Buoyancy

I used to read to my kids from a popular children's book which featured a curious monkey who got
into trouble by grabbing a large bunch of helium balloons from a balloon salesman.  According to the storyline, the monkey was carried out over the city hanging from the balloons.  Now, I'm sure the illustrator for the book was deeply interested in thermodynamics (isn't everyone?) but may have been in too big a hurry to calculate the appropriate number of balloons.  Or maybe artistic aesthetics overruled thermodynamic accuracy.  In any case, today we'll go back and look at the situation in more depth.

Let's assume that each balloon is approximately a sphere one foot in diameter.  We’ll also assume that the balloons are filled with pure helium at normal temperature and pressure, and that the surrounding air is also at normal temperature and pressure.  We’ll neglect the weight of the string in order to get an estimate for the minimum number of balloons that would be needed.  Finally, we’ll assume that the monkey weighed 12 lbs, and the rubber of each balloon weighed about 0.4 oz.

The buoyancy force is equal to the weight of the fluid displaced.  So, from the ideal gas law, the mass of air displaced by each balloon would be:

and the weight of the air would just be:

which comes out to be 0.03934 lb.

A similar calculation for the weight of the helium gives 0.00543 lb with an additional 0.025 lb for the rubber in the balloon, resulting in a net lift from each balloon of 0.00891 lb.  So, it would require at least 1347 balloons to lift a 12 lb monkey and send him floating over the city, which would undoubtedly be awkward to draw accurately, so maybe it was best that the illustrator used artistic license to represent the idea instead of adhering to strict realism.

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