Question

The Hindenburg, the German zeppelin that caught fire in 1937 while docking in Lakehurst, New Jersey, was a rigid duralumin-frame balloon filled with \(2.000 \cdot 10^{3} \mathrm{~m}^{3}\) of hydrogen. The Hindenburg's useful lift (beyond the weight of the zeppelin structure itself) is reported to have been \(1.099 \cdot 10^{6} \mathrm{~N}(\) or 247,000 lb \()\). Use \(\rho_{\text {air }}=1.205 \mathrm{~kg} / \mathrm{m}^{3}, \rho_{\mathrm{H}}=0.08988 \mathrm{~kg} / \mathrm{m}^{3}\) and \(\rho_{\text {He }}=0.1786 \mathrm{~kg} / \mathrm{m}^{3}\). a) Calculate the weight of the zeppelin structure (without the hydrogen gas). b) Compare the useful lift of the (highly flammable) hydrogen-filled Hindenburg with the useful lift the Hindenburg would have had if it had been filled with (nonflammable) helium, as originally planned.

Short answer

b) Compare the useful lift of the hydrogen-filled Hindenburg with the useful lift the Hindenburg would have had if it had been filled with helium. Express the comparison in terms of a percentage or a simple numerical ratio.

Step-by-step solution

Calculate the total weight lifted by hydrogen

The total weight lifted by the hydrogen-filled zeppelin can be calculated using the buoyant force formula: \(F_B = V(\rho_{\text{air}} - \rho_{\text{H}})g\), where \(V = 2.000 \cdot 10^3 \mathrm{~m}^{3}\) is the volume of hydrogen gas, \(\rho_{\text{air}} = 1.205 \mathrm{~kg} / \mathrm{m}^{3}\) and \(\rho_{H} = 0.08988 \mathrm{~kg} / \mathrm{m}^{3}\) are the densities of air and hydrogen, and \(g = 9.8 \mathrm{~m}/\mathrm{s}^2\) is the acceleration due to gravity.

Calculate the weight of the zeppelin structure

We are given the total useful lift of the Hindenburg, which is \(1.099 \cdot 10^{6} \mathrm{~N}\). The weight of the zeppelin structure can be found by subtracting the useful lift from the calculated total weight lifted by hydrogen. So, the weight of the zeppelin structure is \(W = F_B - L\), where \(W\) is the weight of the structure and \(L = 1.099 \cdot 10^{6} \mathrm{~N}\) is the useful lift. b) Compare the useful lift of the hydrogen-filled Hindenburg with the useful lift the Hindenburg would have had if it had been filled with helium

Calculate the total weight lifted by helium

We can calculate the total weight lifted by a helium-filled zeppelin using the buoyant force formula as before: \(F_B = V(\rho_{\text{air}} - \rho_{\text{He}})g\), where \(\rho_{\text{He}} = 0.1786 \mathrm{~kg} / \mathrm{m}^{3}\) is the density of helium.

Calculate the useful lift of a helium-filled Hindenburg

Similar to the hydrogen-filled Hindenburg, the useful lift of a helium-filled Hindenburg can be found by subtracting the weight of the zeppelin structure from the calculated total weight lifted by helium. So, the useful lift of the helium-filled Hindenburg is \(L_{\text{He}} = F_{B_{\text{He}}} - W\).

Compare the useful lifts

Finally, we can compare the useful lift of a hydrogen-filled Hindenburg \((L)\) to that of a helium-filled Hindenburg \((L_{\text{He}})\). We can express the comparison in terms of a percentage or a simple numerical ratio.