A spherical balloon whose volume at sea level is V0, rises slowly in the atmosph

Question

A spherical balloon whose volume at sea level is V0, rises slowly in the atmosphere up to a height h. Assuming the volume of the balloon varies inversely with the surrounding pressure, obtain an expression for the work done, Wb, by the skin of the balloon on the atmosphere when the balloon rises to a height h. The variation of atmospheric pressure P with height z is given by P0 is the pressure at sea level, g is the acceleration due to gravity, and Po IS the density of air at sea level. The expression for Wb must in terms of known variables.

Explanation / Answer

The work can be obtained as W = P(z=h)(V(h)-V_0) = P_0 exp(-rho_0 g h P_0^-1) (V(h)-V_0). Now, V(z) / V_0 = P_0 / P(z). Thus, we have the desired result by inserting V/ V_0 into the formula for W

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