A balloon weighing 100 kg has a capacity of 1000 m^3. If it is filled with hydrogen, how great

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A balloon weighing 100 kg has a capacity of 1000 m^3. If...

A balloon weighing $100 \mathrm{~kg}$ has a capacity of $1000 \mathrm{~m}^3$. If it is filled with hydrogen, how great a payload in kilograms can it support? At sea level the density of hydrogen is $0.09 \mathrm{~kg} / \mathrm{m}^3$ and that of air is $1.3 \mathrm{~kg} / \mathrm{m}^3$.

Key Concepts

Archimedes' Principle

Archimedes' Principle states that any object submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object. This principle forms the foundation for understanding how gases can provide lift in a balloon by displacing the surrounding air, thereby creating an upward force that can counteract gravity.

Buoyancy and Net Lift

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object immersed in it. In practical applications like ballooning, the net lift is calculated by taking the difference between the buoyant force (which is dependent on the volume of displaced air and the density of air) and the weight of the lifting gas itself, as well as any additional structure such as the balloon skin. This net lift determines the maximum payload that the balloon can support.

Density and Fluid Displacement

The density of a fluid is its mass per unit volume and plays a critical role in calculating buoyant forces. In the context of gas balloons, the densities of both the lifting gas and the surrounding air must be considered to evaluate the net upward force. The effective lift is derived from the difference in densities between the ambient air and the gas inside the balloon, multiplied by the volume of the displaced fluid.

Payload Calculation

Payload calculation involves determining the net lift, which is the buoyant force minus the weight of the lifting gas and the weight inherent to the balloon structure. This net lift represents the additional mass that can be carried as payload. The calculation integrates volume, the density differential between the ambient fluid and the lifting gas, and the gravitational constant, ultimately yielding the payload capacity in kilograms.

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