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 -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 $\rho_{\text {air }}=\rho_{0} e^{-z / 8000}$, 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

Archimedes’ Principle

This principle states that any object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. In balloon problems, this concept is used to calculate the net lift by comparing the weight of the air displaced by the balloon to the weight of the gas inside it, allowing one to determine the necessary volume to support a given payload.

Atmospheric Density Variation with Altitude

Understanding that air density decreases with altitude—often modeled by an exponential decay—is crucial in such problems because the buoyant force depends directly on the density of the surrounding air. Lower air density at higher altitudes diminishes the displaced weight and thus the available lift, influencing the design and performance of the balloon.

Constant Volume Gas Behavior

When a balloon is assumed to maintain a constant volume during ascent, the density and mass of the gas inside remain unchanged. This simplifies the analysis since the variation in buoyant force is due solely to changes in the ambient air density, not changes in the lifting gas, thereby streamlining the calculation of the required volume for a specified payload.

Lifting Capacity Calculation

Calculating the lifting capacity involves determining the net upward force provided by the buoyant force, which is the difference between the weight of the displaced air and the weight of the lifting gas. By equating this net force to the weight of the payload, one can solve for the minimum volume of gas required to ensure that the balloon can lift the payload at the target altitude.

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