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We recently awarded ourselves some helium balloons to celebrate something, and they came with about a dozen little balloons inside, filling it to just under half way.

I was discussing with a colleague whether it would be possible to determine whether the inner balloons were also helium-filled, without popping the large outer balloon.

Obviously, if the inner balloons were filled with something moderately lighter than helium, they would float, but they're huddled at the bottom, so it's either helium or plain air.

How about it?

p.s. you guys really need a 'balloon' tag. :)

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  • $\begingroup$ "Obviously, if the inner balloons were filled with something moderately lighter than helium, they would float"... This is not obvious for me! The balloons have mass and it may the buoyancy force cannot cancel their weight for floating. $\endgroup$ – lucas May 27 '16 at 14:21
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One indication whether the 'inner' balloons are filled with helium or something heavier is whether the 'outer' balloon still floats in air or not.

Whether it floats or not depends on whether its overall density is lower than air's density or not. Its overall density $d$ is simply given by:

$$d=\frac{\Sigma m}{V}$$

Where $\Sigma m$ is the sum of all the masses that make up the balloon and $V$ its total volume.

The sum of all masses means the sum of:

  1. Mass of outer balloon material
  2. Mass of helium
  3. Masses of all inner balloons material
  4. Masses of all gas contained in the inner balloons

If the inner balloons are filled with air and there are too many of them the overall density will exceed that of air and the outer balloon will not float.

But even if the inner balloons are all filled with helium, the outer balloon may still not float. That's because a larger number of inner balloons making up the same volume of a smaller number of inner balloons with create more mass because of the increased surface area and the inherent 'cost' in mass of larger number of inner balloons. For that reason:

Obviously, if the inner balloons were filled with something moderately lighter than helium, they would float, but they're huddled at the bottom, so it's either helium or plain air.

... is not necessarily correct.

If the mass of the outer balloon material, the total volume of the balloon, the mass of the inner balloon material and the volume of the inner balloons was known, then depending on whether the outer balloon still floats in air or not, the lower or lower limit of the density of the inner balloon gas could be calculated.

Call $m$, $d$ and $V$, with resp. suffixes $o$ and $i$ for outer and inner balloons, the mass of balloon material, the density of the filling gas and volume of the balloons. Let there be $n$ identical inner balloons. The overall density of the balloon is then:

$$d=\frac{m_o+d_o(V_o-nV_i)+nm_i+nd_iV_i}{V_o}$$

If the balloon just about floats in air, then:

$$d=d_{air}$$

From which we can deduce:

$$d_i=\frac{d_{air}V_o-m_o-d_o(V_o-nV_i)-nm_i}{nV_i}$$

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