Do you know when we place objects in the fluid, some objects float but some sink? You may have observed that a large ship made up of iron floats in water but a small ball made of the same iron shrink in water. Have you ever noticed that while swimming, we feel our body light or while taking out water from the well, the bucket feels lighter when it is partially or fully immersed in water? This is because our body experiences a force from the downward direction in the opposite direction of the gravitational pull. This results in a decrease in weight. These all happen due to the buoyant force. The magnitude of buoyancy force is governed by Archimedes’ principle. In this article, we will discuss more details about Archimedes’ principle and Buoyancy Formula.

Buoyancy Force

To understand this, let us take an example. If we place a solid iron ball inside the bucket full of water, then it will sink. But if we place a piece of wood or cork in a bucket full of water, then it floats. When we push that piece of wood downwards, a large part of the wood goes inside the water. But as soon as we remove the applied force the, wood comes back upwards and floats again. It means that we have to maintain a downwards force on it continuously to keep the piece of wood inside the water at rest.
So, we can say that fluids apply force in the upward direction. This upward force applied by the fluid on the object or the body when an object is submerged fully or partially in the fluid is called the buoyant force. And, this phenomenon of experiencing an upward force is known as buoyancy. The unit of the buoyant force is Newton (N).

The magnitude of the buoyant force is governed by Archimedes’ Principle. So, to get the value of buoyant force, we will have to learn about Archimedes’ Principle. This will also help us to understand that whether a body will float or sink if placed in the liquid.

Archimedes’ Principle

When a body is wholly or partially submerged in a fluid (maybe water), then it displaces water, and it seems to be weightless than when it is in the air due to the upward force exerted by a fluid called buoyant force. According to Archimedes’ principle, this buoyant force is equal to the weight of the fluid displaced by the body. Now let us derive the expression for the buoyancy force formula with the help of Archimedes’ principle.

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Buoyancy Force Formula

As we know that the density \((\rho )\) is defined as:

\({\rm{Density}}(\rho ) = \frac{{{\rm{ Mass }}}}{{{\rm{ Volume }}}} = \frac{M}{V}\)

Then, the mass of the liquid displaced by the object in the above figure, 

Mass\(=\)Density of liquid\( \times \)Displaced volume\( = \rho \times V……..\left( 1 \right)\)

Thus, the weight of the displaced liquid will be:

Weight\(=\)Mass\(×\)Acceleration due to gravity                                               

\(W = M \times g………\left( 2 \right)\)

Now, using equations \((1)\) and \((2),\) we have: 

\(W = \rho \times V \times g\)

Now, according to Archimedes Principle, we can write,

The apparent loss of weight\(=\)weight of water displaced\( = \rho \times V \times g\)

Then, the  magnitude of buoyant force \((F)\) will be given as:

\(F = \rho \times V \times g………..\left( 3 \right)\)

Where,

\(\rho \) is the density of the liquid,

\(V\) is the volume displaced liquid.

Since it is responsible for objects to float so sometimes, the buoyant force is also called thrust force.

Apparent Weight

To understand the apparent weight, let us consider a piece of metal or stone in the figure below that is suspended from the hook of a spring balance by a thin thread, and the weight of the body in the air is measured. Then the body is carefully lowered into the liquid kept in a bucket full of liquid with the setup shown in the below figure. Keep the mass attached to the balance. Its new weight is noted when it is completely immersed. As the mass will dip in the liquid, it will displace the liquid; this displaced water is stored in the beaker put on the weighing machine. Then, the beaker and water are weighed when no more water drips from the spout. We will observe that the loss in weight of the body will be equal to the weight of the water displaced. Then, the difference between the actual weight and the buoyant force gives the apparent weight of an object. This experiment also verifies Archimedes’ principle.

Laws of Floating

When an object is immersed in a fluid, it will either float, partially float, or sink depends on the magnitudes of the actual weight of the object \(\left( {{W_1}} \right)\) and the buoyant force \(\left( {{W_2}} \right)\) that is exerted by the fluid.

Case 1: If \({W_1} > {W_2}\)

This condition arises when the density of the object is greater than that of the fluid. In this case, the object will sink because the resultant force on the object will act in a downwards direction. 

Case 2: If \({W_1} = {W_2}\)

In this case, the object will float at any depth in a fully submerged state because the densities of the object and the fluid both are equal; hence the actual weight of the object and the buoyant force both become equal.

Case 3: If \({W_1} > {W_2}\)

This is possible only when the floating object is pushed more into the liquid)In this case, the object will float such that the weight of the liquid displaced is exactly equal to the weight of the object. Hence, the object will be in a partially submerged condition. In this case, the density of the object is less than the density of the fluid.

Application of Archimedes Principle

1. The hydrometer works based on Archimedes’s principle. The volume or density of any rigid body can be computed using Archimedes law. 
2. Submarines operate using the Archimedes theory. There is a large ballast tank, which controls the depth of the marine. The quantity of water in the ballast tank can be adjusted, due to this, the actual weight of the submarine is varied, and thus the desired depth can be achieved.
3. To make the average density of ships less than the density of water, ships are made hollow. The ship can float in a partially submerged state as the buoyant force will be equal in the magnitude of the ship’s weight.
4. The density of hot air is less than the density of ambient cool air. This concept is used in hot air balloons so that it can float in the air.

Solved Examples on Buoyancy Formula 

Q.1. Dimensions of a block of wood are given as length \({\rm{ = 2}}{\rm{.5}}\,{\rm{m,}}\) width \({\rm{ = 0}}{\rm{.5}}\,{\rm{m,}}\) and height \({\rm{ = 0}}{\rm{.4}}\,{\rm{m.}}\) If the block is in the water completely submerged, what will be the buoyant force acting on it? The density of water is given as \(1000\;{\rm{kg}}\;{{\rm{m}}^{ – 3}}\) and \(g = 10\;{\rm{N}}\;{{\rm{kg}}^{ – 1}}.\)
Solution: The volume of the block is:  \(V = 2.5 \times 0.5 \times 0.4 = 0.5\;{{\rm{m}}^3}\)
Density of water \(\left( {{\rho _w}} \right) = 1000\;{\rm{kg}}\;{{\rm{m}}^{ – 3}}.\)
We know Buoyancy \(F = \rho gV\)
\( \Rightarrow F = 1000 \times 10 \times 0.5\)
\( \Rightarrow F = 5000\;{\rm{N}}\)

Q.2. Find the fraction of volume of the piece of ice outside the water? If a piece of ice is floating in the water of density \(1000\;{\rm{kg}}\;{{\rm{m}}^{ – 3}}\) and the density of ice is given as \(900\;{\rm{kg}}\;{{\rm{m}}^{ – 3}}.\)
Ans: Let \(V\) be the total volume and \({V_i}\) the volume of the ice piece immersed in water. 
Now, for the equilibrium of the ice piece we have:-
\({\rm{Weight of the ice = Upthrust}}\)
\(V{\rho _i}g = {V_i}{\rho _w}g\)
Here, \({\rho _i} = {\rm{density}}\,{\rm{of}}\,{\rm{ice}} = 900\;{\rm{kg}}\;{{\rm{m}}^{ – 3}}\)
And, \({\rho _w} = {\rm{density}}\,{\rm{of}}\,{\rm{water}} = 1000\;{\rm{kg}}\;{{\rm{m}}^{ – 3}}\)
On putting these given values in the above equation, we get
\( \Rightarrow \frac{{{V_i}}}{V} = \frac{{900}}{{1000}} = 0.9\)
Then, the fraction of volume outside the water will be
\(f = 1 – 0.9 = 0.1\)

Summary

When an object is placed or immersed in a liquid, then it experiences a force due to the fluid in the upward direction. This unknown upward force is known as Buoyant Force. And, this phenomenon of experiencing an upward force is known as buoyancy. According to Archimedes’ principle, the buoyant force is equal to the weight of the fluid displaced by the body. The formula to calculate buoyant force is,

\(F = \rho \times V \times g\)

Where,

\(V=\)Volume of the object inside the fluid, 

\(ρ=\)Density of the fluid,

\(g=\)Gravitational acceleration.

Frequently Asked Questions (FAQs) on Buoyancy Formula

Answers to the most commonly asked questions on the buoyancy formula are provided below:

Q.1. How do you calculate buoyancy? Ans: We can calculate the buoyancy according to Archimedes’ principle. The buoyancy force can be given by the formula: \(F = \rho \times V \times g\) Where, \(V=\)Volume of the object inside the fluid, \(ρ=\)Density of the fluid, and \(g=\)Gravitational acceleration.

Q.2. What causes buoyancy? Ans: The buoyancy force is caused by the pressure exerted by the fluid in which an object is immersed. As the pressure of a fluid increases with depth, the thrust force acts more at the bottom in the upwards direction. The difference in pressure force will cause the buoyancy force.

Q.3. How are buoyancy and density related? Ans: The buoyant force is directly proportional to the density of the fluid in which an object is immersed. The upward force exerted on objects submerged in fluids is called the buoyant force, where \(ρ\) is the density of the fluid, \(V\) is the volume, and m is the mass of the displaced fluid.

Q.4. How do you increase buoyancy? Ans: Buoyancy force on an object due to a fluid is directly proportional to the density of the fluid, the volume of the object inside the fluid, and acceleration due to gravity. If the density of the fluid and acceleration due to gravity is fixed, submerging the object more into the fluid increases buoyancy. Maximum buoyancy acts when the object is completely submerged in liquid.

Q.5. Are the buoyancy in salt water and fresh water are same or different? Ans: The buoyancy (upward force) depends on the density of the liquid. Different fluids provide different buoyancies. There will be a greater upward force on objects in saltwater than in freshwater because the addition of salts to freshwater increases the density of the water.

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