A hot-air balloon has a volume of 2200 m^3 . The balloon fabric (the envelope) weighs 900 N . Th

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A hot-air balloon has a volume of 2200 m^3 . The balloon...

A hot-air balloon has a volume of 2200 $\mathrm{m}^{3} .$ The balloon fabric (the envelope) weighs 900 $\mathrm{N} .$ The basket with gear and full propane tanks weighs 1700 $\mathrm{N}$ . If the balloon can barely lift an additional 3200 $\mathrm{N}$ of passengers, breakfast, and champagne when the outside air density is $1.23 \mathrm{kg} / \mathrm{m}^{3},$ what is the average density of the heated gases in the envelope?

Key Concepts

Archimedes' Principle

Archimedes' Principle states that any object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by that object. This principle is central to understanding how lift is generated in systems like hot-air balloons, where the balance between the displaced surrounding air and the lighter heated air inside generates the lift.

Density

Density, defined as mass per unit volume, is a fundamental property that determines how heavy a substance is for a given volume. In contexts such as buoyancy and lift, comparing the density of the heated gases inside a balloon with the density of the surrounding air allows us to assess the net lift available for carrying additional loads.

Force-Mass Relationship

The relationship between force and mass, expressed through the equation weight = mass × gravitational acceleration, is crucial for converting between weight (a force measured in Newtons) and mass (in kilograms). This conversion is necessary when analyzing the forces acting on a hot-air balloon and determining the density of the gases involved.

Thermal Buoyancy

Thermal buoyancy describes the process by which heating a fluid, such as air, reduces its density relative to a cooler surrounding fluid. In hot-air balloons, heating the air inside the envelope decreases its density compared to the ambient air, thereby creating an upward buoyant force that lifts the balloon and its payload.

Equilibrium and Net Lift Calculation

In a lifting scenario, equilibrium is achieved when the upward buoyant force equals the total downward weight of the balloon system, including the structural components and any additional load. Calculating the maximum load a balloon can sustain involves balancing these forces and using the volume of displaced air and the difference in density between the hot air and the ambient air.

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