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enter image description here

This is how many cartoons depict a character about to fall from the edge of a cliff. You can see movie characters do it. You can also see real people doing it. But does spinning arms like that actually help you to gain balance?

It would, ( through conservation of angular momentum) if your arms were like flywheels connected to a motor through an axle.

enter image description here

But they are not. (Try spinning your arms once without letting them flip in their own axis). As your hand completes a revolution about your shoulder, your arm completes a revolution about its own axis. It seems like things are not that trivial here.

Short, concise answers are most welcome, because this question has the potential to save many lives. And if you happen to be on the edge of a cliff while reading this, you are encouraged to perform confirmatory experiments.

Qmechanic
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AlphaLife
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7 Answers7

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The human body is a complex multi-part machine. In this case, windmilling your arms does work (as part of a more complex multi-stage full body motion), and more than the initial forward swing of the arms is useful. It works sort of like a ratchet.

The body can change shape. Static friction between the feet and the ground can be used to produce a variable net rearward force on the body. The magnitude of the rearward force varies based on the body's shape, and applying the rearward force requires the body to change shape (specifically, straightening flexed legs). If the person is able to have their butt behind their toes, they can probably recover even if their center of mass is forward of their toes.

The order of events is like sequential backwards hops, except without leaving the ground. (If you try doing sequential backwards hops without leaving the ground, you will fall over backwards, which is the opposite torque from falling over forwards, which is the point.)

The person flexes the legs (negligible net horizontal force excluding gravity).

The person thrusts the arms forward, then immediately extends the legs (net rearward force excluding gravity).

The person flexes the legs while thrusting the arms back (negligible net horizontal force excluding gravity, because the legs and hips are moving forward while the arms are moving back - all forces are internal).

Here is a crude diagram of this 'ratchet' that I drew in MSPaint. enter image description here

Edit for clarity: the net torque comes from the legs pushing on the ground. The arms just help position the body such that the legs can push in the desired direction (or, as SomeGuy points out in the comments, creates an internal torque that the legs and other muscles in the body can turn into a push in the desired direction by holding rigid during part of the motion). If it weren't for pushing with the legs, the person would (absent gravity) just wobble backwards and forwards as their arms went forward and back. Thrusting the arms forwards allows the hips to move momentarily backwards, during which time the legs can apply a greater horizontal force.

Charles Staats
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g s
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    This answer is based on wordplay and not on physics. It starts with nonsense. There is no net rearward force on any body when it stands on the ground. One might as well as there is a sideways or forwards force. The only forces on a still body are vertical: gravity pulls down; the Earth pushes up. The net force is zero so the body remains stationary. – Anton Jan 01 '23 at 08:33
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    @Anton unless you are standing on ice, the force between you and the ground absolutely can have a horizontal component. It's called friction. – Aetol Jan 01 '23 at 12:08
  • There is no frictional force between two objects in contact. A frictional force between two objects in contact only exists when an attempt is made to push one over the other. As that push increases, the frictional force increases, so the net force is zero and nothing moves. When the frictional force reaches its upper limit (adhesion breaks) and movement starts, the net force is push minus friction, and motion proceeds according to Newton’s second law. – Anton Jan 01 '23 at 14:36
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    @Anton People accelerate all the time with no slippage between their feet and the ground. Static friction is the main way that we get around, by accelerating our center of mass, then lifting one or both feet off of the ground and putting it back on the ground somewhere else so that we go farther than we could get by falling over. I think you may have let a limited understanding of physics based on the dynamics of inert rigid bodies overcome your common sense and prove that humans never move. – g s Jan 01 '23 at 17:30
  • Comments are not for extended discussion; this conversation has been moved to chat. – Buzz Jan 04 '23 at 03:09
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The cartoon version does exaggerate, of course.

Here is what I think: you only have time for one swing of your arms. If that first swing does not save you then by the time you go around a second time it's already too late.

Your sense of balance knows those things. Let's say you are on an edge, and it is half a meter down. You try one swing, and when your sense of balance feels that was not enough then your sense of balance will decide you must jump of the edge. By jumping your motion is controlled and you will land on your feet. If your sense of balance would not make you jump then you would fall over, and likely you will be injured.


I think there is a way to test this. You will need someone to assist you.

You need an edge with a very small height difference, for instance the height difference between road and sidewalk.

Stand with your feet on the edge of the sidewalk, with your heels unsupported (Comparison: the way competitive divers position their forefeet on the edge of the diving board before the jump.)

At that point the person assisting you should push you just a little. In order to be unprepared for the push I expect it will be necessary for you to close your eyes. (If your eyes are open your sense of balance can see your assistent make a move, and your sense of balance will commence countermeasures in advance.)

Under those circumstances: I expect that if it is possible at all to remain on the sidewalk a single swing will do it. If that single swing is not enough your sense of balance will sense that, and you will step down onto the road.

I expect that the help of an assistent is absolutely necessary; your sense of balance will simply disallow you from putting yourself out of balance.


As to the physics:
The physics is the same as the physics of a device that maintains balance by way of rotation of reaction wheels. Youtube video: Self balancing stick

Your feet are your point of contact, and that point of contact does not slide, so it acts as a fixed pivot point. So when the reaction wheel is spun up the stick is reoriented with respect to that pivot point.

Cleonis
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  • Correct. Are human arms like a flywheel connected to a dc motor, though? A part of the question explains how human arms cannot be considered so. For one, we cannot rotate our arms in one complete cycle without tearing the ligaments. There is a second rotation at play here, the rotation of arms about the axis of the bones. Would that play a role here? – AlphaLife Jan 01 '23 at 07:12
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    @AlphaLife I agree of course there is some rotation along the length axis of the arm, but I think the contribution of that is negligable, so I ignored it. The determining factor is distance to the axis of rotation. In the video I linked to, consider the reaction wheel. If that reaction wheel would have a larger diameter a slower angular acceleration would suffice. Magnitude of effect is proportional to distance to axis of rotation. In the case of rotation (component) of the arm around the length axis of said arm: the mass of the arm is close to the length axis of the arm, hence small effect. – Cleonis Jan 01 '23 at 11:54
  • Hmm, that sounds reasonable. – AlphaLife Jan 01 '23 at 12:06
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    @AlphaLife Thinking some more about it: in the case of the self-balancing stick in the video I linked to. The thing that counts is angular acceleration of the reaction wheel, not angular velocity. If the response of the reaction wheel would be to spin at a constant velocity then even a small lean of the stick is unsustainable. In order to have effect the reaction wheel must accelerate. So indeed the cartoonish drawn out swinging is unphysical. Your arms are accelerating only the first quarter to half of your swing. If that initial acceleration does not bring you back you are done for. – Cleonis Jan 01 '23 at 13:51
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    Right, you need angular acceleration to generate torque. You can also put the same argument using the conservation of angular momentum. The initial angular momentum of the body-arm system is zero, and hence any angular velocity of the arm must lead to an opposite angular velocity of the body. – AlphaLife Jan 01 '23 at 14:02
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    You don't need a cliff or partner: just lean forward until you lose your balance, then try to regain it without taking a step forward. I have just tried it many times, and sometimes a second arm swing does help. – usul Jan 01 '23 at 18:12
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    @usul Thanks. Good to know. (I tried some balancing, but I went easy on my arms. I have a mild RSI problem with my left shoulder (workplace related). Not a problem in daily life, but swinging forcefully is uncomfortable.) – Cleonis Jan 01 '23 at 18:18
  • I agree that this (the self balancing stick) is the correct model to use. I disagree that the reaction wheel spinning at constant velocity wouldn't work. Consider a 6 foot tall man who punches an eight foot tall giant, then drops his hand to clobber a four-foot dwarf on the top of his head, then pulls back his arm. Punching the giant provided a smaller torque than pulling back the dwarf clobber did (differing distance from man's feet), so the difference is a small change in the man's angular velocity. He is now rotating forwards, and if he does not stop himself, he will faceplant... – Some Guy Jan 02 '23 at 06:22
  • ... as such, a wheel rotating at constant velocity will produce a constant acceleration of the self-balancing stick around its pivot point (the base of the stick) due to the differing torque provided by the top vs. the bottom of the wheel. If this acceleration was exactly enough to overcome gravity, the stick could stay balanced, even at an angle, with the wheel spinning at constant velocity, since the wheel could produce exactly the constant acceleration needed to oppose gravity's constant acceleration. – Some Guy Jan 02 '23 at 06:22
  • Unfortunately I feel this is simply wrong. You CAN repetitively swing; it's NOT the case that you only get one bite at it. – Fattie Jan 03 '23 at 12:47
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This is just expanding a bit on the answer of g s.

Anyone saying that arm spinning cannot change the (angular) momentum should first remember that it is possible to swing on a swingset.

Here is a simplified model that I think captures the main phenomenon (but I don't know for sure). The muscle group here is a stand-in for the psoas (hip flexor), though of course a lot of other muscles are involved. We consider the center of mass of the entire body excluding the arms, and argue that there is a net backward force on it, as well as other forces.

  1. The arm swings as the leg muscles tense. This produces a net backward force with no or small change in angular momentum.
  2. The arms reset, producing a small net forward force, as well as some forward angular momentum.

enter image description here enter image description here

enter image description here enter image description here

usul
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  • Expanding a comment I posted on another answer: Moving arms forward/backward at head height does not produce the same torque around the feet as moving them forward/backward at waist height, due to differing distance from the point of rotation (the feet). Thus windmilling arms produces a net torque, and hence a net force on the body, despite the fact that the feet are not moving. The body rotates (leans) and hence its center of gravity moves, even if the feet are fixed in place. Bending/straightening knees at appropriate times can enhance this effect. – Some Guy Jan 02 '23 at 05:28
  • Dangling your arms like pendulums from your shoulders and moving them back and forth produces no net change in angular momentum, because the torque from moving them forwards exactly cancels the torque from moving them backwards. The way to get net torque is to bend your knees during forwards motion but not backwards (or vice versa), hence changing the radius from the pivot in an imbalanced way as your arms move perpendicular to it. Windmilling in a full circle is better. But bending your knees in the bottom half of the windmill (effectively increasing the windmill radius) helps even more. – Some Guy Jan 02 '23 at 07:03
  • @SomeGuy Generally agreed. I am not sure the vertical components are necessary for it to work, but it seems like they should help. – usul Jan 02 '23 at 13:00
  • But when I try the experiment, I don't flex my knees up and down repeatedly. It doesn't seem possible without just launching yourself forward and falling over. – usul Jan 02 '23 at 13:09
  • Bending of knees is not required, although it may be helpful in some circumstances if you can make it work. Probably it only works with a couple of big, dramatic forceful arm movements rather than quick circles. Also, the farther you get from vertical, the less helpful bending your knees will be, and at some degree of lean, bending your knees will just pull your feet up or sideways so that even if you do correct your lean, you will already be plummeting off the cliff anyway. Windmilling is the important thing, as that helps you adjust your lean. – Some Guy Jan 03 '23 at 06:58
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Mostly it's the first half swing which helps

There's a reaction mass thing happening with moving the mass of your arms compared to the mass of your body. The force required to move your arms forward applies an equal and opposite force backward on your torso. This will tend to pull your torso back over your feet.

Whilst your body is leaning forwards, your friction with the ground is reduced, and your natural leg movements will tend to push you forwards. If swinging your arms is enough to bring your torso back upright, then friction is improved and your natural leg movements will push you backwards.

Of course if it's not enough then over you go! But there is enough success rate for it to be a good strategy. Learning to use your arms as reaction mass for balance is one of the basic things we develop as babies when we first start walking, so it's a very well-developed reflex. It's not innate (as everyone who's seen a baby learning to walk can testify), it's just something we've used continuously from before we could talk, so the skill becomes reflexive.

Note that this relies on there being a solid surface underneath you, otherwise your equal and opposite reactions will equally well push your legs the wrong way. These reflexes are something you have to consciously unlearn when you start any sport which relies on you standing on a wobbly surface, such as windsurfing or paddle-boarding. For these, you need to develop a different balance reflex where you move your hips to move your centre-mass, pivoting around your ankles. This is actually a better balance technique for when you're stood still, because you're directly moving your body mass, but it doesn't translate well to walking motions.

After that, there will be a measurable effect, but not much

The OP claims that because the arm rotates about its axis as well as about the shoulder joint, it cannot produce a net forward force. This is clearly incorrect though. The net force for continuous arm spinning is of course produced by displacing air, so all we need is more air to be displaced on the back-to-forward part of the spin.

If you try the arm movements for swimming backstroke, you will clearly see that your arm and hand rotate to present your palm, giving maximum cross-sectional area through the air on the half-stroke from behind and passing your waist. As your arm comes back on the other side, you twist your wrist quickly so that your little finger is upwards. Your hand's cross-sectional area through the air is now reduced to only the blade of the hand, and it remains so for the rest of the rotation.

In a more dense medium, this would clearly have significant effects. You absolutely can do backstroke underwater to demonstrate the principle (although of course there are easier ways to swim underwater). In air though, you would need to be spinning those arms really fast, because you need to displace enough air to hold yourself up.

How much? Let's do the sums...

The worst-case of course is countering your entire body mass, in which case you're basically a human helicopter, and then you don't need to worry about falling because you can fly. Let's assume the effect isn't quite that awesome, so you can only support a quarter of your body mass with whirling arms. For an average 80kg adult, that means we need about a 200N force.

Looking at my hand, I can probably cup a half-litre of air, which (at an average air density of 1.2kg/m3) weighs 0.6g. Say 0.5g in round numbers, and for simplicity let's say (because you have two arms) that you're moving 0.5g of air continuously forwards. Let's also simplify the movement and say that you're accelerating this air linearly over a 1m linear distance, and that your arm movement has a 4m circumference (which gives you the speed of your arm, and hence the air). Then for revs/s R and F=ma,

200 = (0.0005 * R) * (((4*R)^2)/2)

8R^2 + 0.0005R - 200 = 0

Solving (and ignoring the negative solution),

R = 80 revs/s

So for your cartoon character capable of whirling their arms like helicopter blades, this could genuinely be a practical strategy. For humans, not so much.

AlphaLife
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Graham
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  • It is comforting to read a sane answer. – Anton Jan 01 '23 at 23:23
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    Sane, perhaps, but not correct. See my comments in other answers. It is not only the initial motion that helps restore balance: windmilling provides a net angular acceleration over time. Windmilling for longer helps more, as long as the net angular acceleration (over a period of time) produced by windmilling is greater than that of gravity (over the same time period). Also, the claim here that the only effect of windmilling is via air resistance is incorrect. – Some Guy Jan 02 '23 at 07:24
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    @SomeGuy Your analysis is incorrect though. You make the assumption that it's also possible to bend knees and pivot your body backwards synchronously with arm movement. This is clearly impossible for any significant speed of arm movement. But also if you can do that then you don't even need to move your arms, because this means you have enough traction with your feet to simply pivot from the ankles and push backwards (as I mentioned). The only reason to whirl arms is if you don't have that grip, and then air resistance is all you have. – Graham Jan 02 '23 at 07:41
  • @Graham You have not understood my argument. Bending your knees is not necessary (although it may help if you can do it fast enough to be in time with your windmilling). Air resistance is not all you have. As you describe, it's not even likely to be very helpful. Windmilling works because torque depends on distance: moving 1kg a foot in a perpendicular direction at 3 feet from the pivot point (the feet) produces a different torque than moving 1kg at 6 feet from the pivot point: Moving arms "forward" at the top of the circle does not completely cancel moving them "backwards" at the bottom. – Some Guy Jan 02 '23 at 07:52
  • @Graham Try to understand how the self-balancing stick works in the video in Cleonis's answer. They don't work by air resistance. They work by differential torque at the top versus bottom of the spinning wheel(s) imparting a net torque on the stick itself. – Some Guy Jan 02 '23 at 07:56
  • There's also the fact that swinging your arms isn't just to maintain balance; It's to stop you falling forward, even at the cost of over compensating. If you're on a cliff edge, the preference will be to fall backwards over falling forwards and that would be quite sufficient. – UKMonkey Jan 03 '23 at 15:14
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I haven't yet seen an answer that clearly states the conservation law in play, or clearly describes the forces of interaction with the environment.

Assuming a simple point contact with the edge, you cannot produce a torque at that point. Thus, the forces you can create do not affect your angular momentum around that point. Only gravity, which acts on your center of mass, does.

Thus, your angular momentum around the edge will increase unless you can get your center of mass behind the edge. To move your center of mass, you need reaction forces with the environment. What forces are at your disposal?

The contact force can be broken into two parts — a normal force and a tangential force from friction.

Ignoring the exact direction of the normal force (up and maybe a little forward), we do know it has to be positive (you can't pull on the edge.) The tangential force is perpendicular to the normal force, and has to be within the limits of friction, or you will slip. Intuitively, it seems like you might be able to create some horizontal force to move your center of mass backwards.

How can arm rotation connect to center of mass motion? Revisiting our "conversation law", angular momentum around the contact point is the sum of angular momentum around the center of mass, and the angular momentum of the CoM around the contact point.

That means that if there's more angular momentum around the center of mass, there's less motion of the center of mass around the contact point. Enough "internal" angular momentum, and the center of mass will move the other way.

That change in center of mass velocity has to be created by the contact forces. In other words, attempting to accelerate your arms will create a reaction force with the ground. That's what actually accelerates your center of mass.

But you have to keep moving them: only so long as you have some angular momentum around your center of mass can you keep your CoM moving back towards safety.

Glorfindel
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Alex K
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  • As far as I can tell, this answer is the only fully correct one here, although it's stated a bit confusingly. – Some Guy Jan 02 '23 at 07:32
  • yes! finally someone giving the conservation of angular momentum. For people who want to see a demonstatation: https://www.youtube.com/watch?v=n_6p-1J551Y – Apfelsaft Jan 03 '23 at 03:22
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The force of gravity pulling you down acts at your center of mass. The sum of the external torques equals the change in angular momentum. If you have no initial angular momentum gravity pulls you over (like a stationary bike turning over). If you have some initial angular momentum gravity tends to turn you (like a bike rolling forward). But I doubt you can create enough initial angular momentum in the correct direction to make much of difference (like a slow moving bike).

An analogy is jumping upwards from an elevator in free fall that has a large velocity downwards. The decrease in your net velocity downwards is too small to save you.

Moving your center of mass (e.g. leaning away from the edge) will help, but this is differnt from swinging.

John Darby
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  • If you are positioned so that you can move your centre of mass away from the edge you are not in danger. – Jasen Jan 01 '23 at 10:35
  • This is a good basis of a sane and physically realistic answer. It is based on sound dynamics involving understood terms such as conserved angular momentum and deserves only a little extra work and clarification. But why bother when so many readers seem satisfied with the vagueness and confused empiricism of other answers? – Anton Jan 01 '23 at 22:32
  • See my other answers. It's not all about the initial angular momentum. Also, a bike's gyroscope spins at right angles to gravity. Not so here. In this case, the faster you windmill your arms (and the more massive your arms are, and the higher and lower the top and bottom of the arc, and the longer you do it), the greater the angular acceleration you produce. If this total angular acceleration (over a given time frame) exceeds gravity's total angular acceleration (over the same time frame), you end up rotating more towards the vertical and can maybe eventually recover your balance. – Some Guy Jan 02 '23 at 06:38
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Assuming you are in an unstable stance where your centre of mass is slightly past the brink - if you do nothing you will topple forwards and fall over the brink.

Throwing your arms forwards does not help at all! Your centre of mass stays in the same place (conservation of momentum), if it was past the brink it is still past the brink you have gained nothing.

When you start spinning your arms that puts rotational momentum into them this puts a brief torque (or an anglar impulse) on your body via your shoulders which angular impulse may be enough to tilt you back onto the safe side of the brink.

When you stop spinning there will be an opposite impulse so be sure to full regain steady balance first.

Your arms don't twist when you spin them but that's not actually a problem, rotational inertia is mass × radius squared, and you still have that.

You can experiment with this at a swimming pool or over a soft mattress. or hay pile etc.

Jasen
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    Moment of inertia has the dimensions of mass times length squared, so your mass x radius is wrong. – Anton Jan 01 '23 at 23:22
  • That is not correct because your legs are attached to the ground and there can be a non-zero net force acting on the center of mass. – John Alexiou Jan 02 '23 at 06:54
  • Jasen seems to be the only one here who understood torques.. @JohnAlexiou what is a non-zero net force? – Apfelsaft Jan 02 '23 at 09:23
  • @JohnAlexiou in all cases where spinning arms is needed your centre of mass is past the brink and so you cant impart any beneficial torque with your feet. ( Unless you are wearing Micahel Jackson's patented trick shoes https://www.youtube.com/watch?v=sFvENQBc-F8&t=240s https://patents.google.com/patent/US5255452A/en ) – Jasen Jan 02 '23 at 10:06
  • @Jasen That's not correct. Just because you are about to fall if gravity continues to accelerate you to tip over (rotate) farther doesn't mean that you have irreversibly fallen yet. You may be still able to exert force on the ground in some direction with your feet to correct yourself. You just need a source of torque to apply such a force. Such as windmilling your arms, to apply angular momentum to them in one direction, and hence apply angular momentum in the opposite direction to the rest of your body (Newton's third law). – Some Guy Jan 03 '23 at 07:32
  • Also, it's not only the initial starting and stopping of the spin that helps save you. Every time your hands go over your head one way and pass your waist the other way, there is a net imbalance of torque (around the pivot point of your feet). If done quickly enough, this can add up to correct your lean faster than the torque from gravity trying to increase your lean. If you could spin your arms fast enough (spoiler: you can't), you could hang out at a 45 degree angle over the abyss indefinitely, if you had enough grip on the cliff edge with your big toe to prevent a straight downwards drop. – Some Guy Jan 03 '23 at 07:46
  • @SomeGuy please re-check SFAIK momentum is always conserved. – Jasen Jan 03 '23 at 20:27
  • @Jasen Momentum is conserved. If you are a robot sitting there windmilling your arms rapidly in order to hang indefinitely at 45 degrees over an abyss, your body is trying to rotate towards the abyss, but can't, because the torque from windmilling is cancelling the torque from gravity, and forcing your toe into the ground harder than just your own weight would. This rotation force is being met by an equal force from the ground holding you up and stopping you from rotating. Momentum conservation says that the earth's rotation, and compression of the dirt under your foot, are therefore affected. – Some Guy Jan 04 '23 at 02:51
  • Consider how this "magic trick" works: https://www.youtube.com/watch?v=-bLjMrqRNfw. In that case, the reverse torque that fights gravity's torque is provided by the fact that the fork/coin center of gravity would have to rise in order to let the whole system rotate (the fork weight counterbalances the coin weight), but the same effect could have been had if you were to angle the handles of the forks out farther and put a spinning motor with a flywheel on the end of each fork instead. – Some Guy Jan 04 '23 at 03:03
  • Sooner or later the robot explodes due to centripetal force. This is because continuous torque requires continuous rotational acceleration. fork motors would also explode. – Jasen Jan 05 '23 at 02:10
  • @Jasen Constant torque does not require continuous rotational acceleration. It only requires constant rotation (enough so that the flywheel torque balance's gravity's constant torque). The torque produced is proportional to the rotational speed of the flywheel, not the rotational acceleration of the flywheel. If the flywheel were to accelerate its rotation, the torque would increase accordingly until the flywheel were to stop accelerating, and then remain at the new value (corresponding to the new rotational speed of the flywheel). – Some Guy Jan 09 '23 at 11:42
  • @Some guy, stop trolling me with your alternative physics. if you think you have an answer post it. – Jasen Jan 09 '23 at 21:53