Wikipedia:Reference desk/Archives/Science/2023 December 8

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December 8

Bee in balloon

If a standard bumblebee was placed inside of a standard party balloon, the balloon was pumped up with air and fully inflated, picked up, and then released in mid-air, could the bee fly into the walls of the balloon and thus move the balloon around the room, as if the balloon was flying around the room? Ignoring any negative effects that the pressure inside the balloon could have on the bee. (I am aware this is a bit of an unusual question.)

Any insights would be greatly appreciated!

User:Heyoostorm^_talk! 04:35, 8 December 2023 (UTC)

According to the Waikato Domestic Beekeepers Association, a honeybee "can bring back half her weight, i.e. 0.05g, though sometimes she brings back only 0.02g." A generic-looking filled balloon weighed in at 3.5 grams.[1] That seems like too much for one bee to manage. Clarityfiend (talk) 09:18, 8 December 2023 (UTC)

As the poor bumblebee bounces back, the balloon will get a jolt forward, but it won't move out farther than its diameter. The centre of mass of the system formed by balloon + bumblebee does not budge and however far the balloon moves, it will keep this centre of mass inside. If the bumblebee is much heavier than the balloon, the centre of mass of the system will in fact be inside the bumblebee, so the bumblebee flying will hardly move the bumblebee itself, but mainly cause the balloon to bumble around while remaining confined to a rather limited space. This is only a first-degree approximation, though. The full story is more complex, because the movements of the balloon will be subject to friction with the air in the room, transferring momentum (and causing air currents), so balloon + bumblebee is not a fully closed system. Because of this friction the balloon will make more of a subtle Brownian motion and – purely theoretically – a fully situation-aware bumblebee might be able to impart a bias on this motion.  --Lambiam 09:39, 8 December 2023 (UTC)

But if the bee stings the balloon in frustration – loud pop and the bee flies away. Martin of Sheffield (talk) 09:49, 8 December 2023 (UTC)

A bee powers its flight by pushing air backwards and down, which pushes itself up and forwards by Newton's 2nd Law of Motion. Here the momentum of the air pushed back is equal and opposite to the bee moving forward, and if the momentum of that air didn't then distribute in all directions (still equal and opposite in all directions btw) it would propagate straight backwards, hit the back wall of the balloon, and bounce forward again (if there's any elasticity in the collision) while the balloon wall is bounced backward. The bee, meanwhile, continues forward to the front wall of the balloon, where is hits with the same momentum and bounces back. Or image the bee pressed up against the front wall of the balloon flapping with all their might -- the backward-moving air will again push off the back wall and counteract the forward force. As Lambiam notes above, this is what a closed system looks like, and we can simplify it by looking only at the center of mass and forces acting outside the system. The bee and balloon can sadly only move relative to each other.

Now let's say it wasn't a bee in the balloon, but a micro spacecraft using some ion thruster where only a the outgoing ions take a long time to all collide with the surrounding air, and they pass cleanly through the balloon. Then your bee-balloon system is no longer a closed system -- your spacecraft pushes out exhaust that travels beyond the balloon walls, giving your spacecraft as much forward force relative to the surrounding air as they have fuel for (and taking the balloon with it when it hits the walls). Note that the ions leaving the balloon-spacecraft system are a loss of mass from the system as well as energy.

(On a mechanical sidenote: from the sounds of it you probably want your balloon floating steady in the air, in which case you'd want it to have equal density (the balloon + the air it contains) which means it has to offset the weight of the balloon by having a bit of helium mixed in the air, or having the outside air be comparatively very cold, or something like that.) SamuelRiv (talk) 11:04, 8 December 2023 (UTC)

A major correction -- an error of intuition we're frequently making with these simple pedagogy problems (which is why we should work them out thoroughly at all angles before presenting them in a classroom). The bee is pushing on the balloon can moving it very slowly back and forth in the surrounding air -- if the surrounding air (the environment surrounding the closed bee-balloon system) were actually a vacuum, everything I said above about it having no net movement would be correct. However, in cases like this, a relatively small object moving through a fluid slowly (so the fluid looks very viscous, very low Reynolds number), when the bee collides with the front wall of the ballon, the balloon is jolted forward ever so slightly (thanks again Lambian) overcoming the static friction necessary to move the air out of the way, but then is ground to a halt from the dynamic friction of the viscous air. Meanwhile, that rearward motion of the bee and the air, with equal momentum, dissipates into a much more gradual push backward which almost certainly will not be enough to overcome the viscosity of the air and move the balloon backward at all (and even if it did, it would be slower and stopped from friction much sooner). Thus the bee, if it is banging on the front wall instead of pushing on it continuously like a tractor, can ratchet it forward millimeter by millimeter using the viscosity of the air as a brake. SamuelRiv (talk) 12:33, 8 December 2023 (UTC)

As an aside, when we first study physics or teach intro classes we are encouraged to abstract into the largely frictionless world of Newton's Laws. But for most normal humans, the everyday experience of air and water means that intuitive physics is fluid physics. So perhaps our intro physics classes -- the ones trying to explain that math describes the world we actually see and feel around us (not just throwing a baseball or trains on a track) -- should actually begin with, dare I say, a modernized Aristotle? Carlo Rovelli 2013 argues a compelling case on the subject. SamuelRiv (talk) 12:39, 8 December 2023 (UTC)

For something similar, once the balloon has landed somewhere, there are Mexican jumping beans, in addition to Hamster balls. Modocc (talk) 11:09, 8 December 2023 (UTC)

Great examples! A jumping bean, when it jumps on the ground, is of course not a closed-bug-bean-system anymore because the force the bug exerts downward is countered by an opposite normal force from the ground (which can be arbitrarily large, thus the bug can jump with all its might and not merely be pushing its the shell backwards). An analogy with our bee-balloon system would only occur if the bee is flying up against the wall of the balloon (or pushing air against it) that is then adjacent to a hard surface. If the balloon is resting flat on the horizontal ground, and the bee is flying against the wall of the balloon in a horizontal direction (i.e. parallel to the ground) nothing changes from the above. However, if the balloon instead flies at any angle toward the ground, diagonal or whatnot, or if the ground were diagonally sloped, then there would be a contribution of a normal force from the ground, outside the bee-balloon system, and the system could actually start moving.

To take the hamster wheel, rolling works on a similar principle of having a normal force outside the system, which is necessary for a wheel to roll on the ground. Given that, the wheel in the abstract can roll smoothly indefinitely (but of course in the real world it has rolling resistance). SamuelRiv (talk) 11:51, 8 December 2023 (UTC)

In the limit we can have the bee enclosed in a very thin rubber suit (super 🐝 with 🅱️ on the chest?) and flying the '🎈 balloon' along so there's more to it than just the conservation laws 😁 NadVolum (talk) 12:07, 8 December 2023 (UTC)

When the bee is encased in rubber, the rubber-encased wings now flap against the outside air and it flies as normal. The point of the exercise of the spherical balloon enclosing the bee is that there is no way for the balloon, with only air and a bee moving slowly inside it, to push the outside air around it with a steady net force in some direction. SamuelRiv (talk) 12:18, 8 December 2023 (UTC)

A common similar "teaching example" is the motion of a balloon in a closed car as the car accelerates and decelerates. DMacks (talk) 12:52, 8 December 2023 (UTC)

A close analogy, perhaps easier to visualise, would be if a person inside a rowing boat, entirely sealed by a tarpaulin (so no interaction with the outside air or water, or ability to throw mass out of the boat), were to try to move the boat by pushing on the inside of the bow, or by bumping themself against it. {The poster formerly known as 87.81.230.195} 90.199.215.44 (talk) 14:34, 8 December 2023 (UTC)

As anyone who has rowed knows, this does significantly move the boat ("check"), and can be used to impart net momentum. Hence my corrected reply above. SamuelRiv (talk) 14:45, 8 December 2023 (UTC)

If the balloon is on the ground and can slide forward or back, then the trick with static friction works. The bee alternates fast rearward accelerations with slow forward accelerations. The reaction force on the balloon alternates between a large forward force, enough to overcome static friction, and a small rearward force, not enough to overcome static friction, so the balloon slides forward. But that only works for a balloon sliding over a surface. In case of fluids, there's no static friction, no threshold before any movement happens. But there may still be a way to move.

Suppose the bee alternates fast forward moves with slow rearward moves, both over the same distance. By conservation of momentum, the balloon makes the reverse movement: fast rearward moves and slow forward moves, with durations inversely proportional to the speed. The drag will, during each move, apply an impulse proportional to drag times time interval. Assuming the drag increases quadratically with speed (high Reynolds number), this impulse is proportional to speed, so more impulse is applied during the rearward move of the balloon, providing net forward propulsion.

The thing breaks down when the drag is proportional to speed (low Reynolds number). Now the impulse applied during the forward move is of equal magnitude and opposite direction as the impulse applied during the rearward move and there can be no net change of momentum.

In general, a bee in a balloon (or a person on a boat) can provide propulsion using drag, without directly interacting with the surrounding medium but just moving back and forth, if and only if the drag isn't proportional to velocity.

PiusImpavidus (talk) 17:45, 8 December 2023 (UTC)

Note that for a boat well below the hull velocity, it's a pretty good approximation to say that drag is proportional to speed – except that for an asymmetrical hull the constant of proportionality is different for a forward move compared to a rearward move. PiusImpavidus (talk) 17:54, 8 December 2023 (UTC)

In a jolt, an object's position undergoes an appreciable change of position in a very small time, as when integrating the Dirac delta function. Δs may be small, but Δt is very small, so Δs/Δt is – very briefly – quite high. Even with the low Reynolds number of air, if the velocity is high enough we may no longer ignore the nonlinearity of the drag. But the bumblebee needs to keep hitting in the same direction for a net effect, which is why I wrote it had to be fully situation-aware.  --Lambiam 18:42, 8 December 2023 (UTC)

I'm surprised that no-one has mentioned the Birds in a truck riddle yet. Mitch Ames (talk) 02:03, 10 December 2023 (UTC)

Cylinder capacity

In an un-reffed source, a 6-cylinder engine is claimed to have a swept capacity of 6.191 litres, but the cylinder dimensions are not stated. How might have this particular figure been arived at?
The engine manufacturer definitely used π = 355÷113, thus π/4 = 0.78539823, but other approximations like 3.14 and 22/7 may have been used. The manufacturer almost always increased bore and stroke in 5 mm increments, eg 100 * 110, 100 * 115, 100 * 120, 105 * 130 etc., and only once used a cylinder of 100 * 106 in various 4, 6, 8, and 12-cylinder designs. Big differences in the dimensions (eg 90 * 140) are unlikely, as are oversquare dimensions (eg 110 * 100).

Capacity in litres = (bore² (mm) * stroke (mm) * π/4 * no. of cylinders) ÷ 1,000,000
The closest I can get to the claimed 6.191 litres is with π = 355÷113 and a bore & stroke of 100 * 130 = 6.126 litres: taking π as 3.14 results in 6.123; with 22/7 = 6.128 litres. (I hate doing maths in public, there are probably some errors). Other possible dimensions taking π = 355÷113:

105 * 110 = 5.714 litres

100 * 125 = 5.890

105 * 115 = 5.974

105 * 117 = 6.078 (unlikely, just to check)

100 * 130 = 6.126, closest to 6.191 litres: taking π as 3.14 = 6.123; 22/7 = 6.128

105 * 118 = 6.130 (again unlikely)

105 * 120 = 6.234

105 * 130 = 6.754

Apologies for any mistakes. Any thoughts? MinorProphet (talk) 15:33, 8 December 2023 (UTC)

If bore and stroke are whole multiples of 1 mm you can only get there (assuming cylinders with a circular cross section) when both bore and stroke are allowed not to be multiples of 5 mm:

(99 mm)² × 134 mm × ((22/7) / 4) × 6 / 1000000 = 6.1914... L.

 --Lambiam 20:30, 8 December 2023 (UTC)

Dimensions don't have to be, and often aren't, whole numbers. I doubt any modern manufacturer takes pi as anything other than whatever their software uses. 21:50, 8 December 2023 (UTC) Greglocock (talk) 21:50, 8 December 2023 (UTC)

Many thanks for your kind reply, L. Going with the pattern of 5mm increments it seems possible that the dimensions were indeed 100 * 130, which might indicate that some error or fanciful notion has probably crept in to the claimed figure of 6.191 litres. Thanks again for all your replies. MinorProphet (talk) 22:46, 8 December 2023 (UTC)

All the above calculations seem to assume that the top surfaces of the piston and cylinder are flat. Is it not the case that they are sometimes somewhat domed? This would introduce small variations from purely cylindrical volume calculations. {The poster formerly known as 87.81.230.195} 90.199.215.44 (talk) 22:29, 10 December 2023 (UTC)

Isn't the swept capacity still the same, by a variation on Cavalieri's principle (divvying up the swept volume lengthwise into many thin columns instead of cutting across into thin slices)?  --Lambiam 23:26, 10 December 2023 (UTC)

Thanks for everyone's continued interest. I have finally found a reliable source which says that the engine in question had a bore & stroke of 105 * 120 mm which results (taking π = 355÷113) in a capacity of approx. 6.234 litres, one of the possibilities I mentioned in my OP. The engine is a Maybach HL62 (Table not updated yet...) designed in 1935: a technical drawing of the similar HL42 shows flat piston heads. Later diesel I-6 and fuel-injected V-12s definitely had (concave-)shaped pistons. MinorProphet (talk) 00:09, 11 December 2023 (UTC)

References

• Zima, Stefan (2021) [1992]. "Hochleistungsmotoren 1933 bis 1950". In Eckermann, Erik; Treue, Wilhelm; Zima, Stefan (eds.). Technikpionier Karl Maybach - Antriebssysteme, Autos, Unternehmen. [Originally published as 'Hochleistungsmotoren - Karl Maybach und sein Werk', ed. Zima & Treue] (in German) (3rd ed.). Wiesbaden, Germany: Springer. ISBN 978-3-658-25118-5. Edit for link if interested.

Resolved

A space probe to move Asteroid Apophis in 2029?

Will NASA or some companies send a space probe to move and guide the Asteroid Apophis away from Earth in the year 2029 ? 45.58.92.17 (talk) 20:53, 8 December 2023 (UTC)

Unlikely, given information at 99942 Apophis. Mike Turnbull (talk) 22:32, 8 December 2023 (UTC)