The mass of a hot air balloon and its cargo (not including the air inside) is 200 kg. The air outside is at 10 oC and 101 kPa. The volume of the balloon is 400 m3. To what temperature must the air in the balloon be heated before the balloon will lift off. (Air density at 10 oC is 1.25 kg/m3.)
Two balloons have been filled up with air under atmospheric pressure to volumes V1 and V2, respectively. They are now submerged under water. A thin string of length L, which is run through a pulley at a fixed depth H, connects the balloons. (The radii of the pulley and the balloons are much smaller than the length of the string.) By setting the initial positions of the balloons, one can achieve a state of equilibrium. Neither balloon is rising or going down. Determine the difference in the depth of the balloons (in terms of H and L) under those conditions. The mass of the balloon skins, of the string, and of the air is negligible. The temperature of the water is constant and equal to the temperature of the air.
h1 + h2 = 2H - L
h2 - h1 = [(1 - (V1/V2))/ [(1 + (V1/V2))][2(Ptop/ρg) + 2H - L]
Here Ptop = 101 kPa and ρ = 1000kg/m3.