Miniature Hot Air Balloons and Archimedes's Principle of Buoyancy

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The history of hot air balloons starts in the Asia. Literary references in China date back to 180ce and cite Chu Ko Liang as the inventor of lighter than air flight (1). Early unmanned balloons were used in China as military signaling lanterns (2). Today ethnic Chinese communities celebrate the end of the Lunar New Year Holidays (3) by launching small hot air balloons ( known as Kung-Ming Lanterns ).

The first recorded mathematical description of buoyancy (and thus hot air balloon behavior) was developed by Archimedes over 2000 years ago in Greece (4). The bouncy force is summarized by Archimedes's principle , “the magnitude of the buoyant force is always equal the weight of the fluid displaced by the object.” (5)

The use of hot air
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The weight of the beach-ball has changed, but the upward force due to the liquid remains the same. The force due to the weight of the beach-ball is now smaller than the force due to the liquid surrounding it, the beach-ball will rise in the water tank.

This is the exact same phenomenon that causes hot air balloons to rise in the atmosphere. The buoyancy force from the atmosphere is greater than the force due to the weight of the balloon. The perceived lighter weight of the hot air balloon is caused by heating the air mass within the balloon. This makes the air in the balloon less dense than the surrounding atmosphere. Just like the air in a beach ball is less dense than the water around it.

The buoyancy force is due to a difference in pressure between the top of a submerged object and the bottom. If the pressure at the bottom of an object is greater than the pressure at the top, the resulting force will be upward. The buoyancy force can be describe by the equation:

B = (Pb – Pt ) A = (Dfluid g h) A = Dfluid g V (1)

Where B is the buoyancy force, Pb and Pt is the pressure at the top and bottom of the object respectively, A is the surface area of the object, Dfluid is the density of the fluid, g is the acceleration of gravity (9.8 m/s^2), h is the height of the object, V is the volume of the fluid displaced by the object, and M is the mass of the fluid displaced by the object.

Equation 1 can be summarized in the form:

B =
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