that if
you “roll yourself up” in the chain a little, and so lift
yourself that way, you must unroll
at some point. When does
that happen? The answer is, as you come down. At the bottom of the
arc, the line of pull is once again through your center of mass. At
you’ve “unrolled” again from the chain and the extra potential
energy has turned into kinetic energy, and the extra bit of torque
from the chain has resulted in increased angular momentum at that
What If You Go “Over the Bar”?
What if you swing so high you go all the way around? Legend has it
you’ll turn inside
For all we know, the legend could be true -- for you can’t
over the bar!
As soon as your arc
reaches 180 degrees, so that the pull of the chain no longer has a
vertical component at the peaks of the arc, the pumping will stop
working. There is no way to “pull yourself up” the chain if the
chain points down
. (But note that there's no way
to verify this with a simple experiment because of the issue
outlined in the next paragraph...)
Furthermore, as soon as
your arc exceeds 180 degrees, your velocity vector as you near the peak
of the arc on the backswing is
actually pointing forward
, and your velocity vector once you
horizontal in front is pointing backward
. As a result, you
won’t come back down a
nice, clean arc – you’ll come down in a parabolic fall until you
use up the slack in the chain, at which point your path will make a
sharp angle; that will cost energy because, for a moment, the pull of
the chain will have a component pointing back along your path. So
swinging is self-limiting:
you can’t go past the horizontal in front or in back.
that this argument only
applies to swings supported by flexible ropes or chains. The rest of
discussion on this page applies, with small changes, to rigid
rod-supported swings, but
this section most assuredly does not: I have actually seen a
rod-supported "swing device" at an amusement park driven a full 360
degrees by determined
occupants! (They didn't turn inside out.)
Page created on 9/5/05. Updated with corrected diagrams