Physics of playground swings

Dan,

If I know my physics right, the phenomona is caused by the same thing that makes figure skaters spin faster when they tuck there hands in close to there body (or is it extend them out as far as they can?)  Regardless, the important thing is the location of the center of gravity.  When you "pump" your legs, your center of gravity shifts forward, creating a force that propels you farther ahead when you are the direction you are facing.  When moving backwards, you tuck your legs - and this causes your center of gravity to shift to the level of the swing, so while you don't gain speed in that direction - you are at least able to conserve it.

If you wanted to create something to simulate this for a science project, or whatever - I can't see a reason why you wouldn't be able to simulate it, assuming you have appropriate materials.

Rob

I think vandyr has it with the swinger.  I think it is tuck there hands in close as opposed to extend them for the skaters.

Scott

Sorry guys I have to disagree.  In fact, you pretty much have said the exact opposite.

Sitting on a swing, you are essentially a pendulum.  Easy enough.  You go higher and higher (up to a point) because you apply a cyclic forcing function at approximately the natural frequency of the you-pendulum (YP).

You get the most effective 'pumping' motion by leaning back in the swing on the forward-downstroke.  You can infact do this without legs, your legs just assist you in applying your inertia quickly by acting as a counter balance.  What you essentially do is shift your center of gravity to the rear of the swing during this stroke (if you remained still, your c.g. would be along the line of tension in the swing's chains and your motion would decay like a normal unforced pendulum).

When your c.g. is behind the swing's line of tension, you have effectively set up a moment on the YP.  This moment force propels you faster through your arc than without it, and thus you go higher with each 'pump'.

<ugh> can i remember a swing? which class can i attend has one handy?!

I like the skater analogy. As I recall, you also tighten with the hands, pulling on the ropes, effectively shortening the distance to the center. Extended feet pushed forward.

What I want to know is,  how many of you made it all the way through the complete circle? (as well as where I can go try one to refresh my neurons).  Any make it over the top?  Anyone know someone who, er, almost made it?

I am too much of a clutz to even try.  I would have ended up killing myself

I recall that you can also pump while standing up on the swing seat.  Think of a circus act where the scantily clad artist increases her speed.  

As I recall, you just crouch down at the top of the cycle and stand up just as you start moving downward.  I think it might simplify the model to think of it that way.

It's something about conservation of angular momentum.

-- Dan

All you are doing is causing your mass to shift.  And considering your on a pedulium, it swings out as the mass shifts out, and in as the mass shifts in.  Being humans with a computer upstairs capable of millions and millions of calcs per second, we can time these swings correctly to actully cause the pendulium to accelerate rather well.

And we never made it over the top, but we did get pretty high with pushing help.  We used to do the texas chainsaw on the swings when we where in the 2nd and 3rd grade.  We would get like an 8th grader to push, and we would go far.

Until the the cheap montgomery ward swingset legs start coming off the ground and the really scary "bumping" begins...

The reason you can propel yourself is that you *raise* your center of mass at appropriate times. More specifically, you bring your center of mass closer to the center when you are moving faster (near the bottom). This is how you convert your potential (gravitational) energy into kinetic. You let yourself fall away from the center as you slow down (at the ends), which will convert some kinetic energy to potential, but not as much as if you did it near the bottom. Repeating this will continue to convert potential to kinetic energy.

By the way, this would not work if you tried it in space or sideways on a merry-go-round. The same goes for the skater example. Only because of the asymmetry of gravity does this work at all. Skaters can speed themselves up by pulling towards the center or rotation, but when they release, they will give up that kinetic energy. The trick it to release when it costs less kinetic energy to do so.

For the mathematically inclined, think about your centripetal acceleration, v^2/r. The work you do by pulling in is proportional to this. When v is smaller, there is less work done by pulling in and vice-versa. Thus you can take advantage of this to repeatedly do more and more work.

Seth

*drools, in thought* ...scantily clad artist...

SethHoyt,
That sounds right on target.   The 'subconcious question' I couldn't put my finger on is that whatever you gain should be lost again when you change position again.  

But I think you're saying it's like the rocket that needs only a small bit of propulsion when out near the apogee of an elipitcal orbit in order to make a major change in the orbit (or is this a different phenomenon?).

The skater example makes sense, too:  If a scantily-clad skater could just "pump her arms" she could spin forever...  but she does not benefit from the gravity asymmetry.

=-=-=-=-=-=-=-=-=-=-=-=
Would the formula also apply on a small scale?  Like an 8-inch high swingset?
=-=-=-=-=-=-=-=-=-=-=-=

Aside:  An scantily-clad astronaut in zero-G gets kicked by another astronaut and is spinning uncontrollably in the center of the cabin.  She can't touch the walls and has nothing to throw.  Can she slow or halt her spin by moving her arms and shapely legs in a certain way?

-- Dan

Hi Dan,

It's more like the way the "slingshot effect" propels rockets. The example you mentioned is a little different because the rocket is pushing off the expelled gases. This has no parallel in the case of the swingset (disallowing projectiles and pushing off).

Interestingly, we cannot start from a stopped state on a 'perfect' swingset without pushing off. The only way to get started without pushing off the ground is if you can get the swingset to bend a little as you move back and forth. This gives your center of mass just enough motion to work with.

As far as scaling goes, this all works perfectly well on a small scale, in case Geico was looking for another ad idea for their gecko.

The scantily-clad astronaut, however, would be in trouble. She cannot stop or progressively slow her rotation without pushing off something. If she could, she'd be able to start rotating without pushing off, and that can't happen. In physics this is referred to as the conservation of angular momentum, which is basically Newton's law that objects in motion remain in motion unless acted upon by an outside force.

The only thing she could do is to alter her angular *velocity* which is how skaters speed up and slow down, but her angular momentum (which is proportional to both velocity and radius) would remain unchanged. Basically you exchange radius for velocity like in the swingset case, it's the same principle.

Seth

It seems to me that she could change the rate of rotation by first curling up, changing body orientation, and then stretching out again.   Imagine being oriented vertically spunning like a top, then curling up and then stretching out 90 degrees from the previous orientation (flat to the "floor").  I think whe would slow down considerably.

And is there no way to convert her angular velocity to to directional motion?   What if she had a string with a weight tied to it?   Throw it out, then let it wind around her like a yoyo.

Dang this is fun.

-- Dan

SethHoyt is pretty close, but...

First of all, if you go to Oktoberfest in Munich, then you can see a
swing where you can actually do the experiment of trying to flip
yourself over the top.  It's fascinating to watch, and you can pay
a few Euros and try it, too.  Good exercise.

It's got two rigid poles in place of a chain or rope attaching
the swing to the horizontal overhead pole, and you stand in "it" as
opposed to sit in "it".  They physically attach one foot of the person
in it to the metal surface that you stand on (it's something that
securely wraps over your foot) plus there is a secondary safety
attachment about your waist that still allows movement.  You hold
onto the rigid poles either side of you that go up to the overhead
(horizontal pole).  They start you off.

So let's consider that a person is a point mass, m.  Not true of
course, but I just want to illustrate the principle (and furthermore,
it may tend to apply to the swing you want to build).  Also, consider
that after they start you off you will reach a height h above
the bottommost point of your swinging where your velocity will
be 0.  Now here the main point: you can "convert" from potential
to kinetic to potential energy all you want, but in a closed system,
energy is conserved.

The ONLY way you get higher is by putting in energy.  So, if we
suppose that when the swing is at its nadir, you instantaneously
increase the "height" of your center of mass by x (say one foot),
and The radius from the horizontal bar to your center of mass is r,
Then the amount of energy you've added to the system is mgx.
Thus, the point mass will swing to a point that is x+h over the
nadir of its previous low point, which was x over the original
base nadir.  That is to say, the point mass will go up to h+2x over
the original nadir.  However, to repeat this process, the point
mass should revert to its original position.  By similar triangles,
this reverted position is at a height h1 (over the original nadir) of
(r-h-2x)r/(r-x) which simplifies to r(h+x)/(r-x).  So h1 > rh/(r-x).
So for the numbers that we're talking about, this would mean
about a 10% gain in the height of h at each pass.

    Now the three major points that were sloughed under the
curtain were friction (a source of energy loss), the fact that people
are not point masses, and the details about moving the
point mass at the swing's nadir.  The implication of the middle
one is that you're actually adding less energy than that m would
imply.  The implication of the first one is that you must keep
adding a minimal amount energy, otherwise you stop.  The
third point is very interesting.  It says that if you want to make
that swing do higher you must exert yourself.  Specifically, as
this rigid swing gets higher on each pass (h increases) it's potential
energy of mgh translates into increasingly higher velocities at
the nadir and the person will feel an increasing force that he must
overcome to add the same amount of energy.  Hand waviness
aside, this is what actually happens.  Try the swing and your
legs become increasingly fatigued as you do squats.

    The upshot is that you could build your desktop swing.  I bet
you could sell it, too.  Get a little switch that actuates right
as it passes the low point, and you'd also have to sense when
the velocity goes to 0 at its apex.  It would be pretty neat.

    Lots more on this such as what to do right as you go over the
top, but that's enough for now.

    Csaba Gabor

Excellent!

Then the device would be mechanically very simple:  
Just a piston that raises and lowers.  It raises to the max when nearing the bottom of the arc and then lowers itself when it senses the motion slowing to a stop.  

It might even work with just a single simple sensor.... if it could know when it was at the bottom of the arc, it could then assume that it was at the top just by timing two such events (am I correct that assuming that the actual point of "crouching down" is not critical to the operation -- as long as it was finished in time to "stand up" at the bottom of the arc?)

-- Dan

Points be hanged.  I want a copy of this thing if you ever build it!

Actually, the actual point of crouching down is critical (although some slop
should be permitted on either the crouching or rising).  Consider,
if the person were to stand up and then crouch down immediately
afterwards, he would be doing useless work.
    What happens is that when the person crouches down, then they
are taking away potential energy from the system (actually, once you
make it past the 50% point you start adding potential energy).  If you
wait till the apex of your swing, then you will minimize the amount that
you lose (or maximize the amount you add if you are past 50%).
    Here's the way to see it.  Potential enery increase or decrease is
proportional to the vertical difference between the point in question.
So if your swing swings 45 degrees, say, then the important thing
as far as potential energy is concerned is that projection of the "x"
vector (now at 45 degrees) onto a vertical line.  This is cosine(45)*x
which is less than x.  Thus, while there was a gain in potential energy
at the bottom of mgx, the loss is only mgx*cos(45) and since cos(any
angle)<1 we lose less than we gain.  If we crouched down earlier,
then we would still gain some energy, but not as much as if we had
waited for the maximum angle.

I agree with the piston part.  You could clothe the piston in a doll
even (or not have any person for more of a new age look).

Hmmm, I have another idea on this.  Whereas you can't cavalierly
time this thing, some judicious timing might work.  Let's assume that
we approximately know how long it will take to get from the nadir
to the apex on a particular swing through.  So we measure the time
at which the guy goes by on his way up, and then again just as he comes
back down (on his way up to the other side).  Well, the time of the
new trip up to the top is going to be slightly more than half the
round trip time of the previous up and back trip.  Good enough
for government work, as they used to say.

I think you'll need a sensor to know when you're flipping over
the top.  Once you flip over the physics change and you don't
do any more crouching.  A simple but precisely timed head and
foot movement provide the energy to overcome the loss from
friction - it's quite amazing really.  Also, as you pass the top,
your velocity slows down to near 0.  It means that it's fairly
simple to reverse direction by just preventing yourself from
crossing over (don't do the appropriate head/foot movements).
Your swing should be able to deal with this.

Csaba

I thought I had answered all of this already... ;)
It's always the things I think are obvious that are what trouble people...

I figured everyone knew that the way to build a pendulum that goes around all the way is to do it with a rigid rod. Of course the same goes for a swing. I'd thought of implementing that in particular when I still used swingsets, good to hear it was done. I had thought of using this principle as a game for intelligent control, when I later heard Rich Sutton discuss the same inverted pendulum idea in a talk he gave at my school. In particular, it's possible to not only reach the top of an inverted pendulum, but maintain it there, although this is not easy to do.

The piston idea is precisely what I had in mind when I wrote my original response, I didn't even think it would be worth mentioning since it seemed to be an obvious implementation of how you bring your center of mass towards the center.

What Csaba has said about the half-way point is incorrect. He is not taking into consideration the rotational motion, which, near the bottom is most significant for adding energy to the system. Gravitational potential is not the only potential energy in the rotating frame of reference. As far as the piston is concerned, all it needs to do is:

"Push in the direction that is most difficult whenever it is most difficult to do so."

This principle applies everywhere in its trajectory. even when the swing is perfectly level, there is a centrifugal potential to overcome by pushing somewhat towards the center. If we constrain the piston to move only radially, it can still get work done here. Flipping over is relatively easy to do if that's what you want. Holding it near the top or other acrobatics are not so simple.

Seth

Remember on your desktop model to include a scantily-clad action figure ;-)

Oh, and it's not shifting your centre of gravity left or right, it's vertically that counts as your adding more kinetic energy at key points in the motion.

Haydn.

This post is getting long and I thought you guys had sorted this out. HaydnH is exactly correct, by reducing the radius of rotation at the bottom of the arc (by inputting energy through a force)and raising it at the top  you increase the kinetic energy of the system. The principle is  quite different to the sling shot effect.

It could be possible to consruct this by having a weight on the bottom of the swing that is raised  a little at the base of the arc (this is were the energy is input into the system) and lowers is itself at the top of the arc. The raising could be close to unperceivable as this is a resonant sysytem.

A design must detect the position of the rotating weight and have an energy soucre to raise it at the base of the arc, possibilities are a clockwork mechanism in the swing weight itself or the source could be a falling counter weight system as in the pendulum clock.

The cool thing here of course is that the period is unaffected by the amplitude, allowing a timing device of some sort. A very simple idea is to some how pull the string in a little and let it out again repeatedly at the natural frequency of the system.

GfW

....position of the weight in the arc is not important in the design as once the weight is moving if is raised and lowered at the right frequency the swing will adjust to the correct timing (phase).

hang a string with a weight over a horiozntal bar (stair bannister say) by holding the other end and swing it to get get an idea of the natural frequency for the length of sting. Start it swinging very gently and  alternatingly pull the rope in and let it out at about the natural  fequency. The weight will slowly increase its amplitude and adjust it's phase to the rope going in and out.

I do believe I was the first to mention that raising one's center of mass when at the bottom is how energy is added to the system....

Time to award the points, Dan!!  ;-)

I stand by my answer.  You raise the point mass by x at the nadir
and that adds mgx to the Total Energy (how you want to claim that
that energy is decomposed is your business).  This translates to
an increase in height of x(h+r)/(r-x) at the apex of the swing over
the height h at the previous apex.

This is achieved by raising and lowering the point mass at the
proper times, namely raise at the bottom of the swing (when velocity
is maximal) and crouch when the when the velocity has dropped
to 0 (at the top of the swing).

Notes:  Amplitude does affect period.  It is only at small angles
where it does not.  Natural frequency of the swing becomes a
moot point in getting to the top.

What we mean by instantaneously moving the point mass by x
is that we displace it without altering its velocity.  Of course in
the real world that can't happen, so you just move as fast as
you can.

Note that at the apex of the swing, we must crouch.  Until we
reach the halfway point (h=r) , this actually causes a loss in
energy (it's also the direction that it's easiest to pull in), but
this is more than offset by the gain we get at the
bottom (and waiting to do it at the apex is where we lose the
least energy).  We have to do this crouching because otherwise
we couldn't rise at the bottom.

A final comment on traditional swings.  People usually characterize
adding energy to them by "pumping the legs", but consider that the
mass you are talking about there is whatever is below the knees.
I don't know how you ride a swing, but I think my upper body is
getting as much play as the part below the knees.  Perhaps the
legs are just along for the ride (so to speak).

Csaba

Csaba,

Your energy calculations are still incorrect...

You are not taking into consideration that the swinging body is in an rotating (accelerating) frame. When you move your center of mass towards the center, you are accelerating it, adding additional energy to the system. F=ma, remember?

The simplest way to look at it is in the rotating frame, where you are overcoming a centrifugal force of mv^2/r in addition to the gravitational force. It is imperative to consider the centrifugal force as its change it is dominant most of the time.

Let theta be the maximum displacement of the swing's angle from the equilibrium position (in radians). The maximum variation in the radial component of the gravitational force is mg*(1-cos(theta)), and for the centripetal force it is exactly twice that. You can verify that for yourself.

But the real proof here is that you can design a swing that constrains the mass to only move vertically, and it still works. Imagine a double-pendulum (two rigid rods with a pivot between), where the outer half of the pendulum is always vertical (this is actually easy to do). A mass moving up and down on this part of the pendulum will be able to add energy to the swing. According to your reasoning, this would not be possible. But if you think about it, you should realize it does work.

Seth

Another thing Csaba...

How can you possibly think you are adding more and more energy to the system through gravitational potential alone? You should realize that the gravitational potential you add by standing up at the bottom is going to eventually be lost. Your analysis does not explain how you can ever place your center of mass higher than it originally was. Considering only the gravitational potential, your crouching won't get your center of mass higher, just the swing. Your explanation is thus very much inadequate.

Seth

Seth, you are right I should have read all the posts more carefully. The energy calculation is very simple since the energy input into the system occurs when the swinger applies a force to raise himself, this is the mass times the sum of the gravitational and rotational accelaretions. The energy input is now the force times the distance the center of gravity the swinger is raised. To avoid integrals since the swinger stands up fairly quickly at the base of the arc the rotational accelaration can be take be approx. that at the base of the arc.

To some extent this will work in space for if the swing in rotating 360o by standing there is energy input from the swinger as a force has to be applied to stand due to the rotational acceleration. By pumping the the swinger can increase the rate of rotation.

Dan, This sounds like the sort of thing the Victorians would have done they loved toys like this. I have visions of a one of those beds on a swing with a couple fornicating up and down at the natural frequency of the system eventually reaching a climax in more ways than one and sending the swing through 360o. Designing a pair of clockwork fornicators could be fun

Oh Gwen, that is too funny!

I have to say, even with my wild imagination, I didn't think of that. Actually, I'm somewhat surprised I didn't....

You have to be a little bit careful, since the frequency of oscillation does decrease as the angles become large, but I think it would be enough to use a clockwork device to 'stimulate' the swing's motion. If you want it to go higher, you'll have to use feedback for the driver, but I think it can be done without any additional sensors. All you need to do is measure energy output of the driving device, which can be done within the circuitry, and try to maximize that for each cycle.

Seth, My mind is in the sewer this morning (if truth be told it usually is), for marketing purposes you could also have His and Hers swings, switching the position of who is on top.
 
Concerning the change in frequency you do have a bit of play here as the pumping action does not have to be bang on the resonant frequency, I have not done the math but instinct tells me you  probably have to be in phase to within 1/2 a wave length ( I could be wrong here). Clearly the closer to being in phase than the greater the resonance.

I have scribbled a few equations down here and the change in frequency could be a problem unless you give the swing a good sized push at the beginning, which would spoil the effect as I think starting with a close to stationary swing that ends up swinging violently is more fun. There are also possibities of mechanical detection devices at the fulcrum (such as in pendulum clocks) that detect the verticel positon and raised the weight dropping as it swings past the vertical. To have a net resonant effect it does not have to dropped exactly at the apex at the arc. There are also possiblities of spinning acceleration like detection devices that are on steam engines (I can not remember what they are called)

The only problem here is that you will get out of phase rather quickly, because when you are in phase the oscillations will increase in amplitude and then cause you to go out of phase. This is identical to a coupled oscillator, you can see this if you hang a mass from a spring connected to a rotating rod. Or even simpler, you can just use a long spring as the pendulum with a mass attached. When you oscillate the mass vertically, you will notice a transfer between the horizontal and vertical oscillations. This transfer will go back and forth as the coupled oscillators go in and out of phase.

This means that timing the oscillations will necessarily result in the amplitude being modulated at a frequency related to the difference between the two oscillator's frequencies. That's why I think it's worth looking at a simple feedback mechanism that attempts to maximize energy output per cycle.

Even with just a single electromagnet doing everything, it's possible to build a circuit that delivers higher voltage when it translates into more power output. All you need to do is measure current for a given voltage and adjust the voltage higher when the current increases. This indicates a larger load (harder to push), and would give the effect we desire without the need to time anything.

Do I really take that long to write a response? I wrote my last response without the benefit of seeing your last one, Gwen. So forgive me if you've already picked up on the problems I mentioned

Oh, BTW that spinning device on a steam engine is a Watt regulator:

http://umlab.ru/index/demo/20.htm

That's it a regulator! I am all for simple elegance and an electro-magnetic method seems a bit of cheat, (you can get those spinning tops that spin on a small platfrom for ever, when in fact there there an electro-magnetic input from within the platform).

I wouldn't say it's a cheat... I'm not talking about an external driving force or anything. I'm talking about a device that can operate independently, so the harder you push on it, the harder it pushes back.

Although... you have a good idea there. Basically, you can use an accelerometer to regulate the push. I think that might work well as long as you don't get into coupled oscillations. A mass on a damped spring would probably do the trick, now how do you use that to regulate push without electronics?

"Would it be possible to make a desktop model of this with a small battery-operated "swinger" that would move up and down at the right times to keep the device swinging until the battery ran down? "

If you want it battery opperated you would just need a solenoid - a device that moves a bolt when an electric current is passed to it and a brush switch that makes a circuit at the correct timing in the motion to move the solenoid up - nothing would be needed to move the solenoids bolt down again as gravity would do that for you when there is no circuit passed to it.

Haydn.

This was essentially my suggestion using the electromagnet. But I think it's better to use feedback in the circuit rather than absolute timings, since the drift will turn out to be fairly unacceptable in my estimation. If you're suggesting an switch that detect's the absolute position, that would work, but I'd still rather see the swinger do it all independently.

I'd place the solenoid in the seat of the swinger, and have it sit on the bolt, so that it was completely autonomous. There should be a bit of a spring loading there, and it would be able to sense an increased acceleration through the solenoid's response. It could, in turn, push itself up and release when the acceleration decreases by sensing the decreased load.

I'm sure that's pretty similar to what you had in mind Haydn, and so I think that's the way to go, even if Gwen prefers to use a steam powered piston... ;-)

Seth

"But I think it's better to use feedback in the circuit rather than absolute timings, since the drift will turn out to be fairly unacceptable"

You could use absolute timings quite easily if you through a capacitor in the solenoid circuit. Essentially the capacitor would only allow the solenoid to move if the brush switch contact is for a certain period of time (i.e: long enough for the cap to charge - cap must discharge while the brush switch is open) - this would avoid putting ever increasing amount of energy in to the system and keep the swing at a certain height regulated by the capacitance of the capacitor - you could even have a variable capacitor so that you can have fun changing the swing height via a dial.

Haydn.

*cough* I meant the absolute position - not timing

When I was young, my grandmother took us to a park down the street from her house.  It had the traditional swings, but it also had these cool swings.  They where like a tiny roll cage, which you sat in the middle, and you had big handles for your arms.  To swing you pulled back on the handles to lift yourself forward, then pushed out on the handles to go back.  It was the coolest swing I have ever been on.

Reading back through the posts I just came across this:

"Actually, the actual point of crouching down is critical (although some slop
should be permitted on either the crouching or rising).  Consider,
if the person were to stand up and then crouch down immediately
afterwards, he would be doing useless work."

Actually this isn't true... it's standing at the lowest point of the swing where the kinetic energy is the greatest that increases the energy by increasing the centre of mass along the length of the rope and thus decreasing the moment of inertial (to use a previous example: as demonstrated by an ice skater when he/she tucks up). No matter how long the person stands for this decrease in moment inertial will add kinetic energy to the system. The point of squatting only has an influence if the duration of standing does not increase the kinetic energy more than you actually lose in the up swing.

i.e: the time the person squats doesn't matter as long as the duration of standing creates a greater amount of energy than the energy lost by the system.

Haydn.

    If you are going to go to an electronic system since the vertical position is easily determined (as the top of he swing at the fulcrum will brush past something) then determining when to drop the weight a little can be based on the time intervals between successively passing through the centre. My electronics is a little rusty but I do not see too complex a circuit for this.

     For mechanical devices a pendulum clockwork mecahnism would look cool, these are not all that difficult, in fact you can make them out of cardboard, (ever tried one of those cardboard clock kits, really cool). These mechanisms can be very pleasing to look at and an up market version would be neat, (although a couple of fornicators may spoil the effect).

     Finally I would not mind betting that if you put a couple of clockwork fornicators going at it hammer and nail on a swinging bed that they will get that thing "just a rocking" within a fairly broad range of  frequencies. (This is calculable but am to busy today, perhaps later).  

Actually, Csaba is correct here, Haydn.

Motion towards and away from the center are equally important to maximizing energy transfer. You are able to transfer increasing amounts of energy by taking advantage of the asymmetry in the centripetal acceleration. When you pull away from the center, you are converting centripetal potential energy to rotational kinetic energy. The trick is to minimize this amount and maximize the the transfer the other way. That's why you need to wait before you pull away from the center, you'd like to do it when your velocity is as small as possible.

Another way to think about that is when you are on a typical swing, you pull continuously through the fastest part of the cycle. If you let go during that time, you'll slow your motion significantly. In fact, if you could push outward hard enough, you could stop your motion altogether.

This is in direct contradiction to much of the analysis presented here, since virtually everyone had overlooked the importance of centripetal acceleration. You are doing negative work when you go away from the center of rotation, and you can do as much of it as you want by applying a large enough force. But if the most you could do by pushing out prematurely is to stop yourself in place, what happens to any excess negative work done? It either goes into heat when you crash against the swing, or goes with you when you fly off of it.


One thing I've been wanting to mention about this problem is that prior to reading Dan's post, I'd never given any serious thought to why a swing works in the ten years that I've had enough physics to perform the analysis. But it took only about ten minutes or so for me to figure out why it did work, and I was quite amazed that I'd never seen that analysis done before, because it's a really wonderful example of how the real world and theory come together. So I'd like to thank Dan for making me think about this, because otherwise I would have gone much longer without understanding this, I'm sure.

Seth

     You could change the concept and have swing that goes through 360o all the time, the nadir is easily detected mechanicallly for raising the weight as is the horizontal (or any other angle) for letting the weight out. The swing would be started by having it vertical and giving it a slight push to send it through 360o.

     The weight raising mechanism could be in the swing bed, (energy stored by a clockwork mechanism or battery), the detction devices at the fulcrum would be minimal so adding to the mysterious effect  

It's so much more fun than calculating friction coefficients on the playground slides :)

That swing described by rrhunt28 could be an interesting implementation.  Rather than raising a piston, you just rotate an object from being vertical to horizontal.  It still moves the center of mass.

There are large swings where two peole sit facing each other and alternately pull the handle... again, it seems that these work because during the pulling, you sort of lean back, thereby raising your feet.

However, if you put two fornicators on a swinging bed, I'm afraid that you'd have at least one small problem:  As the swing goes higher and higher, the frequency, F,  of the pendulum decreases, such that kinetic enegry (k) in the up position (u) position could reach an absolute maximum, so that our two lovers would end up going slower and slower as time progresses.   In my experience, this is the opposite of the normal situations where F(u)=>C(k).  Correct me if I'm wrong.

-- Dan

Eureka!!

I think I've just hit on a really neat (and very marketable) version of this thing.

For the swing, just hang a string down with a hook or loop at the end, nothing special here. What you're really marketing are the action figures you can hang from it, like gymnasts. They can hold on by a hand, a foot or whatever. You could even make a monkey that hangs by its tail. Each one just pulls up and down using the feedback mechanism I described before. They could even involve a little body wiggle to get them going and give them a more fluid motion. I think this is really cool, and I could definitely see it becoming a hot item, especially since you'd want to collect all the different 'swingers'.

How does that sound to you guys?
Seth

LOL, what country are you in, I am unsure it would sell here in america.  I seem to think I might have seen something similar before.

Hey, I'm from America too.... but it's a pretty big place, and you never know how it might sell elsewhere (Japan?)

Anyway, it's all about the design and marketing. They need to move fluidly, and seem like they understand what they're doing to make it sell. A little AI wouldn't hurt here, especially to get them to do some acrobatics. I'm serious!!

Seth

Well yes, if you put a bit of AI in and made it fancy maybe.  It would probably end up being a fad toy around christmas.  

BTW Kansas here.
LOL
And the swings I saw where in Arkansas

Not from America,
 
Dan:  From experience there is no normal way of fornicating :-)

What's in my mind doesn't always come across on paper (err.. html) too well, so just bear with me on this.

What I see is not a swing to sell, but figures capable of swinging themselves on virtually anything that hangs freely. I'm certainly not aware of this being done, and the physics is sufficiently tricky that it probably hasn't been fully exploited yet. I always picture AI in everything, since that is my field, so I might neglect to mention certain "abilities" I picture them as having. Obviously, you can't make a hit toy nowadays without some realism. But the use of feedback as required to swing oneself is something we don't typically associate with machines, and would thus appear intelligent, even if there was no 'real' AI in it at all.

Seth

Oh, about the two seat gliders, the reasoning is the same for a normal swing. However, I believe I've been forgetting to mention how you actually go about raising your center of mass. Looking back, I never did state that the purpose of leaning back is to *bend* the rope, as well as raising your legs. Bending the rope shortens the radius, so that you are closer to the center. If you just leaned back without pulling on the rope, your center of mass wouldn't actually move much at all, since you're just rotating your whole body. Pulling on the rope pivots your body from where you're holding on rather than from where you're sitting.

I can't believe I forgot to say this that whole time, no wonder people are still talking about lifting legs and not mentioning bending ropes... Well I guess better late than never.

Seth

Ummm, I've made two mistakes so I may as well correct them.  They don't affect any of my conclusions, but I still don't like to see my formulas wrong.
1.  First is my calculation of the new height.  I was adding an extra x to the height (just prior to recrouching), plain and simple.  Don't know what I was thinking (actually, I had changed my reference frame and neglected to subtract off the resulting change in potential energy that implied).

2.  Subsequent to Seth's saying that I wasn't taking certain things into account, I examined in detail the net work done in moving the point mass by a distance x.  After doing the appropriate integrals (since both velocity and forces acting on m are changing under this analysis), I came up with a different value.  I had left out the conservation of angular momentum.  Postulating that I could instantaneously move the point mass by x toward the center at the nadir of its path without altering its velocity was incorrect because that would violate conservation of angular momentum.

Thus armed, we can easily work out what the new velocity and Energy are.  If it starts at a 0 velocity height of h, at the nadir of its path (just before rising), the point mass m has a velocity, v, given by v^2 = 2gh and the total energy, E(0), of the system is mgh [nothing new there].  Angular momentum is r cross p (where p is the momentum vector, mv) and since these are perpendicular, we can just take the magnitude of their product: rmv.  Since angular momentum is conserved, right after the instantaneous move we must have (r-x)mv' = rmv.  So (v')^2 = v^2 * (r/(r-x))^2.  Therefore, the total energy of the new system is now mgx + 1/2m(v')^2 = mgx + E(0) *(r/(r-x))^2.  This leads to a conclusion that the height change (after subsequent crouching at the top) is 3rxh / (r-x)^2 + h(x/(r-x))^3.

> You are not taking into consideration that the swinging body is in an rotating (accelerating) frame. When you move your center of mass towards the center, you are accelerating it, adding additional energy to the system. F=ma, remember?

The frame [of reference] I chose is not accelerating, with its origin at the nadir of the point mass right before lifting.  It would also not be accelerating if I chose any other fix point such as where the swing is attached to the horizontal pole.

> The simplest way to look at it is in the rotating frame, where you are overcoming a centrifugal force of mv^2/r in addition to the gravitational force.
> It is imperative to consider the centrifugal force as its change it is dominant most of the time.

I'm pretty happy with the conservation of (angular) momentum argument.  However, I initially did this the long way (calculations on request) explictly taking all the forces into account.  Happily, they match.

>  But the real proof here is that you can design a swing that constrains the mass to only move vertically, and it still works. Imagine a double-pendulum (two rigid rods with a pivot between), where the outer half of the pendulum is always vertical (this is actually easy to do). A mass moving up and down on this part of the pendulum will be able to add energy to the swing. According to your reasoning, this would not be possible. But if you think about it, you should realize it does work.

No.  This is a nice apparatus, but it does not illustrate your point under my original incorrect analysis.  This is because when the point mass crouches when its velocity reaches 0, its effect will not be to lower itself, but rather to pull the bottom of the (weightless) swing to itself!  The same principle would apply even if the swing's bottom or the rest of the swing had weight, but in that case they would meet somewhere in the middle.  Interestingly, this crouching motion would imply a lateral movement for m so the swing could accomodate it.

> Seth

> Another thing Csaba...

> How can you possibly think you are adding more and more energy to the system through gravitational potential alone? You should realize that the gravitational potential you add by standing up at the bottom is going to eventually be lost. Your analysis does not explain how you can ever place your center of mass higher than it originally was. Considering only the gravitational potential, your crouching won't get your center of mass higher, just the swing. Your explanation is thus very much inadequate.

Seth, do me a favor and don't worry about how I think or how I should think.  Chances are you'll be wrong, and anyway how often do you pay attention when somebody makes an authoritative pronouncement about what you ought to do?  You can repugn what comes out of my head all you want, but about what goes on inside - don't go there.  As far as the substantive part of that paragraph, I've already explained this, and my earlier mistakes have not altered my conclusions nor this explanation: the rising and crouching are done at judicious times so the amount of energy gained from rising is less than that lost from squatting.  This is true even when the word 'energy' in the prior sentence is replaced with 'potential energy'.  In particular, since the swing is not in motion at the time of crouching any energy lost at that time is only potential energy and this energy (loss) must be less than the (potential energy) gain because the amount lost can never be mgx since only the vertical component of the mgx "crouching vector" is lost (I.e. mgx*cos(theta) is lost and cos(theta) is never greater than 1).

Spouting principles is nice but before calling me wrong next time, why not actually take a concrete position with your own equations describing the motion of the swing or energy or something?  There's nothing like the feeling of cold hard equations to give lie to a person's postition.

HaydnH,

Thanks for that web site you mentioned near the top.  I found it very interesting reading.  The eighth sentence in the 2nd paragraph in the introduction there seems to be an approximation you get under the assumption x << r in the analysis above.  (I.e.  1+2x/r approximates r^2/(r-x)^2).  The author's email no longer works.

Csaba

Here we go again...

Csaba,

If if makes you happy, I could have replaced the word "think" with "conclude", and to me it means the same thing in that context. There's no need to use that many lines to debate that one word.

I was actually thinking (long ago) of mentioning that you were forgetting about conservation of angular momentum (which you were), but then I realized that ANGULAR MOMENTUM IS NOT CONSERVED for the swing apparatus alone. You'll have to include the earth in your analysis if you want it to be correct now. So I'm perfectly fine with the fact you got the same results, because they are still wrong.

My point about the mass being in an accelerating frame was not to tell you to use that frame correctly, but to tell you NOT to use that frame at your convenience, which is what you were doing. You most certainly did ignore the fact that in YOUR frame, you ARE accelerating the mass when you move it! (Check it for yourself) Only in the rotating frame can you ignore this fact and instead use the centrifugal force. You were taking both sides here. The ONLY exception to this is when the pendulum is not moving (at its apex).

The counterexample I gave to your argument may not be perfect, but I could care less now, since the point of that was just to show you that you were wrong, which you now admit. Any student of logic knows that once you assume a fallacy (your argument), you can prove anything you like, so I'm not even going to bother thinking about whether it was "correct", since that is moot now. Its purpose was solely to help you realize that what you were doing was incorrect.

"Cold hard equations".... I'll have to remember that one!

Anyone who thinks physics is about equations is sorely mistaken, and not a very good physicist I might add.

If my tone bothers you, it's because you keep insisting you are correct, which necessarily implies that I am not, however, you haven't disproven any of my (much simpler) analysis. It was so darn simple that I DIDN'T NEED to do any integrals for this, because I know how to use symmetries and approximations correctly. Your reliance on equations has continually led you in the wrong direction.

Now, I wouldn't even put up with you if it weren't for the fact that my goal is to TEACH. This is why I'm here in the first place. I also learn at the same time, but I do love explaining things to people, especially when there is an intuitive, conceptual explanation that does not rely on equation after equation of nonsense. I say nonsense because you are so happy to incorporate conservation of momentum, which you must like due to the fact that it leads to "cold hard equations" such as omega=constant when there is no applied force. Well that would be great if it weren't for the fact that now you're forgetting about both gravity and the fact the whole thing is attached to the ground.

Could you please explain how, if angular momentum is conserved here, that each time you repeat the process you end up with more angular momentum?? I've seen so many students make these kind of mistakes by relying so blindly on equation that they lose all concept of what's going on.

I'm generally quite patient with students, but I'm having difficulty dealing with the level of disrespect I've been experiencing in this case. If I were in your position, I'd be showing a bit more respect now that you've already realized your errors, after you stood so boldly by your position. All I ask of you is to give respect in advance to those who may very well deserve it. I began to lose respect for you when you kept insisting my analysis was flawed without any proof whatsoever. When I very clearly disproved you, you simply rejected it at first. Most other people would not even try to help you understand it after that, because you aren't showing a desire to learn what you didn't understand, just a desire to prove yourself correct, irrespective of any misunderstandings.

I think I've said all I need to say about this. And if you still refuse to even consider that my analysis just might be correct, don't count on me to point out your errors in the future, because it is a waste of my time.

Seth

fight! fight!

LOL... I'm glad at least somebody's getting a kick out of this.

Gwen, somebody, please help me out here!

I really can't deal with this sort of thing any longer. Do we have to start posting our degrees on the site to earn any respect? I would drop it if it weren't for the fact that it's in a public forum, and I'm trying to clarify the issues for everyone here. Everytime Csaba posts this nonsense, I have to undo the damage, otherwise it defeats the purpose for me being on here at all. I don't really know if anyone believes this stuff, but I still feel the need to respond in order to clear things up. I hate for this to have to come down to a popular vote, I prefer the more scientific method of argument. But for some reason, this argument doesn't seem to end!

Sorry Gwyn for the multiple misspellings, it just occured to me... 8-)

All right,

Just to satisfy those of you who insist on using symbols for an explanation, I have gone to the trouble of doing some totally unnecessary calculations. I say unnecessary because their agreement with my original analysis is pretty much redundant, given the fact that nobody had actually disproved it in the first place.

I calculated the recurrence for the height, a la Csaba's notation, using cartesian coordinates. I did so by carefully examining the effect of applying a large, vertical impulsive force on the mass within a region infinitesimally close to the nadir. I could have done it all instantaneously using dirac delta functions, but I didn't feel like doing any integrals for this, so I did everything as a first order approximation in the small angle through which the force was applied.

If you don't understand what I just said, don't worry, my original analysis also works and is easy to understand - but don't take up a career in physics antime soon ;-)

Although it was tempting to use the fact that angular momentum is approximately conserved within the small angle, I decided against it since I wasn't sure how large the correction term would be. Instead I used conservation of linear momentum, as that is the most general, but a real pain to work with here. To find the maximum transferrable energy, it is necessary to apply a very large force right around the nadir so that the mass will shift upwards as quickly as possible. To prevent the mass from going skyward, the pendulum should be made of a rigid rod with the mass located inside a box at the end. The impulsive force will slam the mass against the top of the box, so its ceiling should be 'sticky' in order to hold the mass in place. By computing the momentum transferred in the collision, we obtain the following relationship between the initial and final velocities for the mass:

v1 = v0 * (1 + x/r)

where v0 and v1 are the initial and final velocities, respectively; x is the height raised and r is the distance from the pivot to the floor of the box.

It is worth noting here, that the angular momentum has now decreased by (x/r)^2 times the original value. This shows that using conservation of momentum as an approximation here is only valid in the small x approximation (x << r).

Beyond this, it is fairly trivial to show that:

h1 = h0 * (1 + x/r)^2 + x

This assumes that the mass is brought rapidly back into place as soon as it reaches the apex, even if that occurs for h > r. Obviously this equation fails when the pendulum loops over, but similar equations for energy can be obtained easily that will still work in that case.

I don't believe this result involves any approximations at all, it should represent a tight upper bound on the height in terms of the previous one a half-cycle earlier.

For small x (x << r, h), this can be approximated as:

h1 = h0 * (1 + 2x/r), which I believe is the result stated earlier from another site.

I haven't checked any of these equations independently, nor have I seen any other solution to this problem. I like to remain unbiased in my analysis until I'm positive everything checks out. But that does imply the possibility that I made an error somewhere in the calculations, so I wouldn't be terribly surprised if that happened. However, the result looks good enough for me to say that I'm fairly confident it's correct.


Now, using my earlier analysis in the rotating frame we can quite easily obtain this same equation in the small x approximation. Since gravitational potential energy depends only on height, it is conserved for any given cycle and can thus be safely ignored for this purpose. This leaves centrifugal force as the only external force acting on the mass. Calculating the work done through a small distance x against this force, we obtain

W = mv^2/r * x

Since this is the energy added on each pass, we can easily calculate the corresponding change in height, and solving this for the new height in terms of the old yields:

h1 = h0 * (1 + 2x/r)

which is the small x approximation shown earlier. I leave the trivial details of this derivation to the reader.

If I really wanted to, I'm sure I could also show the more general result from this analysis, but that isn't really necessary to prove my point, and isn't quite so trivial, so I'll leave that to anyone who feels the need to do it.

I sincerely hope that closes the book on this particular issue for now. But I still can't see why I had to spend more than 10 times as long doing something the more difficult and less intuitive way, when the simpler, easier to understand method should have been just as good.

Seth

Energy input to system when swinger rises =    ER
Energy absorbed from system when swinger squats = ES

ER - ES > 0

Hence swing swings.

GfW

Interesting is that for a seated swinger the energy input by sitting upright is less than for standing but the the energy loss by leaning back proportioionally is less as the force is applied mostly in the direction of travel, (ie back into the system), as opposed to radially for the standing swinger and mostly having to be absorbed by the swinger.

I think you people have put wayyy to much time into swings.  

I think you many be right about that...

Unless my marketing idea actually takes off, I would have been completely satisfied with the 10 minutes of staring into space it took me to solve this the first time around. :P

GwynforWeb wrote:

> ER - ES > 0
> 
> Hence swing swings.
> 
> GfW


Hey Gwyn, I'm with you there!
I prefer explanations that are simple and to the point.  (and also valid...)

Seth

The calculations are way beyond me.  I don't understand the points being pressed by SethHoyt and kerringo.

I've increased the points.  Can someone explain the differences between these two theories in concreate terms?  Can you plug in some real testable values?  Like ...

"If you use keringo's equations it means that a man who weighs 50 Kilograms, swinging on a swing that is 5 meters tall, will need 20 cycles in order to reach a height of 2 meters.   On the other hand, if you use SethHoyt's equations, that same man would need only 10 cycles."

=--=-=-==--=-=-==--=-=-==--=-=-=
I once had my daughter calculate how fast I'd need to go on my motorcycle in order to jump 100 feet with a 30° ramp.   But she gave it to me in KPH so I overshot the ramp and died (just kidding!  But I did stick the scratch paper with her calculation to the refrigerator with a magnet I got out of an old microwave oven.  It looks nice next the the finger paintings).

-- Dan

I dont see what is so hard to understand, you are simply moving your mass back and forth, and timing it to achive maximum distance.

rrhunt28:- correct, but people are trying to track where all the energy is coming from and going.

Well, the energy is coming from ATP and going to kinetic engery,  that was pretty easy, now give me a hard one. hehe

I guess it's just so hard to understand, you can't even see what's hard about it...   ;)

No, it's just that the effect is really subtle, and it's very easy to miss what's going on there. The reason it seems simple to us is that we are intelligent beings (most of us anyway...) and can quickly learn how to make it do what we want, even if we don't understand exactly what we're doing. It's one of those things that people know how to do but can't really explain how. It's sort of like trying to explain to someone how to stand up without falling. We all learned how to do that so long ago, we can't remember not being able to. But this is actually quite difficult to do if you try to make a machine that does it.

Seth

Did that guy just call me simple?

If it is so simple explain how the ATP comes into play.

Any science is a search for understanding of some system, be whether it be biological, physical, chemical or of any other nature.  While it's impossible for one to know what any person really understands, the way understanding is demonstrated to another person is to present some model (a theory) that can be used in predicting the behaviour of another similar system.  Such a theory is called scientific if there is some experiment that can be performed that will DISPROVE the theory.  For example, conspiracy theory is wonderful, because no matter what another person demonstrates, the conspiracist will always that the vague and nebulous persons behind the conspiracy are even more clever.  It cannot be disproved, hence it's not a scientific theory.  What about the earth revolving around the sun?  Is there any experiment that you would accept that would disprove it for you?  If not, it's not a scientific theory for you - it's an article of faith.  So, physics is certainly not about equations.  It, like all sciences, is about advancing disprovable models, and equations provide a wonderful basis for it.

There was one interesting comment in the Marvin Minsky approach to physics above which I figure deserves some explanation.  I talked about using conservation of angular momentum as the basis to calculate the new velocity (hence energy).  In general, angular momentum, L, is conserved when there are no external forces acting on a system, but that's not the case here - there is a uniform gravitational field.  Angular momentum increases from 0 (where the swing has 0 velocity) to its maximum of rmv (measured from where the swing hangs).  And only at the nadir and 0 velocity point is dL/dt= r cross F = 0.  But anyway radial movement of a point doesn't contribute change to angular momentum since r cross F[radial] = 0.

Csaba

rrhunt28, I was responding to your post just prior to that one

Seth

NO worries I am just messing with you.

So neither one of you can offer a concete example showing you are correct?   Neither can predict the height of a frictionless system after 10 cycles of a given wieght on a given swingset.  

I'm beginning to like rrhunt28's comment -- "well, the guy is pumping, that's why he goes higher..."

-- Dan

Now I am going to piss everybody off.

I must retract part of my previous answer, I just did the maths more rigorously (which is euphomism for doing it corrcetly this time). Czaba is right there is not a change in angular momentum as there is no torque acting on the weight by raising it,  this can be used as the basis for calculating the new kinetic energy of the system.

So in short we are still arguing

My basic argument still holds though,  a lot of work to raise the weight and only a little lost when it is dropped.

Csaba,

I agree with most of what you've said here, however understanding is much more qualitative than quantitative in nature. This is why humans excel at understanding things and are terrible at arithmetic and exact computation when compared to computers. The disparity is many orders of magnitude different.

Computers can be programmed with an axiomatic formulation of Newton's Laws and simulate them with precision, but they don't understand anything about what they're doing. This is why if you don't input everything in correctly, they will not notice the error.

I have observed so many students in physics, math, and comp. sci. who were "programmed" in their courses to solve particular problems really well, but give them anything novel or qualitative in nature, and they virtually crash like a computer. But that makes sense, since they are doing nothing but executing an algorithm. They are unable to go beyond that, unless they are one of the few that can acquire an understanding even when there was none to be found. Those are the ones with the ability to explain things to themselves and others. And if they are good at doing that, they will be able to convey their understanding to others.

Being able to follow the execution of a piece of code, and understanding what it does are very different things. Understanding is a higher level concept, and relates to natural language. Axiomatic formulations and algorithms are lower level, related to machine language.

In most cases, understanding cannot be conveyed by those who do not themselves have an understanding. But sometimes, by arguing with and explaining to one another, multiple people can come to an understanding that none of them had originally. And that process has little to do with who can predict a given result. The only thing predictions can do is discredit those that are incorrect. But that's not the only way to show correctness or the lack thereof. Predictions can only get you to a certain level of confidence, but can't provide understanding on their own. Understanding comes via a completely different mechanism, and predictions are used to help guide that mechanism in the right direction. But as you understand something more and more, those predictions become increasingly less necessary, and eventually, you are able to function without them.

The same goes for teaching by example. Just giving someone many examples will not impart understanding onto students unless they are able to figure it all out on their own. The purpose of having a teacher is that the teacher is assumed to have an understanding of the subject, and is able to convey that to the students. I found that as a general rule, the more my professors did not understand a particular subject, the more symbols they needed to write on the board to say the same thing. Those that have a deep understanding can generally solve problems in ways that make the concepts more understandable to the students.

There is an interesting parallel here between computer languages and natural language. Have any of you ever wondered why it can take so much code to do something very easy to describe in English or other human languages? It is basically the same idea, there is a fundamental difference between having exact knowledge of something and having an understanding of it. Most of computer science continues to overlook this point, and tries to specify everything with precision. There is this notion that programs should be "bug-free". But for most really critical tasks, where lives can be at stake, we still rely on humans, who are quite far from being "bug-free".

Natural language is inherently ambiguous, and that ambiguity is critical for expressing concepts that would be extremely difficult or impossible to describe using a precise syntax. I find things much easier to understand when we allow this uncertainty into our language. Understanding allows me to fill in unspecified details and resolve apparent conflicts that are not formally resolvable. What we give up in return for this power is a much higher chance of making errors within axiomatic domains, but that's what computers were designed for, to compute. If we insist on treating axioms and computation as fundamental, we will not be able to transcent that domain into the space of unprovably true facts. Godel showed this space is non-empty, and I think most of what we consider to be common sense facts are actually in that domain.

Seth

Dan,

    Everybody in this thread is saying that the reason you can swing higher and higher is because you add energy to the system.  And I think people have pretty much come around to agreeing that you can add the most energy when the swing is at its low point, and to recover this position you wait until the swing is at its next 0 velocity position to crouch back down (since that leads to the least energy loss).

    OK, so the (physics part of the) squable is about exactly how much energy is being added (and the nature of this energy).  The swing will eventually go over in any of the analyses because they all conclude that energy is being added to the system.  This is in distinct contrast to what happens at Oktoberfest.  There, most people wind up guessing incorrectly about what to do (or not doing it well or quickly enough) and at some point the energy they add to the system is offset by how much they take out and they don't get any higher.  This almost always happens below the half way point.
    But you want concrete conclusions so here are mine after three disclaimers and a derivation.

    First off, you gotta start the system off with some energy.  So let me assume that the swing gets drawn back so h, the initial starting height is 2x (you start off at a height that is about twice the distance you can crouch).  That's my ballpark guesstimate for how far back they pull people at Oktoberfest.  If you don't like this, just say how you want to change it.
    Second, I don't know how much friction is coming into play here.  For the system you want to build, probably not much.  For the one a human rides, I'm guessing that it adds only an extra two or three cycles since people who knew how to work the swing were pretty handy in getting it to flip over.
    The most important thing, however, is that in the system that a real person rides, you can't assume the 0 mass swing.  The mass of the bars is significant, and I don't have an estimate for their weight.  I'll think about that one separately.  So what follows is for your neat little desktop model.  Let's assume the swing is 10 inches (r=10) and let's assume each rise/crouch is 1 inch (x=1).  And we've already assumed h=2x.  I would multiply these lengths by 12 for the one at Octoberfest.

    So, we express h(n) (the height at the end of swing n) in terms of h = h(0):  Under my analysis, h(n) = ((r^3)/(r-x)^3)*h(n) = (r^(3n))/((r-x)^3n) * h
What we want is to achieve a height 2r in n swings (and solve for n).  So:
From (r/(r-x))^(3n) * h = 2r, by taking logs we get:
3n * ln(r/(r-x)) = ln(2r/h) or
n = 1/3 * ln(2r/h) / ln(r/(r-x))

    I would venture to say that using Seth's formula, we get h(n) = ((1+2x/r)^n) * h which, in similar fashion, yields
n = ln(2r/h) / ln(1+2x/r)

    Let's plug in the numbers:
According to my method, I expect it to be just shy of the midpoint in 5 swings, and to the top in 8 swings
According to my reading of Seth's forumla, it should be at the midpoint in 9 swings and to the top in 13.


Puuuhhhhleeeezzzzzeeee build this thing.  I am Sooooooooooo curious about these predictions.

Csaba

No Gwyn, you're not pissing anyone off, at least not yet anyway... ;)

I'm glad you brought this up, this is important for us all to understand. I've been wrong on many occasions, but my goal is to find the right answers, and not worry so much about having been wrong in the past. There's nothing wrong with being incorrect once in a while, as long as you're able to admit it and move on. Only with an open mind and the courage to fail are we able to find the truth we seek.


Now, on to angular momentum...

I never attempted to do a direct analysis of the exact angular momentum. I conserved linear momentum, which, assuming my calculations are correct, yields a small change to the angular momentum. This is quite possible, since it does take at least some time to raise the mass, even if it is infinitesimally small, during which time angular momentum is not being exactly conserved. The reason I don't think the change in angular momentum goes to zero as the time to reposition the mass goes to zero is that the forces required to do this become infinite, and so does the momentum of the mass as it is repositioned. Thus, for conserving momentum, as the mass's moving time goes to zero, we cannot ignore the small changes in orientation of the apparatus, because they are multiplied by the large momentum.

Let me know if that helps to clarify things, or if you find something wrong with my explanation.

Seth

No Gwyn, you're not pissing anyone off, at least not yet anyway... ;)

I'm glad you brought this up, this is important for us all to understand. I've been wrong on many occasions, but my goal is to find the right answers, and not worry so much about having been wrong in the past. There's nothing wrong with being incorrect once in a while, as long as you're able to admit it and move on. Only with an open mind and the courage to fail are we able to find the truth we seek.


Now, on to angular momentum...

I never attempted to do a direct analysis of the exact angular momentum. I conserved linear momentum, which, assuming my calculations are correct, yields a small change to the angular momentum. This is quite possible, since it does take at least some time to raise the mass, even if it is infinitesimally small, during which time angular momentum is not being exactly conserved. The reason I don't think the change in angular momentum goes to zero as the time to reposition the mass goes to zero is that the forces required to do this become infinite, and so does the momentum of the mass as it is repositioned. Thus, for conserving momentum, as the mass's moving time goes to zero, we cannot ignore the small changes in orientation of the apparatus, because they are multiplied by the large momentum.

Let me know if that helps to clarify things, or if you find something wrong with my explanation.

Seth

Grr.. posted twice accidentally....

I am falling farther and farther behind with my responses...

Next thing I was going to do was Dan's calculation. Well, at least it's done now. But my uncertainty as to where to start it was why I hadn't done it earlier, and I'm still not sure where would be a "natural" starting displacement. I think it would have to do more with the unknowns like friction and the accuracy of the timings involved.

One thing to consider here would be using friction and lateral motion to get at least enough oscillatory motion to then propel yourself. The initial oscillations would need to be enough to overcome any friction in the system as well as inaccuracies in executing the procedure. But it's probably not worth the effort to calculate that. Better to start with a displacement that is much greater and easily measurable, similar to what Csaba chose.

Seth

One other thing I forgot to mention...

Even my exact analysis (pending review) is only an upper bound. What actually happens depends on how close we can get to the exact figure. It would seem to me there is a fairly large range of angles where we can come close to the optimal transfer, but I'm not sure if we can come close enough to verify any of our analyses. Something to think about...

Seth

Hey Gwyn, we must suffer from the same affliction because I'm the same way...

Why else would I spend so much time proving something I already knew intuitively? But at least I have some quantitative results now. This is why I prefer to hand-wave whenever it's justified. The trouble with this beast is that if you want to do it exactly, you either need to solve the system of ODEs, or go crazy keeping track of what order to approximate everything to. Since me and diff. eqs. don't mix too well, I chose instead to formalize all of the approximations I needed to make. That took quite some time to do, but to understand it once you see it all drawn out is not nearly as bad, I'd imagine.

Maybe I can do a little drawing of my analysis for you so you don't have to do it in a more complicated way. Although, having independent confirmation of the results would be nice.

Seth

Where did we come up with this concept of falling related to swings?  

Are you certain about that 5g figure?  
That's equivallent to a putting 5 huge sacks of potatoes on your shoulders -- I believe it is similar to the weight felt by astronauts on takeoff.  Only Arnold Schwartzenegger could do squats with that kind of weight.   Of course, it would be momentrary in cycles as opposed to continuous...

-- Dan

I gotta tell you, this problem is not this difficult.  Instead of me having to read the dozens of pages of info in this thread, can someone define what questions still remain (if any)?  I'll be happy to make some graphs and such if that will be helpful.

Keep in mind, the Oktoberfest (rigid swing) phenomena and the typical playground swing phenomena use different methods of applying the energy to the system.  Which one do we want to wrap up here?

>> use different methods of applying the energy to the system.
In what way?  I think we are assuming that the swinger is standing on the seat of the playground swing and just bobbing up and down at the right times.

>>Which one do we want to wrap up here?
The ridgid swing would probably be the coolest to implement, since it could go full circle, but a desktop model of the playground swing would appear more magical (the rigid swing could just [appear to] have a motor in the hub and the viewer would think 'so what?').

-- Dan

Hi Dan,

    Well, I share your concern about the 5g and Arnold.  In particular, it does make me leary about the whether I've done my calculations correctly.  But as far as your potato analogy goes, it's more like 6 sacks of potatos chained to your feet.  The acceleration felt falls off rapidly as you go towards the center.  But no matter how I work the numbers, (if my formulas have merit) it seems like your center of mass is going to have to deal with pushing against about 3.3g (that's assuming you push at the bottom only while the maximum swing angle is below 120 degrees).  And when you are swinging down from the very top, your center of mass will have a bit less than a 4.5g force on it.
    As a quick reality check, if we consider our old friend the point mass (and massless swing) right when it flips over has a potential energy of 2mgR.  At its nadir, this energy is now completely kinetic energy so we have mv^2/2 = 2mgR => v^2 = 4gR.  So centripital acceleration at the nadir, given by v^2/R, is 4g, so total acceleration felt by the point mass is 5g.  So it seems like my earlier calculations might be in the right ballpark.

    Really, I'll have to stare at this ride some more this fall if I make it over to Munich and see if these guesstimates I'm making jibe with reality.

    By the way, remember the series of 5 steel marbles where you would pull one back to illustrate how momentum gets transferred.  Or pull two back and then the far two would pop out the other side?  Well that's sort of a swing if you take just one steel ball.

    Csaba

HEH I can see little kids on swings with 5g's being applied and them being thrown to the ground.  

There is no way you pull 5g's on a swing.  

The 5g is for mass=1 kg.  If mass is less than 1 kg, you will have less than 5 g, but obviously, no less than 1 g.

Is that supposed to be a joke?

-> obviously <-:   A word best left at home.

Actully you do experance negative g's on a swing.  And while the g force does change I have no doubt about it, I contend it could not be 5g's.  Its not like you have to put a grunt suit on to swing eh...

I explained that for a typical playground swing, the maximum is 2-3 G's. You cannot exceed 3G without going over the top, and rope or chain swings can't make it to the top while remaining taut.

I can see maybe 2 g's but anything more just sounds odd.  How many G's does the average car produce on take off?

Well, when I said 2-3 G's, I meant total. So 1-2 G's centrifugal force in addition to the usual 1 G of gravitational force. But the typical would be about 2 G; 3 is not attainable for the playground swing, just a theoretical maximum.

That seems to make sense.

>>>>Is that supposed to be a joke? <<<<<

Yeah.

Your maximum g load at the bottom of the swing is dependent on velocity and radius of the swing.  So if you assume that the maximum height most people can reach on a traditional playground swing is level with the rotation point (and even that's a stretch for most people), then:

Assuming r=2 meters

Atotal=Ac + g
Ac=sqrt(2gr)/r= 3.132 m/s^2
Atotal=3.132 + 9.81= 12.94 m/s^2= 1.32 g's

Now, if you can have any input to the swing you want and you don't mind spinning around the top, then you can achieve any g-load you bloody well desire.

alexcvt,
Can you compare that to the scenario were were discussing?
Assume that the "chains" are rigid bars and that you have just enough rotational speed to *barely* make it over the top.

Also, it makes sense to me that the g-force at the level of the person's head could be considerably less than that at his feet.  Can you verify and quantify that?

-- Dan
P.S.  In case everyone is getting bored here, be assured that I will make another 500 pt question to divy up among the assists.

Apparently you were serious, alexcvt...

Unfortunately, you are very far from correct here, which is why I posed the "joke" question.

If you check your work, you will find your dimensions do not even agree. What you've done is take v/r as the centripetal acceleration instead of v^2/r. The correct result does not depend on r at all, and is as I stated previously.

-Seth

Nor, by the same argument, does it depend on the mass.

But since we are here, let's explore the g forces felt just a little bit more.  In particular, under the conclusions reached earlier, the g forces on the outside are even higher than what we get for the point mass (although they drop off quickly moving towards the center), and my initial reaction was similar to most peoples', I think, along the lines of "Oh, go on, I don't put up with forces like that."  But the question remains of why should it be?

Let's consider a point mass m on a pole of radius R.  Only this time, the point mass isn't going to be at the end of the pole.  Let's say it's 1/4 of the way in.  Then the acceleration that it feels at the bottom is still (4+1)g (if started from the top) or (2+1)g (if started from halfway).  But the velocity of the outside point has got to be faster than that of the point mass so something at the outer end of the pole would feel more g's (4/3 more centripital acceleration than the point mass feels).  This is essentially what is going on in my earlier analysis.

Suppose we now ditch the point mass and say that the swing's pole has mass M, and we start it at the top.  The total energy of the system will be (3R/2)Mg.  When the swing gets to the bottom, the total energy is: (R/2)Mg + Integral((1/2)*(((R-y)v/R)^2)*(M/R)dy;y=0;y=R) = (R/2)Mg + (M/6)*v^2 where v is the velocity at the end of the pole.  Equating, we get:  v^2 = 6Rg => Total acceleration at the bottom is 7g!  (If it's started from the midway point, it's 4g).  The "effective center of mass" is between the end of the pole and the center, so the end feels a higher g.

The conclusion is that if you want to get the outer gs to a minimum (which we've already seen is 5g for a point mass started from the top), distribute as much of the weight as you can towards the outside.  Keeping in mind that adding weight increases the amount of energy needed to get the swing to the top.

Csaba

Seth, get a grip, man ;)

I have no idea who's 'right' on this questions, but it sure has been fun discussing it.  Thanks to everybody!
-- Dan

LOL, Thanks dan.

Thanks Dan, let us know if you build it.