Compute payload at sea level

Question

A modern day zeppelin holds 7290 m^3 of helium. Compute its maximum payload at sea level. (Assume the helium and air to be at 0 degree C and 1 atm.

Answer 1

Let

V = 7290 m³ be the volume of the zeppelin,

g be gravitational acceleration,

h = 0.18 kg/m³ be the density of helium at 0°C,

a = 1.29 kg/m³ be the density of air at 0°C,

m (to be calculated) be the mass of the zeppelin structure plus maximal load.

The term "payload" normally excludes the mass of the zeppelin itself.

But we are not given the mass of the zeppelin, so the best we can do

is to calculate the maximum mass, m, the zeppelin can carry, including its own mass.

But m does not include the mass of the helium.

The zeppelin is subject to two forces: gravity and boyance.

The force of gravity is due to its mass m plus the mass of the helium:

g(m + Vh)

The force of boyance is due to the weight of air displaced by the zeppelin:

gVa

The maximum m is achieved when the two forces are in balance:

g(m + Vh) = gVa

m = V(a - h)

Substitute actual numbers

m = 7290(1.29 - 0.18) = 8092 kg