# Balloon on a string; which moves first?

1. Jan 23, 2010

### chudd88

Picture a helium balloon on a string. I hold one end of the string, and the balloon rises until the string is taut. With the balloon hovering at its location, and with tension on the string, I cut the string near the bottom. The balloon will rise, taking the string with it.

So, the question is, what rises first, the balloon, or the bottom of the string?

The balloon can't rise while the string is holding it down. But the string can't rise unless the balloon pulls it up. Assuming there is no elasticity in the string, it seems that neither the balloon nor the string can rise.

I guess this is similar to the idea of pushing one block against the other, Block A into Block B. Block A can't move while Block B occupies that space, and Block B can't move unless something else applies a force to it. So, how can to surfaces in contact apply for to each other, with one object displacing the other, if they can't occupy the same location at the same time?

2. Jan 24, 2010

### A.T.

That is the key of the issue. If the string was perfectly rigid, the lower end of the string and the upper end (balloon) would accelerate simultaneously. But perfectly rigid materials don't exist.

Another common reason for confusion is naive cause-effect reasoning (A pushes B, B pushes C,...) applied to forces. http://www.youtube.com/watch?v=k-trDF8Yldc".

Last edited by a moderator: Apr 24, 2017
3. Jan 24, 2010

### clem

They rise at the same time. Get a balloon, a string, and a scissors, and see.
It's not rocket science.

4. Jan 24, 2010

### Cleonis

As A.T. points out, the assumption of no elasticity is what introduces paradox here.

Take the opposite extreme, a big balloon held down by a bungee chord. Cut the bungee chord at the lower attachment point.
The bungee chord will contract, a contraction that will tend to move all parts of the chord/balloon system towoards the common center of mass.

The contraction of the bungee chord may actually pull the balloon down a bit, initially. So the bottom of the string/chord isd the first to move up.

5. Jan 24, 2010

### GRDixon

The balloon is deformed (elongated) while held down. If your question is, "which moves first, the top of the balloon or the bottom of the string?" then I would say it is the string, which is pulled upward while the balloon becomes more spherical, but while its top is still at rest.

6. Jan 24, 2010

### diazona

That I agree with. The fastest process that occurs when the string is cut is the contraction toward the center of mass, which is due to the release of the tension in the string.

7. Jan 24, 2010

### Prologue

How are you so sure that the elastic process wins against the buoyant process?

I think you can dream up situations where it does win, as you say, but also where it doesn't. I haven't tried to do the math,maybe I will later, but I would be surprised if there was a sure rule that it always did this or that. The contraction will be instant but so will the displacement upward of the whole system. The question is, does the center of mass rise faster than the contraction? I bet that it depends on the masses, densities, gravitational constant, spring constant, a bunch of stuff.