It has been proposed that we could explore Mars using inflated balloons to hover

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 4.60 g . We inflate them with a very light gas whose mass we can neglect.What should be the radius of these balloons so they just hover above the surface of Mars? What should be the mass of these balloons so they just hover above the surface of Mars? 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? If on Mars these balloons have five times the radius found in part A, how heavy an instrument package could they carry?

Explanation / Answer

1.

Buoyant force is equal to the weight of air displaced by balloon

B = rho)air*V*g

g = gravity on mars

for a sphere V = 4*pi*R^3/3

Surface area of sphere = 4*pi*R^2

mass pf balloon = (4.6*10^-3 kg/m^2)*4*pi*R^2

B = m*g

rho)air*V*g = m*g

rho)air*V = m

rho)air*4*pi*R^3/3 = 4.6*10^-3*4*pi*R^2

R = 3*4.6*10^-3/0.0154 = 0.896 m

2.

Mass = 4.6*10^-3*4*pi*0.896^2 = 0.0464 kg

3.

Fnet = B - m*g = m*a

B = rho)air*V*g = rho)air*4*pi*R^3*g/3

B = 1.2*4*pi*0.896^3*9.81/3 = 35.47 N

a = (B - mg)/m

a = (35.47 - 0.0464*9.81)/0.0464 = 754.63 m/sec^2 in upward direction

4.

B = M*g

rho)air*V*g = (Mb + Mload)*g

Mload = rho)air*4*pi*R^3/3 - 4.6*10^-3*4*pi*R^2

Mload = 0.0154*4*pi*(5*0.896)^3/3 - 4.6*10^-3*4*pi*(5*0.896)^2

Mload = 4.64 kg

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