Ballooning on Mars. 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 kgm3 (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 5.00 g. We inflate them with a very light gas whose mass we can ignore. (a) What should be the radius and mass of these balloons so they just hover above the surface of Mars? (b) If we released one of the balloons from part (a) on earth, where the atmospheric density is 1.20 kgm3, what would be its initial acceleration assuming it was the same size as on Mars? Would it go up or down? (c) If on Mars these balloons have five times the radius found in part (a), how heavy an instrument package could they carry?
Solution 56P Step 1: a) Provided, the atmospheric density of mars, = 0.0154 kg/m 3 mars 2 2 The mass density of the balloon, balloon 5g/m = 0.005 kg/m Therefore, the total mass of the balloon, M = × surface area balloon balloon 2 2 M balloon 0.005 kg/m × 4r 3 Volume of the balloon, V balloon (4/3) r Then, the density of the atmosphere in mars can be given as, mars= M balloonV balloon 3 2 2 3 0.0154 kg/m = [0.005 kg/m × 4r ] / (4/3) r 0.0154 kg/m = [0.005 kg/m × 3 ] / r 2 2 3 Rearranging, we will get, radius of the balloon, r = [0.005 kg/m × 3 ] / 0.0154 kg/m r = 0.974 m