viernes, 12 de mayo de 2017

Physics and movies: Up

I don't know anyone who dislikes Disney-Pixar's UP. This story of friendship between an elderly man and a boy is very inspiring. The movie deal with a physics principle: buoyancy. The movie tells us how a recently widower man tries to make his wife's dream real by traveling to South America. To do so, he uses his house as a mean of transport by making it float in the air. According to the plot, the house goes up thanks to the buoyancy force produced by thousands of helium balloons. Would that work? A National Geographic TV show tried to recreate the set-off of the house and got the result you can see in the video below.


Despite it is a lightweight house specially designed for this experiment, a lot of helium was used. Could something like that be done with an actual house like the ones that people live in? How much helium would be needed? According to an estimation by, a 150-squared-meter house may have a mass of roughly 100 tons. Are you able to estimate the volume of helium that could make a house fly?

Let's do the math.

First, we can get some info from the video. They use 300 balloons. Each of them has a diametre of 8 ft (2.44 m). The reporter says that it was almost freezing, so I will assume they are at STP conditions. We also need additional data, such as helium and air densities:
He: 0.179 kg/m3
Air: 1.295 kg/m3.

How much helium do they use?
Each balloon is a sphere with a radius of 1,22 m. Its volume can be easily calculated: 7.59 m3. All the balloons together add up a volume of 2277 m3. Multiplying by its density we figure out that the mass of helium inside the balloon is 407.5 kg which has a weight of 3994 N.

What buoyancy force can be obtained from that mass of Helium?
The buoyancy force is the weight of the fluid displaced by the balloon. In this case we have to calculate the weight of 2277 m3 of air.
E = 1.295 kg/m3 · 2277 m3 · 9.8 m/s2 = 28897N.

This buoyancy force has to be strong enough to rise not only the house but also the helium itself. The weight of cannot be greater than 28897 N - 3994 = 24903 N, so it has to be lighter than 2541 kg, roughly two and a half tons. To rise an actual house, like the one you may live in, with an estimated mass of 100 tons we would need forty times more Helium, which is 91080 m3. That volume is the equivalent of a sphere with a radius of 28 m. Take the volume of a party balloon (0,91 according to the website a party stuff provider company) into account we conclude that 100000 balloons would be needed. I honestly was expecting an even higher number of balloons. In fact, we would need much more balloons, at least to set the house off. Don't forget the force we have just calculated is the one required to lift the house but, to set off we have to pull it out from its foundations and higher buoyancy force would be required.

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