Question

It has been proposed to move natural gas from the North Sea gas fields in huge dirigibles, using the gas itself to provide lift. Calculate the force required to tether such an airship to the ground for off-loading when it arrives fully loaded with \(1.17 \times 10^{6} \mathrm{~m}^{3}\) of gas at a density of \(0.796 \mathrm{~kg} / \mathrm{m}^{3}\). The density of the air is \(1.21 \mathrm{~kg} / \mathrm{m}^{3}\). (The weight of the airship is negligible by comparison.)

Short answer

\[ Force_{tether} = Weight_{air} - Weight_{gas} \]

Step-by-step solution

Calculate the Weight of the Gas

First, calculate the weight of the gas inside the airship. We can do this by multiplying the volume of the gas by the density to get the mass, and then multiplying by the acceleration due to gravity (9.81 m/s^2), using the formula: \[ Weight = Volume \times Density \times Gravity \] So, \[ Weight_{gas} = 1.17 \times 10^{6} \, m^{3} \times 0.796 \, kg / m^{3} \times 9.81 \, m/s^{2} \]

Calculate the Weight of Displaced Air

Next, calculate the weight of the air which is displaced by the airship. Use the same formula but this time using the density of the air: \[ Weight_{air} = 1.17 \times 10^{6} m^{3} \times 1.21 \, kg/m^{3} \times 9.81 \, m/s^{2} \]

Calculate the Tethering Force

Finally, calculate the force required to tether the airship, which equals the difference between the weight of the air and the weight of the gas. The formula then is: \[ Force_{tether} = Weight_{air} - Weight_{gas} \]

Key concepts

Archimedes' Principle

Understanding Archimedes' principle is essential when exploring buoyancy in fluids, as demonstrated by the dirigible exercise. This principle states that any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. Imagine you're submerged in a pool; the water you displace is pushing you up—that's buoyancy at work.

The principle helps us calculate the upward force on the dirigible in the air, which is essentially the weight of the air displaced by the dirigible. As the airship floats, just like a boat in water, it exerts a force on the air equal to the weight of that volume of air. This concept is pivotal because the tension in the tether holding the dirigible is directly related to this buoyant force. So, Archimedes' principle isn't only a historical footnote; it's a practical tool in engineering and understanding the behavior of floating objects in various mediums.

Density Calculations

In our dirigible example, density is a key parameter in understanding what makes the airship float. Density is the mass per unit volume of a substance, measured in this case in kilograms per cubic meter \( \text{kg/m}^3 \).

For a given volume, higher density means more mass and therefore more weight. In the force calculations for our dirigible, the two densities involved are those of the transported gas and the surrounding air. The idea is that if the gas inside the dirigible has a lower density than the air outside, the overall weight of the displaced air will be greater than that of the gas. By comparing these densities, we can determine whether the dirigible will rise, fall, or stay neutral in the air. Profound understanding of density is not only crucial for this exercise, but also for a wide range of other applications, like designing ships or understanding atmospheric phenomena.

Force Calculations

Finally, comprehending the force calculations completes our trinity of concepts needed to solve the dirigible's tethering conundrum. The force that needs to be exerted to keep the dirigible grounded – the tethering force – is essentially the buoyant force acting to lift it, which we found using Archimedes' principle. It's necessary to perform two crucial calculations: one for the weight of the gas inside the dirigible and another for the weight of the displaced air.

Each weight is calculated by multiplying the volume of the dirigible by the respective densities of gas and air and then by gravity \(9.81 \text{ m/s}^2\). The difference between these two weights gives us the magnitude of the upward buoyant force, and consequently, the required tethering force. Mastery of force calculations allows not only to solve this exercise but also to understand many real-world situations where forces play a crucial role, like in the design of elevators, understanding the motion of planets, or the functionality of vehicle brakes.