But not long enough to cross a bridge, even if the amount was significant.
The jar increases in mass once the fly is added. The fly is obviously denser then the air it replaces, so that causes a net increase in mass (how much matter is present). But don’t get confused by the fly being in flight - that makes no difference. Newtons Third Law Of Motion states that for every force, there is a equal force exerted in the opposite direction. So… every flap of the fly’s wings will exert a slight downward force on the air around the fly, which will actually push on the bottom of the jar, making it heavier. The upward flap of the wings will also exert a force on the top of the jar, pushing it up, making it lighter… But, the downward force of the wings has to be greater then the upward force (to counteract gravity), or the fly would not stay aloft. So the net downward force from flying will increase the measurable weight of the jar. OR at least that’s how I remember it from H.S. Physics…
The motion of the fly makes a difference in the scale reading, but not the weight. “Weight” is the force exerted on the body by gravity, not the force between the body and the scale (which is what the scale reads). The two are only equal when the center of mass of the body isn’t accelerating vertically. That’s why you have to stand still in order for a scale to accurately measure your weight.
akrako1’s explanation only applies when the fly hovers stationary. In that case, the center of mass of the fly-air-jar system doesn’t move, so the scale is exerting a force on the bottom of the jar that just counteracts the weight of the fly-air-jar system. Therefore, the scale reading equals the weight.
But if the fly moves, all bets are off. Suppose the fly moves up. Then the center of mass of the fly-air-jar system also moves up, so a net upward force must act. The force of gravity doesn’t change, so that means the force exerted by the scale on the bottom of the jar must exceed the weight of the jar. Thus, the scale reads a higher value than the weight of the fly-air-jar system. It is the same as if you stood on a scale and then “pushed off” against it. The scale reading momentarily exceeds your weight.
The tricky part is that the jar+fly and the truck+bird systems are not rigid bodies. Even though the jar itself isn’t moving, the motion of its contents is making the system’s center of mass move. Likewise, the motion of the birds doesn’t change the weight of the truck+birds system, it only causes the center of mass to move around. When that happens, it takes a force different from the weight to hold the truck itself (or the jar itself) stationary.
To be a trifle more precise, they must be changing their velocity (up or down) to have an effect on weight - constant velocity yields no effect (until they hit the top or the bottom of the jar, which causes a change in their velocity).
Yours is the only reply that makes clear sense to me.
What if we consider that the fly found an escape. Would the jar be lighter after the escape? Heavier when he returns again ?
I would hypothesize that the moment an escape was available for the fly in a sealed jar a specific number of moles of air** equal to the weight of the fly **would escape first as long as the fly was flying in the jar. Not only would my WAG allow for a specific change in the weight of the jar for the two situations but also makes sense when you consider that increased motion of air molecules under confinement is what pressure is all about and that pressure will try to equalize when the system is no longer closed.
Is the jar on a treadmill?
heh. that was my first thought as well 
This, like the infamous treadmill thread, is one of those questions confounded by ambiguous wording - the fly doesn’t increase the mass of the jar - because the only way to do that is to add more glass.
But of course that’s not what the question is really about - restated in (hopefully) less ambiguous terms, the question is something like:
Is a jar containing only air less heavy than an identical jar into which a heavier-than-air fly has been introduced, when the fly is hovering.
And the answer is: Yes. Adding a fly to the system increases the mass (and therefore the weight) of the system by the mass of the fly, minus the mass of the air displaced by adding it. How could it be otherwise?
Change ‘weight’ to volume. A floating object displaces its weight; a submerged body displaces its volume.
I put my radio in the bath and turned up the volume - it didn’t displace any extra water.
Well, are you sure? How long did you weight?
About a foot.
[Colonel Graham Chapman]Allright! That’ll be enough of this! It’s getting way too silly.[/CGC]