My birthday was two months ago, and SteelyKid’s was the weekend before last, so we’ve had balloons running around the house for a good while now. Meaning that when I came into the library yesterday, I saw the sad little image on the right: a half-deflated Mylar balloon floating at about chest height.
Now, the first thought of a normal person on seeing this would be “Why didn’t we throw this away a while ago?” My thought, since I’ve been on a bit of an everyday physics kick for a little while now, was “Hey, physics!”
“What do you mean?,” you ask. “What physics is there in the sad balloon? It floats because it’s still got some helium in it, making it lighter than air. It’s elementary buoyancy– even SteelyKid probably understands that.”
“Ah, I reply, but why is it floating at that height?” I reply. “After all, if it was really lighter than air, it should keep going up until it’s not lighter than air any more– until the density of the air outside matches the density of the air inside. But I can assure you that our ceilings are not nearly high enough for the air density to vary appreciably over the height of a first-floor room.”
“Hmmm….” you reply. “Hey, physics!”
So what’s going on, here? Well, the balloon itself isn’t the only factor in this problem– there’s also that long ribbon tied to it. For the balloon to float, the mass of the displaced air has to be equal to the mass of the balloon plus the ribbon. And, as you can see, a bunch of that ribbon is trailing on the floor.
And that’s what makes the balloon float at chest height. It’s not light enough to lift the entire ribbon, so it ought to be pulled down to the floor. But part of the ribbon is trailing on the floor, and thus being supported by a (really small but non-zero) normal force from the floor. So the balloon is really only supporting the weight of part of the ribbon– the bit that is vertical.
The height at which the balloon floats is determined by the addition of three forces: the buoyant force from the displaced air, the weight of the ribbon, and the normal force due to the bit of ribbon that is resting on the ground. When those three add to zero, the balloon is in equilibrium, and floats at a constant height.
This is, of course, easy to test. If I snip off most of the ribbon lying on the floor, the balloon remains at more or less the same height:
The weight of the remaining ribbon just matches the buoyant force from the displaced air, so the balloon hovers with only a tiny bit of ribbon drooping onto the floor.
And when I snip off a bit more, it rises all the way to the ceiling again:
The buoyant force from the displaced air is now greater than the weight of the shorter ribbon, so the balloon rises until something else makes it stop– in this case, the ceiling of the room.
So, you see, there’s physics even in a sad balloon.
Extra Credit: the ribbon is just under half a centimeter wide, and started at a length of roughly 234 cm (3/16 in and 92 in, if you want to do your own unit conversions). The length of ribbon on the floor in the first picture is around 29 cm. Using these figures, estimate the volume of helium in the balloon. Remember to show all your work.