How many cubic meters of helium are required to lift a...

How many cubic meters of helium are required to lift a balloon with a $400-\mathrm{kg}$ payload to a height of 8000 $\mathrm{m}$ ? (Take $\rho_{\mathrm{He}}=0.180 \mathrm{kg} / \mathrm{m}^{3} . )$ Assume that the balloon maintains a constant volume and that the density of air decreases with the altitude $z$ according to the expression $\rho_{\text { air }}=\rho_{0} e^{-z / 8} 000$ where $z$ is in meters and $\rho_{0}=1.25 \mathrm{kg} / \mathrm{m}^{3}$ is the density of air at sea level.

How many cubic meters of helium are required to lift a balloon with a $400-\mathrm{kg}$ payload to a height of 8000 $\mathrm{m}$ ? (Take $\rho_{\mathrm{He}}=0.180 \mathrm{kg} / \mathrm{m}^{3} . )$ Assume that the balloon maintains a constant volume and that the density of air decreases with
the altitude $z$ according to the expression $\rho_{\text { air }}=\rho_{0} e^{-z / 8} 000$ where $z$ is in meters and $\rho_{0}=1.25 \mathrm{kg} / \mathrm{m}^{3}$ is the density of air at sea level.

Key Concepts

Net Lift Force Calculation

The net lift force available from a balloon is computed by subtracting the weight of the lifting gas from the weight of the air displaced by the balloon. This net force must be greater than or equal to the weight of the payload for successful lift. The calculation involves taking the difference in densities (between the outside air and the helium inside) multiplied by the volume and gravitational acceleration, highlighting the importance of both local atmospheric conditions and the properties of the lifting gas.

Constant Volume Balloon Assumption

Assuming that a balloon maintains a constant volume during its ascent simplifies the analysis by keeping the volume of displaced air and the mass of the lifting gas constant. This assumption allows us to focus on the change in surrounding air density with altitude when determining whether the buoyant force remains sufficient to lift the payload.

Exponential Atmospheric Density Variation

The density of air decreases with altitude, and an exponential model is often used to describe this decrease. This model typically takes the form of ?(z) = ?? e^(?z/H), where ?? is the sea level air density, z is the altitude, and H is the scale height. This variation is crucial when calculating the net buoyant force at different altitudes, since the amount of displaced air (and hence the buoyant force) depends on the local air density.

Buoyancy

Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object. In the context of lifting a payload with a gas-filled balloon, the buoyant force is determined by the weight of the displaced air, which is calculated as the product of the air density, the gravitational acceleration, and the volume of the balloon.

Archimedes' Principle

Archimedes' 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 that is displaced by the object. For a balloon, this means that the lifting capability is derived from the difference between the weight of the air displaced and the weight of the gas (helium) inside the balloon.

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