# A flying fly inside a sealed box on a scale

suppose there is a scale able to measure weight with an uncertainty of $10^{-9}kg$ . On the scale, an airtight plastic chamber is placed. Initially, a fly of mass $10^{-5}kg$ is sitting at the bottom of the chamber, which sits on the scale. At a later point in time the fly is flying around the chamber. Will there be a difference in the observed weight as measured by the scale when the fly is sitting at the bottom of the chamber compared to when it is flying around the chamber at some point in time? If so, what does the value of this difference depend on (I am most concerned with the case where the fly has not touched any surface of the container in enough time for the scale to reach some equilibrium value (or do the pressure variations induced from the flies wings cause constant fluctuations in the scale)?

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If you had a perfect scale, the reading would fluctuate based on

$$\delta w = m\ddot{x}_{cm}$$

$\delta w$ is the size of the fluctuation in the reading, $m$ the total mass on the scale (including fly and air), and $\ddot{x}_{cm}$ the acceleration of the center of mass.

Integrating over time,

$$\int_{time} \delta w(t) = m\Delta(\dot{x}_{cm})$$

Here, $\Delta(\dot{x}_{cm})$ is the change in velocity of the center of mass over the period you observe the readings. Because the velocity of the center of mass cannot change very much, if you integrate the fluctuations over time, you wind find that their average tends towards zero. If the fly begins and ends in the same place and the air is still, the fluctuations integrate out to exactly zero.

Whenever the fly is accelerating up, we expect the reading to be a little higher than normal. When the fly accelerates down, we expect the reading to be a little lower than normal. If the fly hovers in a steady state, the reading will be the same as if the fly were still sitting on the bottom.

A real scale cannot adjust itself perfectly and instantaneously, so we would need to know more details of the scale to say more about the real reading.

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interesting... this also seems like something that could be experimentally verified – Timtam Jul 25 '11 at 20:25
that seems a bit odd to me though, because if you replaced the fly with a cat that could jump very high, you would think that after the cat loses contact with the ground the scale will quickly lose all the force from the cat and should equilibriate to just the weight of the chamber by itself. – Timtam Jul 25 '11 at 20:31
@Timtam Yes, and the cat would be accelerating down. That would make $\delta w$ negative, just like I said in the post. When the cat takes off and lands, there will be big spikes in the weight so that the integral of the fluctuations over time is zero. – Mark Eichenlaub Jul 25 '11 at 20:38
this looks like a fairly convincing argument... funny that in college me and a team of physics undergrads couldn't solve this in an hour... and somebody here just answers it like it is a nothing problem – Timtam Jul 25 '11 at 20:42
@TimTam - well 'fly-in-a-box' is a rather specialzed area of physics, a branch of the more common 'cat-in-a-box' problems – Martin Beckett Jul 26 '11 at 3:10