# The Physics of a Sad Balloon

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.”

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.

## 17 thoughts on “The Physics of a Sad Balloon”

1. In fact, thanks to your ribbon-ectomy (and presumably the air currents in the house), the sad balloon had made it all the way to the upstairs bathroom by this morning.

2. Reading the first part of your post, I thought, “Hey, why didn’t I ever wonder about that?!” and felt kinda dumb. (Like my math students feel sometimes?)

Then, when you explained about the ribbon, I thought, “No way! I’ve seen balloons half-way up with nothing trailing down to the floor!” But maybe I haven’t… Maybe my brain just created that because my (subconscious?) hypothesis was something like thinner air (not exactly that, but…).

Interesting…

3. I’ve long been fascinated by the meta-phenomenon of phenomena that are hard to give popular explanations for because people consistently misremember them.

4. Chuk says:

That seems like an elegant and satisfying explanation.

Also, man those mylar balloons hold their helium a long time. We’ve had them hanging about for months after a birthday.

5. I actually have been able to make helium balloons float with neutral buoyancy for a short time by weighting them carefully with paper clips, but it’s an unstable situation.

6. Jim says:

I’ve seen mylar balloons float for extended times without anything trailing. Many (many) years ago, my mother came home from Christmas shopping with 2 “Space Pets”. These were helium filled mylar balloons with cardboard tabs lightly stuck to one side (no trailing ribbon). You peeled off tabs until it achieved neutral bouyancy, then let it go. They would float a few feet off the ground, drifting from room to room and following people around.

Every couple days, you would peel off a couple more tabs to compensate for helium loss.

These were floating around our house for over a month. (One was floating around the living room when it decided to commit suicide by suddenly diving into the lit fireplace.)

Might I suggest some more experimentation is in order?

7. @Jim #6

The important part for Chad’s balloon is not the part of the ribbon trailing on the floor. It’s the weight of the *vertical* piece of ribbon. The part of the ribbon on the floor exerts no force on the balloon.

The cool thing is that the height of the balloon will change until there is just enough vertical ribbon to balance the buoyant force of the helium. It’s just like your “space pets,” except that you don’t have to peel the tabs manually. Perhaps one could market space pet “leashes” (pieces of ribbon) that automatically tether them to the ground.

8. TheBrummell says:

OK. Now how about the escape of helium from the inside of the balloon. Is that *really* primarily the result of He atoms being really small, smaller than the pores in the mylar? Or are there other factors at play, too?

9. Robert S says:

TheBrummell, If I remember correctly Helium will slowly diffuse through most everything even if there are no pores or cracks. I don’t know even the order of magnitude that it takes place on, likely much slower then a balloon deflates at, but even a helium balloon with perfectly pore and defect free walls would eventually lose its helium.

10. CCPhysicist says:

Why did the balloon go up between pictures 1 and 2?

11. There is a significant force missing from the description. There is a weight force on the balloon and its contents. The four forces (normal on the ribbon, buoyant force, and two weight forces) must balance to maintain a particular height. It is not correct that the balloon rises because the buoyant force exceeds the weight of the ribbon. It must exceed the combined weight of the ribbon, the balloon, and its contents.

12. Kevin says:

10 CCPhysicist: The ribbon in the second photo is hanging straight down; it’s not in the first. Same weight of ribbon above the floor, therefore same length of ribbon, but it follows a different path so the height is different.

13. There’s a little variation in the height, basically a very slow oscillation about an equilibrium point. It’s also easily perturbed by air currents in the room, such as those caused by a large physicist putting the balloon in place, then stepping back to take a picture.

It’s also damnably difficult to get the balloon in focus, so I ended up using a picture that was slightly sub-optimal in terms of the height.

14. This is amusing because my fifth grader did the same experiment last time we had a balloon. He had to work quickly because it was just a cheap balloon, not mylar. He did not offer me the extra credit assignment of helium calculation, though. Also, I think he left the discarded ribbon pieces lying on the floor at the end of the experiment.

I’ll tell him the dog-physics guy said he wasn’t normal.

15. “Not normal” isn’t necessarily a Bad Thing…

16. Oh yeah, he will took it as a complement, especially considering the source.

17. CherryBombSim says:

Heehee! I remember doing this when I was a little kid. My first Physics experiment.