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Archimedes' Law

The force exerted by a fluid on a floating or submerged object is equal to the weight of the water displaced.

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Hot Air Balloons

Hot Air Balloons

Credit: Carolyn Conner
Source: http://www.flickr.com/photos/42966376@N02/4267826080/

One of the earliest forms of flight, hot air balloons demonstrate how Newton's 2nd law and the principles of buoyancy can work together.

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• Newton's 2nd law states that the sum of all the forces acting on an object is equal to the mass of that object times its acceleration. Therefore, for a hot air balloon to ascend, the net force on it must be point upward:

FupFg=ma\begin{align*}F_{up}-F_g=ma\end{align*}

• Fup\begin{align*}F_{up}\end{align*} is the buoyant force directed upward, and Fg\begin{align*}F_g\end{align*} is the force due to gravity, which points downward. The buoyant force, from Archimedes principle is shown in the following equation:

Fup=ρVg\begin{align*}F_{up}=\rho Vg\end{align*}

ρ\begin{align*}\rho\end{align*} is the density, V\begin{align*}V\end{align*} is the displaced volume, and g\begin{align*}g\end{align*} is gravity.

Credit: Hans
Source: http://pixabay.com/en/balloon-hot-air-balloon-balloon-glow-8478/

The part of the aircraft that looks like a balloon is called an envelope [Figure2]

• Since the air inside the hot air balloon is heated by burners, it is less dense than the cooler air outside the balloon. When enough lift is generated, the buoyant force is greater than the weight of the balloon and the balloon can begin to accelerate upwards.

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Using the information provided above, answer the following questions.

1. Show mathematically what Newton's 2nd law says when the balloon's burners are heating up the air inside the balloon but there is no upward lift.
2. What is the relationship between temperature and density?
3. If the density of the air inside the balloon was greater than the density outside of the balloon, would lift be generated?

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