Question

A spherical helium-filled balloon is used to lift a load of 5320 N (in addition to the weight of the helium). Suppose the density of the air is 1.27 kg/m3. Find the minumum necessary radius for the balloon

Answer 1

Balance condition: $F = mg$

$$(\rho (\text {air}) - \rho (\text {He})) \mathrm {Vg} = \mathrm {mg}$$ $$V = \frac {m}{\rho (\text {air}) - \rho (\text {He})}$$ $$\frac {4}{3} \pi R ^ {3} = \frac {m}{\rho (\text {air}) - \rho (\text {He})}$$ $$R = \sqrt [ 3 ]{\frac {3 m}{4 \pi (\rho (\text {air}) - \rho (\text {He}))}}$$ $$R = \sqrt [ 3 ]{\frac {3 \times 5320}{4 \times 3.14 \times (1.27 - 0.18)}}$$ $$R = 10.52 \, (\mathrm{m})$$

Answer:

$$R (\min) = 10.52 \, (\mathrm{m})$$