Up

You can learn a lot from Pixar’s film Up.  For example, there are apparently some circumstances when it’s OK for a creepy old man to make off to South America with an adolescent boy.  But what physics did we learn from Up?  How many balloons would it take to lift the old man’s house?

In Reinforced Helium Balloons, I calculated how much helium you’d need to lift a concrete balloon.  The physics is the same here.  The helium in the balloons is lighter than air, so there’s a “buoyant” force that pushes the balloons up.  Buoyancy arises because the gravitational force from the Earth pulls more strongly on heavier objects, so the heavier air will be pulled closer to the Earth forcing the lighter helium to be squeezed upwards.  This buoyant force can be approximated by the equation¹,

force = [(density of air) – (density of helium)] · (volume) · (gravitational acceleration).

In order for the house to float, this force must be at least as large as the gravitational force on the house, which is given by

force = (mass of the house) · (gravitational acceleration).

By setting the buoyant force equal to the gravitational force, we can calculate the total volume of helium needed to float the house.  Assuming the house weighs about 300,000 kg,² we can compute the total volume of air needed,

volume = (mass of the house) /  [(density of air) – (density of helium)]

= (300,000 kg) /  [(1.2 kg/m³) – (0.17 kg/m³)]

= 300,000 m³.

If each balloon took up a cubic foot of space, it would require 11 million balloons to lift the old man’s house.  That means you’d need a cluster of balloons with a diameter of about 250 balloon widths.  Judging from this photo, it seems Pixar had way too few balloons, but in principle this idea could work.  For example, just this past week Jonathan Trappe set a world record by becoming the first cluster-balloonist to cross the English Channel.  For those that are skeptical, check out this article in Wired.

[1] In principle, I should also include the mass of the balloons here since heavy balloons will be more difficult to lift, but balloons are pretty light so I’m going to neglect their mass.

[2] To get this number, I assumed a cubic two-story house with dimensions 40 ft by 40 ft by 40 ft.  I assumed the walls and floors are 1.0 ft thick and that the house was made of some combination of wood and brick that had an average density of 1000 kg/m³.