(II) How many helium-filled balloons would it take to lift a...

(II) How many helium-filled balloons would it take to lift a
person? Assume the person has a mass of 75 $\mathrm{kg}$ and that each
helium-filled balloon is spherical with a diameter of 33 $\mathrm{cm} .$

Key Concepts

Archimedes' Principle

Archimedes' Principle states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid that the object displaces. This principle is foundational in understanding how a helium-filled balloon can generate an upward force sufficient to counteract the weight of an object like a person.

Buoyancy

Buoyancy is the upward force that acts on objects submerged in a fluid, resulting from the pressure difference at different depths. It explains how a balloon filled with a gas less dense than air, such as helium, can float or even lift additional weight by displacing a volume of air that weighs more than the helium inside.

Spherical Volume Calculation

Calculating the volume of a sphere is crucial for determining the amount of air displaced by a balloon. The volume is found using the formula V = (4/3)?r³, where r is the radius of the sphere. This calculation helps in establishing the magnitude of the buoyant force available from each balloon.

Density

Density, defined as mass per unit volume, plays a key role in buoyancy calculations. By comparing the density of air to that of helium, one can determine how much net lift a helium-filled balloon can provide, since the buoyant force depends on the weight of the displaced air relative to the weight of the contained helium.

Gravitational Force and Weight

Understanding gravitational force is essential, as it is the force that must be overcome for lifting an object. Weight is the product of mass and gravitational acceleration, and calculating this force allows one to determine the total upward lift required from the balloons to counteract gravity.

Net Lift Calculation

The net lift provided by a helium-filled balloon is calculated by taking the difference between the buoyant force (determined by the weight of the displaced air) and the weight of the helium within the balloon. This net lift tells us how much additional weight can be supported, which is critical for determining the number of balloons necessary to lift a given mass.

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