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 why 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.

Energy

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 be lifting 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 Motions

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 rotation.

However, either 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 your lower 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 obvious.

Consider what’s going on: At the top of the arc, you are momentarily stationary; in particular, your body is not rotating. Your angular momentum is zero.

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, which is the pull of the chain.  It must not be pulling through your center of mass.  In figures 2 and 3, 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.

Figure 1:

Figure 2:

Figure 3:

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 of 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).

Figure 4:

Let's go over that again a little more carefully.  The chain is pulling 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.

Figure 5:

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 that point, 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 point.

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 out.

For all we know, the legend could be true -- for you can’t go 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 pass the 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.

But note that this argument only applies to swings supported by flexible ropes or chains. The rest of this 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 on 9/6/05.