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The size or the shape of the object is not necessarily the deciding factor for determining buoyancy. It is the relation between the object’s weight compared to the weight of water it displaces.  Buoyancy has become an important factor for engineers and designers in designing boat, ships, and seaplanes to keep them afloat. It is also used in many water sports.  Buoyancy not only explains as to why ships and vessels float in water, but also explains the rise of a balloon in the air and the apparent loss of weight of objects underwater. In other words, buoyancy = weight of displaced liquid.
The tendency of a fluid to exert an upward force on an object immersed in it is called buoyancy.   
Archimedes’s was the first one to show that when an object is floating in water, the weight of the displaced fluid can be found out mathematically.  This principle was known as Archimedes’s principle and according to Archimedes's principle, the volume of the displaced fluid is directly proportional to the weight of the displaced fluid.

In other words, the principle states that buoyant force is equal to the weight of the liquid displaced by the object. Buoyancy of an object can also be found by finding the weight of that particular object in air and then finding the weight of the same object in liquid. This final result would be measured in newtons.

Liquid Buoyancy

Pressure pushes cork upwards when it try to pushed down into water

Buoyancy Force

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When we draw a bucket of water from the well, it appears to be lighter as long as it is under the surface of water but the moment the bucket is brought above the surface of the water it appears heavier. This means that an upward force acts on the bucket when it is immersed in water. This upward force make the bucket weigh less. The upward force that acts on an object immersed in a fluid is called the upthrust or buoyant force.

The buoyant force depends only on two factors
  1. The volume of the object immersed in the liquid
  2. The density of the fluid
The greater the object's volume and the surrounding density of the fluid, the more buoyant force it will experience. If the buoyancy of an object exceeds its weight, it will tend to rise. The object whose weight exceeds its buoyancy will always tend to sink. Buoyancy also depends on volume and so an object's buoyancy reduces if it is compressed and increases if it expands.

Buoyancy Formula

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The buoyant force can be expressed using the following equation:

$F_{buoyant}$ = $-\rho V g$

$\rho$ - It is the density of the fluid
$V$ -Volume of the object submerged

The negative sign must be used since the buoyancy is opposite in direction with the acceleration due to gravity.

It is found that the buoyant force exerted by the gas is less than than exerted by the liquid.

Buoyancy of Air

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The air exerts a buoyant force on everything in it, not just balloons. This force, 0.08 pounds for each cubic foot, is so small it can be ignored except when dealing with other gases.

For example, the volume of a person's body is typically around 2 or 3 cubic feet. This means that the buoyant force on a person because of the air is between 0.16 and 0.24 pounds.

An object surrounded by air is buoyed up by a force equal to the weight of the air displaced.

Applications of buoyancy are abundant. People enjoy hot-air balloon rides in which the heated air within the balloon is less dense than the cooler air outside the balloon. Water therapy is used by physical therapists not only because water offers greater resistance than air, but also because the buoyant force reduces the amount of stress on patient's joints.

Neutral Buoyancy

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Neutral Buoyancy is a condition at which a physical body's mass equals the mass it displaces in a surrounding medium. This negates the effect of gravity that would otherwise cause the object to sink. An object that has neutral buoyancy will neither sink nor float. Instead, it will remain at its current level in the medium that surrounded it.