My intuition leads me to believe that the balloon would have negative weight? and I have no idea how we could measure the mass
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Well, yeah it would more or less have a negative weight, in that when you put something on a scale you normally neglect effects of buoyancy.
Strictly speaking it does have weight. Just that weight is being applied to the ground through the air it is sitting on top of.
Measuring it’s mass is less ambiguous though. At least with respect to what mass means. One way would be to calculate it indirectly. You can get the force from buoyancy by any number of methods, one silly one would be to flip a scale upside down and put the balloon under it.
Next you can measure the local absolute air pressure with a barometer and that gives you the density of the air.
After that you measure the volume of the balloon say by sinking it in a tub and measuring the amount of displaced water, or some other more accurate method that avoids compressing the balloon.
Now you have the volume and density of air displaced by the balloon, and the volume of the balloon. Since the bouyant force upward is a result of the difference in mass of displaced air and the mass of the balloon you can then calculate the mass of the balloon.
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