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Let's say that there is a helium balloon attached to a large, sealed box. The box is off the ground and there is neutral buoyancy. The box + balloon are simply floating in air, not moving.

Inside the box is a fan. The fan is bolted to the floor of the box. If the fan turns on, will the box move in the opposite direction to the airflow?

If there were holes in the front and back of the box, yes, the box would move. The movement would be in reaction to the momentum of the air being pushed through the fan, not because the air is pushing against other air. But what if the holes were sealed? There would still be momentum. The air would then hit the inside, back of the box. I suppose the force against the back wall of the box would cancel out the forward force caused by the flow of air through the fan.

But what if the back wall of the box were shaped like a pyramid, with the apex pointing inwards, towards the fan? The air would deflect off the four surfaces of the pyramid and hit the four sides of the box, thus lessening the backwards force. (If I'm right, the force against the back would now be 1/4 of what it was if there were no pyramid.)

But the air would still have to circulate round back to the front of the fan, so presumably that would still cancel out the momentum generated by the fan in the opposite direction. So still no movement, right? (Ignoring the tiny effects of heating the box caused by the impact of the air against it.)

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The motion of the center of mass (CM) for the sealed system (no mass flow in or out of the system) depends only on the net external force applied to the system. There will be no movement of the center of mass in your sealed system, regardless of how you orient the internal fan and its air flow. The change in the center of mass due to movement of air with the fan is negligible, so the box will not move.

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  • $\begingroup$ But if there were holes in the front and back of the box, the centre of mass of the box / fan will not have changed and the mass that flows in will equal the mass that flows out. And yet the box would of course move, just like a jet aeroplane. $\endgroup$
    – the_ether
    Apr 11, 2022 at 17:16
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    $\begingroup$ Yes, you can push on air (prop plane) or eject mass (rocket), but your question focused on a sealed box. $\endgroup$
    – John Darby
    Apr 11, 2022 at 19:42
  • $\begingroup$ No. An aeroplane 's motion doesn't depend on "pushing on air". That's not how the physics works. Aeroplanes move because of Newton's laws: action / reaction. It's from the momentum of the air: its velocity and volume. $\endgroup$
    – the_ether
    Apr 11, 2022 at 21:39
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    $\begingroup$ For a propellor, the work done by the propeller creates a pressure difference that provides the thrust; what I mean by pushing the air is the propellor doing work on the air. See grc.nasa.gov/WWW/k-12/airplane/propth.html $\endgroup$
    – John Darby
    Apr 12, 2022 at 2:50
  • $\begingroup$ When the fan first turns on, it will increase the pressure behind it, as it has just moved more of the air mass there. This will result in a net movement of the center of gravity, even if only for a moment. Bare minimum the box should lunge forward proportionate to how much air has been pushed backwards, yes? $\endgroup$ Sep 13, 2022 at 20:08

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