# Can the fish topple the bowl?

A man is standing inside a train compartment. He then hits one side wall of the compartment with his hand (or you can assume he kicks the wall with his leg). Will the compartment begin to move? I don't think so. (If it happens, there will be no need of fuel.)

I mentioned this incident as a background for my question.

Suppose there is a fish bowl (totally round) on a table and a fish swims inside it. It wants to get out from the bowl to achieve freedom {yes, it won't be the freedom :) }. But it cannot jump over the edge. Then it tries to topple the bowl by hitting the wall of the bowl (Assume that the fish has the enough strength).

Is it possible?

I thought 'no' until I read this question (It might be better to say that I am still in my opinion): Could a fish in a sealed ball, move the ball?

I doubt whether this is slightly different from my question. Can someone explain?

EDIT: I got some answers here those state that the swimming of the fish can move the bowl. While googling I found a similar question on another website. Instead of comforting me, it doubled my confusion :( . Though, causing my consolation, it says what I previously thought. You can find it here.

• Is the train compartment sitting on the ground? How heavy is it compared to the man? Is it attached to the rest of the train? (if so then he'll never move the train, but if it's just a box on the ground and not too heavy, he has a chance) Commented Aug 3, 2021 at 8:49
• so you can assume a 'box'.
– ACB
Commented Aug 3, 2021 at 9:15
• When you say the fishbowl is "totally round", do you mean a perfect sphere? Because that is not how most fishbowls are shaped. They generally have a flattened bottom, specifically to prevent them from tipping over. If the bottom of the bowl lacked this flat side, I see nothing that would stop a sufficiently active fish from tipping it over, even by accident. Commented Aug 3, 2021 at 16:59
• @DarrelHoffman, I thought that if the bottom was flat, it would be difficult for the fish to topple the bowl (if it could). That's why I used the phrase 'totally round'.
– ACB
Commented Aug 3, 2021 at 17:11
• Alright, just clarifying because that is not the shape of a normal fishbowl, for exactly that reason. I'm pretty certain a fish could move a perfectly spherical bowl on a smooth flat surface even without intending to (which would of course be disastrous for both of them). Commented Aug 3, 2021 at 17:17

### Summary

The fish can only move the bowl horizontally, at all, if he can use the force of internal momentum transfers, $$F= m \tfrac{dv}{dt}$$, to overcome the force of static friction with the table, $$F= \mu_s Mg$$, and can only move it vertically if he can overcome the force of gravity $$F= Mg$$. Same is true for the man if the car is stationary and we replace coefficient of static friction with static coefficient of rolling friction. “Perfectly round” is saying more than people often realize, it is an impossible theoretical case, so I mostly address a real-world round bowl.

Furthermore, as discussed at the bottom, NASA slosh scientists who analyze slosh in liquid nitrogen tanks for the mentioned applications generally disregard anything inside the tank, like a mixer, because it is very hard to create any (net) momentum in the liquid from the inside; it usually takes external motions and forces to get any meaningful sloshing, also for the reasons discussed.

### Man in Train Car

For the man in the car, there are three cases to consider:

1. No friction between the wheels and the track

2. A constant “coefficient of rolling friction” between the wheels and the track

3. A coefficient of static friction and a coefficient of rolling friction that are different.

Case 1:

He can never change the center of mass (com) of the man/car system.

This is because he cannot apply any net force on the man/car system. There is no external friction nor any other source of external force. Anything he does to push on the wall will push on him with equal force, and the net resultant force on the man/car system will be zero.

But even in this case he can cause slight temporary back-and-forth motions of the car from the outside, but he cannot change the com of the whole system so he won’t be able to move the car on a continual basis or for a significant distance. The way he can move it a little is by moving in the car. For any motion of the man relative to the car along the direction parallel to the track, no matter how he moves (slowly, quickly, pushing on walls, walking), the car will move in the opposite direction enough to keep the system com in the same place.

Case 2: (constant coef of friction):

He can start the car rolling and actually move the system’s com, but due to friction it will stop again. But then he could do it again. He walks slowly in one direction while it is stopped, keeping his momentum transfers low enough to not overcome friction with the track. Then moves the other direction with enough force to overcome friction and start the train rolling (perhaps by jumping and pushing on the wall, or just sprinting the car length).

Case 3: (Static friction is more than kinetic):

This doesn’t change things a ton, but it matters. It makes it much easier and more forgiving when starting out, and remember we must start out over and over on our journey. We have a differential friction gain. Friction differences in each direction provide the external force that moves com.

——

### The Fish No Friction

Our case: If the bowl is perfectly round (as nothing is), this is a frictionless case when we consider rolling. Again, perfectly round is no friction. He will be able to tip the bowl using even minute waves, or while motionless using his innate differential density capacity, discussed below. One may think it’s not frictionless because the bowl and surface will give a little, but there’s still maximum material stress and force at the very bottom, decreasing radially from there, which only requires any positive leverage however small. If there is any flat, or even imperfections in the roundness, will the bowl move at all in the frictionless case? The remainder if this section consider frictionless, but with a flat rather than the trivial perfectly round:

The fish faces the same general problem with the added complication of the liquid. But case 1 (no bowl/table friction) remains generally the same. This may seem surprising, but every little motion of the fish that changes com and even small currents will be balanced from a center-of-mass standpoint by the bowl moving on the table. For normal (slow) swimming, whenever he swims to one side, the bowl and water move a little the other direction $$d_{\text{bowl}} m_{\text{bowl}} = ( d_{\text{swim}} - d_{\text{bowl}} )$$ $$m_{\text{fish}}$$ where $$m_{\text{bowl}}$$ includes the water.

There’s a twist. This equation gives no motion if his density equals that of the water. Fish have internal pockets of gas that they can compress or expand to help them go up and down, in addition to just swimming up and down. Gas, unlike water, is compressible. So one surprising result is that if the fish is off to the side, and the water is still, and he merely changes the volume of his gas pouch without swimming... the bowl will move on the table. That’s because he has changed his density and the overall com as seen from the bowl. To see this note that when he is as dense as water, the com is on the vertical centerline of the bowl, and when he is not, it is not. But the com viewed from the table cannot move, so the bowl moves. (Frictionless worlds are impossible and sometimes counterintuitive.) The man in the train does not have a density similar to air, but the fish does to water and this makes it even harder. This all means he needs to use the water to transfer momentum, as this gas-pouch effect is small. However, it alone is enough to tip the bowl because the bowl is perfectly round and only requires any amount net leverage. A more interesting question is whether he can get anything with a small bottom to slide or tip:

If he swims fast there will be water motion to consider too. Yet, he also can’t set up big currents and sloshing and do a lot; the location of the bowl on the table depends only on where everything inside it is, not how fast anything is moving, and the maximum distance moved (with the same location for the center of mass of the system) depends only on how much mass he can get to one side in a peak of water, which is very limited operating from within the tank.

There are sloshing problems even without friction (such as in space), but they require external motions and forces to generate the oscillations, not a fish or even a mixer etc from the inside. NASA has sloshing scientists who analyze how liquid nitrogen tanks can affect things and how to control for it. It can only cause a lot of back and fourth, but that can be a big problem when positioning things in a space station. And the sloshing inside as mentioned comes from motion caused externally. You can probably find NASA sloshing analysis papers online. They use computational fluid dynamics and try to estimate what the maximum short-term momentum change through time, $$F_{max}= m \tfrac{dv_{cog}}{dt}_{max}$$, could be to give that as an upper bound for other engineers to know, and how best to reduce sloshing, with dampers or springs or whatever, which change automatically with tank level because sloshing dynamics change with that. There can be something akin to a resonate frequency, and I think that can even be estimated classically (?).

Even if he makes a little net sloshing (note that things like a whirlpool have no overall effect, and sloshing scientists don’t even call them sloshing), he is limited to motion that corrects com, so the sloshing fish will have to use friction also to get across town.

### Fish with Friction

Because perfectly round is frictionless for rolling, this case is not that. It will be much harder for him. And not just because of what was mentioned in the frictionless case (that he can’t get large amounts of water to one side from the inside). To overcome friction and move consistently, he has to get some mass (in the form of himself and some water) to one side, slowly (ie without exceeding the force of static friction with his rate of transferring mass), and then immediately move it quickly the other way generating large momentum transfer rates (and hence external force). Why must he go the other way immediately and not just quickly? Because it is a liquid and won’t stay to the side. If he stops and takes a breath (😉) the water will begin to move back but not fast enough to do what he needs it to: overcome friction. This detail helps explain why the NASA sloshing engineers as above don’t worry about internal mixers in the nitrogen tanks. If the fish sloshes back and fourth overcoming friction each direction, they largely cancel out. He needs rapid then slow momentum changes. So you see that tipping, even with a very tiny flat on the bottom, or a nearly round bowl, seems impossible from the inside, even for a strong fish, even if he wasn’t so close to the density of water.

Tipping with a flat bottom is even harder because $$\mu_s$$ is much lower than one, usually ~$$0.2 - 0.3$$. Good luck lifting, tipping, or even moving horizontally without getting outside at all

• Comments are not for extended discussion; this conversation has been moved to chat. Commented Aug 4, 2021 at 23:23
• We’ve noticed that you’ve edited this post many times. Keep in mind that each edit you make will push someone else’s post off of the front page, so try to prefer fewer, more substantial edits over many minor changes.
– rob
Commented Aug 7, 2021 at 23:45
• @rob Ok i see. Will do Commented Aug 8, 2021 at 1:35

Yes, the fish can topple the bowl, exploiting friction.

You can find robots based on gyroscopes that can topple their case, like the Cubli (you can find a video here). They can do that from the inside of the case, without external moving parts. The box could even be closed and opaque, hiding the "magic" behind it: it would still work.

That would be very difficult to do for a fish, but you said your fish is strong enough, thus it could hypothetically push a large amount of water and make it spin very fast inside the bowl; the friction between the high-speed water and the glass of the bowl will make the bowl rotate, too.

About the other question (whether the center of mass can move): assuming no friction between the perfectly round sphere and the external environment, you're right: it cannot move without external forces.

## Further notes about the Cubli and similar robots

These robots have 3 gyroscopes on 3 different axes. A gyroscope is essentially a wheel with a relevant mass that turns very fast. An important feature of gyroscopes is that they tend to keep the orientation they already have (try to look for the subject "moment of inertia") and this behaviour can be used to increase orientation stability; you can see an example of gyroscope in this video or this one or you can search for "gyroscope" and find lots of examples and other videos.

The Cubli is also relying on another fact: when the brakes of each wheel are activated, the inertia of the rotating wheel is transmitted to the whole robot, making it turn in the same direction. You can clearly see this in the video of the Cubli: when it "stands up" the wheel suddenly stops or slows down its rotation, while a few instants before it was rotating very fast (so fast that we couldn't see its spokes).

• Does this 'Cubli' use something like a fan?
– ACB
Commented Aug 5, 2021 at 5:11
• No, those are not fans, it would even work without air nor other fluids. Those are gyroscopes! I added further details to my answer. I hope I clarified a little bit the physics behind the Cubli! Commented Aug 5, 2021 at 8:48

In practice probably not, but theoretically for a very strong and clever fish, maybe actually.

Consider the following human hamster ball that you can conveniently buy of Amazon. It sorta conforms to the problem you are describing. As long as there is friction between the ball and the ground, moving weight internally can in fact give you motion. Btw the ball can be found here1, (not an affiliate link)

In practice of course it would be very hard to move the steel box as a person. However, With enough friction, long enough box , and if you are super man, you could run and then hit the box until it starts rolling. This does however require friction between you and the box, and the ground and box.Momentum is still conserved because while you and the box move forward, the earth moves backwards a bit. It's an indirect way of pushing off the ground (much in the same way you push off the ground indirectly via your shoes).

For the fish, he would have to do something similar and make use of friction, being a fish this is likely going to be difficult. At a minimum, for a very strong super fish, he could push off the bottom and jump up hitting the top of the bowl(if covered and if bowl is super strong) and cause the bowl to fly. In fact because there is friction (shear) between water and the bowl, you can eventually impart the momentum on the earth (similar to the person in hamster ball).

For the answers making the center of mass argument, this only true for the center of mass of person (fish)+ container (bowl)+ Earth is roughly static. You can still move relative to the earth.

• I thought a hamster ball moves because the person inside it walks. According to your answer, my confusion is, will the ball move if the person bend forward, without walking?
– ACB
Commented Aug 3, 2021 at 4:47
• You are correct, they are definitely walking inside to move. If the person just bends forward slightly without actually fallowing forwards, I don’t believe there will be much motion. Net horizontal motion will always depend on your ability to push off the ground with friction. Without doing that no net movement will occur. Though now I’m tempted to rent out in the city just to try it out! Commented Aug 3, 2021 at 4:58
• @acb Yes it will move. a tiny bit, exactly enough to move the ground contact spot under the new center of mass caused by the man bending over. And then stop in the new position. Commented Aug 3, 2021 at 7:59
• @acb bending forward will move his center of mass forward a bit, shoving his feet back a bit. This will move the train a bit, assuming it has frictionless wheels. The amount is miniscule though, as the man cannot move his mass much without moving his feet, his own center of mass must be somewhere between his heel and toes, otherwise he will fall over on his face. That does not leave much room for movement. Commented Aug 3, 2021 at 8:05
• Also, you don't need friction between person and box: The person could simply propel themselves from side to side, accumulating speed. If there wasn't friction between box and ground you could still do that; conceivably the person could topple over the box (or the fish the bowl) but not move the overall center of mass in the process -- for that you need friction or a ledge to "push off of", that is, transfer momentum to Earth. Commented Aug 3, 2021 at 10:53

A slow unintelligent fish which just moved gradually from one place to another would not move the center of mass of the water plus fish, so would not topple the bowl.

However I think a clever fish might be able to set up a kind of sloshing motion of the water which could eventually result in enough water on one side to topple the bowl. Such a fish can cause the center of mass of water plus fish to be displaced, because there are reaction forces from the bowl.

• Thanks for your nice answer. But I don't think, as you say, if the fish is intelligent, it will need to get out of the bowl : ) .(I have mentioned that fish is strong only) :D
– ACB
Commented Aug 3, 2021 at 4:52
• What is a strong fish, does that include heavy? His mass matters. But shape does too. But as mentioned slosh scientists think is very hard to get much slosh from the inside Commented Aug 3, 2021 at 13:58
• @AlBrown, By 'strong' I meant, if the fish wanted, it could topple the bowl.
– ACB
Commented Aug 3, 2021 at 17:14

YES the man will move the train, and the fish will move the ball/bowl.

NO it is not done without fuel. The movement requires the use of metabolic energy, which is derived from fuel(food)

NO this will not make a miracle reactionless space-drive, it works because it is not a closed system. The train, and the bowl, are in gravity field, supported on a surface, and that surface is not frictionless.

In effect, both are examples of an automobile. An engine burns fuel to provide power, which applies torque to parts of the construct, which causes it to roll in a direction.

• But aren't they internal forces? So can they move the centre of mass?
– ACB
Commented Aug 3, 2021 at 12:03
• @acb You are in a car, with engine running. You floor the accelerator. Plenty of internal forces, but you cause a rotation of the wheels, transferring this internal force to the outside via friction. Same with the man in the train, and the fish in the bowl. Because both are on a surface that will resist relative motion(friction), they are able to apply forces to the surface, affecting the inside of the containers. If there was NO friction, you could shuffle internal layout of the mass (thus causing temporary rotation), but not any movement of the center of mass. Commented Aug 3, 2021 at 12:59
• The fish will only move the bowl horizontally if he can overcome static friction and vertically if he can overcome gravity. Same with the train if it is still. Commented Aug 3, 2021 at 13:38

The man can move the train compartment. The magic word is: friction (static friction or "stiction" to be more precise).

Friction is a non-conservative force. What defines a force as being conservative is this behavior: When you move around an object along a closed path (a loop), and when this object at the end has the same position as before, then the object also has the same energy as before. So, the total work that is done along a closed path is always exactly 0.

But friction is not such a force. When you drag a heavy stone in a circle across the floor, you need a lot of energy to drag the stone. But when the stone at the end is where it was before, it still has the same gravitational and kinetic energy as before. So, the energy you invested in dragging the stone has gone. (In fact it has turned into heat, but it no longer is available as mechanical energy.)

Knowing this, the man in the train compartment can to this:

1. Slowly walk to one end of the compartment and stop there.
2. Walk or run with constant, but small acceleration (not with constant speed! You need to accelerate constantly) towards the other end of the compartment
3. Hit this other end and bounce back with the same speed that you had immediately before hitting the wall, just in the opposite direction.
4. Run or walk to the other end, becoming slower and slower with the same constant acceleration you used before, until you stop exactly where you started.

Why does this work?

This works better the longer the train compartment is. Ideally, you should be able to use the entire train car.

The wagon does not move in step 1 due to static friction if you move to the end without phases of high acceleration.

Step 2: When you accelerate to one side with a small amount of acceleration you create a small force on the wagon that wants to move it in the opposite direction of your own movement. But this force is too small to overcome the static friction. The compartment will not move.

Step 3: When you hit the wall and bounce back, you transfer a lot of kinetic energy in a very short amount of time onto the wagon which results in a force big enough to overcome the static friction. The railcar will start moving.

Step 4: While you walk back to the other side with decreasing speed, the wagon will slow down too, but because it has a much bigger mass, it still keeps rolling along the track when you stop at the other end, being where you started from.

At the end you are at the same position within the wagon as before, but the wagon with you inside keeps rolling.