# Oscillate a swing without touching the ground

Is is possible to set a swing into oscillations without touching the ground? This occurred to me while watching the second pirates movie. There is a scene where the ship's crew is suspended in a cage from a bridge in between two cliffs. They escape by swinging the cage towards one of the cliff. Is that even possible?

Update: From the answers, it is clear that it is possible to make the swing oscillate. Assuming the model Mark has proposed would there be limits to much you can swing? Is it possible to quantify it?

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Never played on a swing, did ya? You can use your body to displace the center of gravity of the system swing+body. This will set the swing in motion. –  Raskolnikov Dec 21 '10 at 12:38
Isn't this what we call "pumping"? –  Jeremy Dec 21 '10 at 15:16
Is pumping to higher and higher heights dependent on friction? –  endolith Dec 21 '10 at 22:24
@endolith No. As long as you add energy to the system, you'll start going higher. –  Mark Eichenlaub Dec 22 '10 at 5:48

## 4 Answers

Yes, you can start swinging.

Imagine a swing in a two-dimensional world. The swing consists of a stiff rod hanging down from a ceiling rigidly attached to a bowl with a ball inside. The rod can swing freely about its connection point to the ceiling, so it's a pendulum. The bowl is a section of a circle whose radius of curvature equals the length of the rod. The ball slides frictionlessly inside the bowl, except that it can exert a force on the bowl at any time if it wants to.

The reason for choosing this system is that as the bowl swings, the physical space it occupies doesn't change much. A swinging motion is as if a little bit of the bowl gets lopped off one side and pasted on to the other. This means that from the ball's point of view, as long as all oscillations remain small, the bowl is essentially stationary, and the ball oscillates like a pendulum.

The bowl itself also oscillates like a pendulum, but it does so at a different frequency. If you write down the equations of motion for the system, you find that the ball oscillates with angular frequency $\omega_0 = \sqrt{g/l}$, but the bowl oscillates with angular frequency $\omega_0 \sqrt{\sin\theta_0/\theta_0}$, with $\theta_0$ half the angular size of the bowl.

Suppose we start from the equilibrium position with the bowl and ball at rest at the bottom of the swing. If the ball momentarily exerts a force on the bowl, they will begin oscillating in different directions. Because they have different frequencies, they will eventually both be displaced in the same direction. When that happens, have the ball latch on to the bowl and cease all motion relative to the bowl. At this point you have a swing that's displaced from equilibrium - you're swinging.

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thanks for the answer. Would there be limits to how much the oscillations can be? (I've updated the question, have a look) –  Pulkit Sinha Dec 22 '10 at 8:28
@pulkitsinha The oscillations could only be large enough that the ball stays in the bowl. However, once you get started swinging, it's not hard to get to bigger and bigger swings, for example by parametric pumping mentioned by Colin K. –  Mark Eichenlaub Dec 22 '10 at 9:09

It is, because you are always touching the ground. You are sitting on the bench, which is attached to the swing's frame, which is touching the ground.

By pulling the chains the right way you can displace your centre of mass, which will cause the swing to swing. The reason you can displace your centre of mass is that you+swing are not an isolated system. The swing is attached to the ground, and so you can displace yourself without moving the swing.

If you somehow managed to build a swing without attaching it to the ground one of the below would be true:

• it would be attached to something else, and the same thing would happen;
• it would be free-falling, and so it would not work as a swing (there would be no tension in the rope/chain);
• it would be on a gravity-less system, and it would be identical to the free-fall case.

Edit (idea inspired by Mark's answer)
The third possibility would be to build a swing that rests on the ground without being attached to it. The swing could be built on top of a huge bowl that rests on the ground. In that case the answer is still yes, you can.

• If there is friction between the swing/bowl and the ground, then it's basically the same case at the beginning of my answer. Having friction would be the same as being attached to the ground.
• If there is no friction, then you still can. When you pull yourself to the right (by pushing your hands against the swing itself) the swing will go left. That will lead to a complicated movement, which will depend a lot on how your swing rests on the ground.
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Thanks @Raskolnikov, I have a terrible habit of swallowing not's. =( –  Malabarba Dec 21 '10 at 17:57
I don't disagree with anything you've written, but the bowl wasn't supposed to be resting on the ground. It's hanging from the ceiling and swinging like a pendulum. –  Mark Eichenlaub Dec 21 '10 at 22:31
Supposing swing is on frictionless ice, it will oscillate with an opposite phase, making the CoM stay in one place. It is a nice experiment with Newton's craddle on wheels. –  mbq Dec 21 '10 at 22:56
@Mark: True, I was in a hurry and misread your answer. –  Malabarba Dec 21 '10 at 23:01
thanks for the answer. I've updated the question, have a look. –  Pulkit Sinha Dec 22 '10 at 8:33

Yes. It is certainly possible to make a swing oscillate without touching the ground.

An important note (particularly with respect to some of the other answers that have been posted) is that I'm talking about periodic changes in the state of the system, not necessarily motion of the center of mass of the whole system. In other words, a swing-set which is not coupled to the ground (I'll say it is on a frictionless surface, for example) can oscillate, although the frame will move opposite the swinging child such that the CoM of the whole system is stationary. For the CoM to move, the swing-set does indeed need some coupling to the ground, be it frictional, rigid, or other, but in either case there may be oscillations and the system can posses non-zero kinetic and potential energy.

First, from a conservation of energy argument, there is no contradiction. Before even considering the system dynamics and its equation of motion it is obvious that if the child on the swing moves around he/she will influence the state of the swing system. If you are having conceptual issues because no force is being exerted on the ground, and it therefore seems that no work is being done, consider that the child represents a bank of stored energy.

Now, the question is whether the child on the swing can initiate oscillations without touching the ground. Clearly, yes he can. By shifting his CoM, the angle of the swing is forced from its equilibrium, which means the system will now oscillate. For the swing-set on a frictionless surface these oscillations will include the counter-motion of the frame, but they will be oscillations none the less.

Interestingly, once oscillations are initiated, they can be amplified without the child exerting any force along the direction of his motion. Simply by raising and lowering his center of mass along the direction towards the swing's pivot, the oscillations can be amplified via parametric excitation, on which wikipedia has a good page.

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Parametric excitation should be at the heart of any real answer to this question. Anything different simply disregards what every child knows, that they needn't roll on the bench to make the swing go, and that they're quite capable of producing a significant amplitude of oscillation. (Which is to say, it's a shame that this answer wasn't accepted.) –  Jonathan Y. Aug 22 '14 at 0:25

This reminds me of one of my favorite physics articles, called "Swimming in space-time", which shows that in any space-time metric with intrinsic curvature, simple periodic oscillations in isolation can lead to translational motion.

http://www.sciencemag.org/content/299/5614/1865.full

http://arxiv.org/abs/gr-qc/0510054

http://arxiv.org/abs/gr-qc/0612131

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Could the Pioneer and other "anomalies" all be extended body effects of this sort? Just some speculation. –  user346 Dec 21 '10 at 15:30
Interesting suggestion space-cadet. Are the Pioneer data accessible online? This might be an interesting research project. –  Raskolnikov Dec 21 '10 at 20:45
@Jeremy: do you know what the "Wisdom" is they are refering to in the article? –  Raskolnikov Dec 21 '10 at 20:47
@Raskolnikov: not "what" but "who": Jack Wisdom, MIT prof and author of the Science article. –  Jeremy Dec 21 '10 at 21:18
@space_cadet: nope; even if Pioneer were specifically designed to swim in space-time, the effect would still be about 20 orders of magnitude too small to explain the observered anomoly. –  Jeremy Dec 21 '10 at 21:22