# Weight of a box with a flying bird

## Main Question or Discussion Point

Hello! This is my first post here, and I look forward to many more! This is a sort of riddle I was told today, and left me full of doubts.

So, the 'puzzle' was the following:

" There's a man carrying behind his truck an enclosed trailer full of birds inside it. All the birds are standing on the floor, none are flying. The weight of the trailer plus all the birds standing exceeds the permited amount the truck can carry (determined by law). At some point, the truck has to go through one of those things that check that the weight of the trailer doesn't exceed the permitted amount, so the man decides to scare away the birds so that they fly inside the trailer " How much will the trailer weigh?

So the riddle's answer provided is " it'll weigh the same because the air pushed down by the birds' wings will create the same equalizing force by the trailer's floor as it did when the birds were standing".

And I was thinking, won't some of that air pushed down go to sides and end up colliding with the walls instead of the floor? And if not, if still all the air pushed ends up colliding with the floor, will the trailer still weight the same as it did before?

I hope you understand my question, I have tried to translate the quiz as well as possible, since it was in Spanish originally.

If you don't understand it well enough, ask me and I can try to explain it in other ways :)

## Answers and Replies

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jrmichler
Welcome to PF.

Good question, it shows that you are thinking about the concepts in the riddle. You are thinking of a "vortex ring state" (google the term), which almost certainly does not apply to the stated problem. The turbulence and resistance (drag) of the square corners and walls will prevent a vortex ring state from happening. Also, the birds will be spread out, so there will be no place for the air to get into a ring state. And if there was a sustained updraft, some bird would find it and get a free ride.

Open the box, let the birds out, then the truck is lighter.

Welcome to PF.

Good question, it shows that you are thinking about the concepts in the riddle. You are thinking of a "vortex ring state" (google the term), which almost certainly does not apply to the stated problem. The turbulence and resistance (drag) of the square corners and walls will prevent a vortex ring state from happening. Also, the birds will be spread out, so there will be no place for the air to get into a ring state. And if there was a sustained updraft, some bird would find it and get a free ride.

Open the box, let the birds out, then the truck is lighter.
Thank you so much for answering! I think that clears things out, I am going to check out the vortex ring state to know more about it! Greetings!

rcgldr
Homework Helper
You are thinking of a "vortex ring state" (google the term), which almost certainly does not apply to the stated problem.
Vortex ring state occurs when a hovering object descends fast enough that the vortex rings at the tip of the lifting surfaces significantly reduce lift. Even in the case of humming birds, I doubt they could descend fast enough to achieve vortex ring state. Vortex ring state is a condition that affects helicopters, depending on the article, part of the issue is the descent rate is fast enough for there to be a relative updraft through a portion of the rotor plane, but the key issue is that the descent rate is great enough that the vortex rings significantly reduce lift, requiring about double the power it takes to hover in order to recover from without having to translate sideways to use wind shear to scrub off the vortex rings. The next step of descent rate is an autogyro state, assuming that the rotor can enter this state during a vertical descent.

birds flying in truck
First consider how the weight of the air affects the truck. There is a pressure differential within the assumed to be sealed truck, less pressure at the top, more pressure at the bottom, with the result that the net downforce exerted by the air inside of the truck is equal to the weight of the air inside the truck. If birds are flying inside the truck, and assuming no net vertical component of acceleration, then the pressure differential within the truck is increased so that the net downforce exerted by the air inside the truck equals the weight of the air and the birds.

Consider the case of a helium filled balloon, deflated with a high pressure tank, and initially resting on the bottom of the truck. As the helium is transferred from the tank to the balloon, it expands, and the balloon eventually hovers, and the increase in volume of the balloon also increases the overall pressure and pressure differential within the truck, and once again, the downforce exterted by the air onto the truck equals the weight of the air and the balloon. This still holds true if the balloon is expanded further, rises to the top and presses upwards on the ceiling. The pressure differential will be increased, and the net downforce within the truck will still equal the weight of the air and the balloon.

An alternative way to look at this is that it's a closed (sealed) system. Gravity pulls the same on all the objects within the truck, air, birds, balloon, regardless if the objects are at rest or hovering, so the total weight of the truck is constant as long as there is no vertical component of acceleration.

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FactChecker
Gold Member
Good point. That is an interesting second step in thinking about the puzzle. But go one step further and consider a few cases:
1) Suppose the air hitting the walls were first bounced off the floor. Then the weight did not decrease from that.
2) Suppose the birds try to fly sideways and all the air from their wings hits the walls directly. Then the birds would fall to the ground. Their "flying" was completely inefficient. The weight will not decrease.
3) Suppose the air is a mixture of cases 1 & 2. Then the birds are flying without complete efficiency. They must fly harder and get enough air moving down to keep in the air. That would make enough air hit the floor to keep the weight from decreasing and the inefficiency of the flying would cause other air to hit the walls.

So, assuming that the birds fly hard enough (considering efficiency) to stay up, the weight will not decrease.

Delta2
Homework Helper
Gold Member
I agree to the answer provided with the riddle but as long as we consider some extra assumptions:

1. The birds fly stationary, that is they are just flapping their wings and floating, without travelling inside the box
2. The box is air sealed
3. The density of air inside the box remains unchanged (assumptions 1 and 2 also helps in this)

Then the air exerts to the birds a force equal to the total weight of the birds (since each bird is floating in air). By newton's 3rd law, the birds exerts to the air a net force equal to their total weight $\sum B_i$. Now given the assumptions 1-3, the centre of mass of the air inside the box does not change, hence the total force exerted to the air is zero. The total force exerted to the air is the force from the box plus the force from the birds (which as we argued is equal to the total weight of the birds). Hence we can write the equation for the total force exerted upon the air inside the box:
$$\vec{F_B}+\sum\vec{B_i}=0 \Rightarrow \vec{F_B}=-\sum\vec{B_i}$$

where $F_B$ is the net force from the box to the air inside the box. So by newton's 3rd law the air exerts to the box a force $-\vec{F_B}=\sum\vec{B_i}$.

Thanks @rcgldr for that very detailed answer, so all in all, the riddle's provided answer seems to be correct. The gravity example really explains everything more clearly if you take into account the balloon example which is simpler than the birds one.

Also appreciate your answer @FactChecker I think that all those suppositions are fulfilled by the riddle, so again the conclusion should be that the answer provided is correct!

And thanks @Delta² for your more detailed and mathematical approach. It seems that this problem explains the third law principle pretty well, but can be examined very thoroughly if wanted.

Thanks everyone for helping :)

rcgldr
Homework Helper
The density of air inside the box remains unchanged ... the center of mass of the air inside the box does not change.
The overall density doesn't change, but any increase in pressure differential due to the air supporting one or more objects within the truck means the center of mass of the air is lowered slightly, since the density at the top will be a bit less and the density at the bottom will be a bit more.

Delta2
Homework Helper
Gold Member
The overall density doesn't change, but any increase in pressure differential due to the air supporting one or more objects within the truck means the center of mass of the air is lowered slightly, since the density at the top will be a bit less and the density at the bottom will be a bit more.
Yes you are right on this. I think I should replace my 3 assumptions with one overall assumption that
1. There is no vertical acceleration of the center of mass of the air inside the air sealed box.

Then my analysis for the forces holds for the vertical component of the forces.

And we need to see what assumptions we should make in order for assumption 1 to hold. I suppose for example that the birds can fly inside the box as long as their fly doesn't have a vertical component of acceleration so then 1. still holds.

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jbriggs444
Homework Helper
2019 Award
vertical component of acceleration
The average vertical acceleration over time cannot vary greatly from zero. Otherwise the center of gravity of the truck contents would move outside the truck. The truck is, by assumption, sealed.

Delta2
Homework Helper
Gold Member
The average vertical acceleration over time cannot vary greatly from zero. Otherwise the center of gravity of the truck contents would move outside the truck. The truck is, by assumption, sealed.
Yes that is correct for the average vertical acceleration. But if a bird for example is flying up and down (with an acceleration) all the time , then the average acceleration would be almost zero, however for half the time (the time that the bird is flying up with acceleration) the weight of the box will increase, while for the other half of time, the weight of the box will decrease.
BUT yes the average weight of the box would remain almost the same...

Yes that is correct for the average vertical acceleration. But if a bird for example is flying up and down (with an acceleration) all the time , then the average acceleration would be almost zero, however for half the time (the time that the bird is flying up with acceleration) the weight of the box will increase, while for the other half of time, the weight of the box will decrease.
BUT yes the average weight of the box would remain almost the same...
The bird acceleration in vertical or horizontal or whatever directions results NO net downforce is equal to zero ,since those forces are internal forces and mutually cancel out ,the truck will be the same wether birds are flying or sitting .

A.T.
The bird acceleration in vertical or horizontal or whatever directions results NO net downforce is equal to zero ,since those forces are internal forces and mutually cancel out
What force cancels out what force exactly?

,the truck will be the same wether birds are flying or sitting .
What about the bird sitting vs. the bird pushing off with the legs to get airborne?

Delta2
Homework Helper
Gold Member
The bird acceleration in vertical or horizontal or whatever directions results NO net downforce is equal to zero ,since those forces are internal forces and mutually cancel out ,the truck will be the same wether birds are flying or sitting .
They cancel out if we make the sum of all forces acting on the system of box+birds. If we consider only the box the force from the birds on the box (it is actually the force from the birds to the air inside the box, and then this force is transmitted from the air to the box, the air is playing the role of the man in the middle to transmit the force from the birds to the box) is not cancelled by anything.

Oh my god, this explanation is so brilliant and simple at the same time, my respects to Stanford haha, thanks for sharing this, and to all of you who tried to solve the riddle with your own hypotesis and methods. Although I'd looked up the riddle on the internet I'd found nothing before, so it's good that it is now clear. It seems to even make sense that they have to make something like twice the force down to then mantain flying with no effort for a little time.

Is the truck covered with canvas, is it sealed, are the birds in a cage like structure that the truck is carrying allowing air flow, etc. These might effect the answer