Question

It has been proposed that we could explore Mars using inflated balloons to hover just above the surface. The buoyancy of the atmosphere would keep the balloon aloft. The density of the Martian atmosphere is 0.0154 kg/m3 (although this varies with temperature). Suppose we construct these balloons of a thin but tough plastic having a density such that each square meter has a mass of 4.90 g. We inflate them with a very light gas whose mass we can neglect.

Part A

What should be the radius of these balloons so they just hover above the surface of Mars?

Part B

What should be the mass of these balloons so they just hover above the surface of Mars?

Part C

If we released one of the balloons from part A on earth, where the atmospheric density is 1.20 kg/m3, what would be its initial acceleration assuming it was the same size as on Mars?

Part B What should be the mass of these balloons so they just hover above the surface of Mars? m =   kg   SubmitMy AnswersGive Up Part C If we released one of the balloons from part A on earth, where the atmospheric density is 1.20 kg/m3, what would be its initial acceleration assuming it was the same size as on Mars? a =   m/s2   Part D Would it go up or down?

Answer 1

Here we have to found for what radius od the balloon will the weigth be equal to the bouyancy force. This is

Where g_mis the gravity in Mars.

The since we have the surface density of the balloon (d_s=4.90 g/m² =4.9 x10^-3 kg/m²) , we can find the total mass of the balloon multiplying the total surface of the sphere by the surface density.

In other hand we have the bouyancy force, woch is given by the Archimedes' principle. The volume of displaced fluid is the same volume than the sphere, . The mass of the displaced fluid is , and therefor the weith of it . The bouyancy force is

And to find the radius

b) Like we said before the mass of the balloon is given by

c) By the second Newton's law the total force in the balloon will be the difference between the bouyancy force and the weigth

(note that here the gravity is the gravity on earth)

The bouyancy force for the balloon in the earth would be given by the same expression than before but with the density of the earth atmosphere and the earth gravity. This is

Therefore, the net force

Now

d) We choose the sign in the sumatory of forces taking the positive sign pointing upward, like the accelereation is positive the balloon on earth will go up