Explain the kinematics of a playground swing.

Factual Questions
1

A child’s playground seems simple enough: sit down, start swinging your upper body back and forth, and you swing back and forth. It helps to get things started with a push from your dad or with your own feet on the ground - but properly executed, it’s possible to begin the oscillation from a dead standstill without ever touching the ground or receiving outside assistance.

So how does this work? Suppose we have an idealized swing with these characteristics:

-frictionless pivot at the top
-massless chains connecting pivot and swing
-zero aerodynamic drag

We start with the rider at a dead stop, the swing hanging motionless directly below the pivot point.

Now the rider begins pumping the swing.

Can someone explain what’s happening as far as locations/movements of the rider’s center of mass, in terms of distance from the pivot and fore/aft movements relative to the pivot point?

2

My WAG is that with these perfect conditions, the center of gravity would never move. I think we’re relying on friction to keep the pivot motionless for a moment, so that changes induced by the kicking have a delayed effect, allowing some actual motion. But that’s just a guess.

3

I don’t understand some of the explanation, but this has a lot of words and pictures that might help…

http://www.physicsinsights.org/up_in_a_swing.html

4

From that link: “The only place we can reasonably expect to find the energy coming from is gravity”

I call BS: the energy is coming from your muscles. Gravity defines the potential differences, and you have to put energy in to go from a lower PD to a higher one.

5

But the claim that it’s torque that’s at work (as you rotate your body, the system response with equal/opposite angular movement), that seems plausible.

6

The swinger (or the swinger/swing system) is in “equilibrium” when your center of gravity is at its lowest point, which is generally when it is directly underneath the top bar. If you start at rest sitting upright in the swing, your center of gravity is in this stable position. When you lean back, you shift your cg backward and out of the equilibrium position. You will swing slightly forward to bring your new cg toward equilibrium. Your momentum will carry you past equilibrium a bit. Then you lean forward, moving your cg even further away from equilibrium so that you gain a bit more velocity on the downswing. Lather, rinse, repeat.

Each time you shift your weight, you are converting energy from your body into mechanical energy (gravitational potential and kinetic) in the person/swing.

7

I think it would work without friction, as long as there’s mass - when you shift your centre of mass, you 're acting on the rest of the system, and that action does not create instantaneous movement.

In the same way, the weight of a closed system can fluctuate about the actual weight - if you’re inside a large closed box that is set on measuring scales, jumping up and down will register changes on the scales, but they net out to zero.

8

The pivot point (where the chains attach to the top bar) could be frictionless and it should still work, but the person will need friction between themselves and the swing in order to shift their weight.

9

On second thought, in idealized situations you could probably accomplish the weight shift without friction, too.

10

True, or else they’re just propel themselves off the seat when they extend their legs.

‘Frictionless’ is difficult to deal with anyway - if I wrap my fingers around the chain so as to completely enclose it, is it friction by which I am gripping it? If so, aren’t the closed links of the chain joined together by friction too?

11

Even if the pivot point is frictionless, you can still pull on the chains in such a way that they come in to that pivot point at an angle, and at that point tension in the chains can result in net horizontal motion.

12

If you’re a sphere of uniform density, you can’t do it. It’s possible for humans because we can relocate our centre of mass by altering our posture.

13

That is a very nice page with good text and pretty pictures. Unfortunately, I think it is wrong.

If you look at Figure 4 of that page, it shows the person on the swing at the top of the backswing, as they lean back to start the swing pump forward. If you look at the line of the chain, it bends at the hands. I agree this happens. However, I believe he has the bend backwards.

When one leans back on the swing, one is not pushing the chains to allow you to lean. Instead, one pulls on the chains to keep from falling out the back of the swing.

Instead of pushing the center of gravity up (with respect to the ground), what you are doing is moving the center of gravity in front of the line of pull, the chain from hands to bar. This provides a torque on the chains driving you counterclockwise (forward). Momentum pulls you past vertical.

At the top of the forward swing, you shift your head and torso forward and legs back, and lean on the chains forward. This shifts your center of gravity backwards, again in front of the line of pull. This applies a torque clockwise (backwards).

Thus one can rock from a standstill to a full swing without pushing off or being pushed. However, it is a slow build up that is shortcut by a pushoff.

You can reverse swing and slow down by inverting the motions. When you reach the top forward of the swing, keep leaning back like before. As you reach the top of the backswing, shift your weight forward and legs underneath. You will rapidly slow down.

14

The pulling and bending of the chain is inconsequential. Even the author of that link notes, but doesn’t see the consequence of, that you can get a rigid pendulum to not only swing, but to go all the way around. I’ve been in one of those amusement park cages… it’s all about yanking the sides with a shift of body weight. A chain-swing does the same thing.

Look at it this way. Plant your feet firmly in the ground at shoulder’s width and bend them. Now make your belly button go in a circle by leaning right, up, left down, repeat. You’re making your center of gravity go in a circle. You remain stationary because of friction with the ground. If you were on ice, you’d be sliding left and right. If you were on a skateboard, you could make the skateboard go left and right from that shifting of center of gravity. And if you were in a steel cage on a rigid pendulum, you could get the pendulum swinging.

Swinging a rigid cage pendulum requires the large body shifts of going side to side. It’s a bit hard to shift center of gravity front and back unless you place one foot out front and another behind you in the lunge position. On a chain swing, bending the chain helps a bit in that shift of center of gravity. And once you start swing high, that shift gets easier and easier.

And so… it’s all about getting your center of gravity to go in a circle. When you do that, you shift around. By yanking or leaning on a pendulum, you transfer the motion to pendulum.

I leave it as an exercise for the reader to determine how this information makes a hula hoop spin.

15

Intuitively, I think you’re right. But this is exactly the sort of case where I fear to trust my intuition. Let’s analyze it:

Here I am, in that box, standing but with bent knees. I push down with my feet. Intuitively, inertia causes my CG to stay motionless for a moment, despite the fact that my feet are trying to push my CG upwards. But because inertia is preventing my CG from rising fast enough, the net result is that my feet have successfully pushed downward, and the scales register a greater weight.

That all makes sense, but the tricky phrases are “for a moment” and “fast enough”. Is it really true? Well sure, anyone can do this experiment and see it happen. But that’s in the real world. Would it happen without friction and such? That’s what I’m not convinced about. On the other hand, I must admit that I don’t see a whole lot of friction in this particular closed system. Where is it? Me and the air around me? That can’t possibly make a significant impact on the scales, can it?

16

moriah, I agree it is shifting of center of gravity that is driving the swing. Let’s look at the case of a rigid arm swing, sitting in it like a chain swing.

At neutral at the bottom, your cg is directly below the bar. Gravity pulls you down directly in line with the bar. You lean your body back. That puts your cg behind the line of the “chains”. Gravity pulls down, but wants to pull the cg directly under the bar. That induces a torque on the swing to move your cg forward. Momentum carries it slightly past the centerline, at which point you quickly shift your weight forward. This shoves the cg farther in front of the centerline, so gravity wants to pull the cg the other direction back under the centerline. Torque the other direction, momentum carries through, shift weight back again, repeat.

The thing is, what are you doing with your legs? You are moving them in the opposite direction of torso, which shifts the cg in the alternate direction, against the effect you are generating with your torso. Why is that? I think that it helps you stay anchored to the seat and apply torque to the seat, it’s a body position thing, but that is a guess.

I think with a chain, the bending aids that process, but is not necessary to drive it. But my main point about the bending is that his description is wrong. He has you pushing on the chains to lean back. He’s clearly not ridden a swing in a long time. You hang on with your arms and pull on the supports. At the front of the swing, when you shift forward, you apply force to the back of the chains as you lean against them.

17

We have a problem already. relocating your CG laterally from its neutral position requires the application of an external force; the laws of physics demand this. At start, the line from your CG to the pivot point is vertical, exerting no horizontal force on your CG. If you move your head/torso one way, your legs/pelvis move the other way, and your CG stays in the same location in space, so still no lateral force. What this means is that my initial assertion (that it’s possible to start a frictionless swing from a dead stop without an externally applied force) was wrong. :smack:

After more poking around on the interwebz, I found this:

http://en.allexperts.com/q/Physics-1358/2008/5/Physics-swing-motion.htm

If he’s right, then a frictionless pivot makes it impossible to start a swing from a dead stop without outside assistance, but pivot friction plays no particular role in swing behavior once it’s already started oscillating.

Check out the real physics similator he links to:

http://www.sciences.univ-nantes.fr/sites/genevieve_tulloue/Meca/Oscillateurs/botafumeiro.html

It shows that the swing pumping action is not at all related to the lateral displacement of the center of mass relative to the chains. Instead, it’s about changing the length of the pendulum by moving your center of mass toward or away from the pivot point:

  1. As you’re swinging backwards, you are sitting straight up, spine parallel to the chains. COM is high, i.e. short pivot length.

  2. When you reach maximum height, you pivot your torso back, spine perpendicular to the chains. Your COM hasn’t particularly changed height above the ground (it can’t, according to conservation of energy), but you now have a longer pivot length, and so on the forward swing, your mass can fall farther than the height it ascended during the back swing, collecting additional kinetic energy.

The simulator suggests a technique for inserting energy twice as fast as most folks do it:

-standard technique: when swinging forward, most folks wait until they are at maximum height before they go to short pivot length (spine parallel to chains). Then they hold that short pivot length for the entire back swing. So no kinetic energy collected during the back swing.

-improved technique: sit up rapidly (i.e. go to short pivot length, spine parallel to chains) at the moment when the swing is directly below the pivot point. Hold this position until swing reaches peak altitude, then go to long pivot length. This applies when swinging in BOTH directions. Now you’re collecting kinetic energy on the back swing as well as the forward swing.

The simulator demonstrates this. Switch it to manual forcing mode (i.e. to where you can pull on the rope with your mouse), then give the pendulum a very small starting oscilation. When it’s at its maximum swing in either direction, rapidly let out the rope to give max length. As the pendulum passes directly below the pivot point, rapidly take in the rope to give min length. Repeat ad nauseam, and in fairly short order you’ll get the pendulum swinging over the top of the pivot point.

The simulator also suggests that most people’s leg action is counterproductive: when people lean back (spine perpendicular to chains), they tend to swing their legs up. If the purpose of leaning back is to move your COM away from the pivot point (and if the purpose of sitting up is to move your COM toward the pivot point), then leg action should be the other way around:

-when you are sitting up, your knees should be straight, getting your leg mass as close as possible to the pivot point.

-when you are leaning back, your knees should be bent, getting your leg mass as far as possible from the pivot point.

18

The initial movement separates the center of mass from the line of the bar not by moving the center of mass, but by moving the bar. Once that’s done, it’s possible to exert an outside force through the pivot and bar.

19

By your own admission, your center fo mass remains directly below the pivot point. The support chain or bar may not be vertical anymore, but the line connecting the pivot point to your COM still is. If you push off on the bar or chain, you are just pushing the bar/chain farther away from your COM; your COM won’t move anywhere, it’ll just stay directly below the pivot point.

Unless you put your feet on the ground or have an assistant give you a shove.

20

Raise your hand if you are now sitting at a desk, moving your belly button around, trying to imagine this.