The buoyant force exerted on the objects in the fluid medium is applied upward on the fluid and ensures the object floats, sinks, or rises on the fluid.

**The buoyant force exerted on the object follows Archimedes’ principles: several entities like weight, density, and nature of the fluid medium are involved. Using all these entities let us learn how to calculate buoyant force and the problems regarding the buoyant force.**

To calculate the buoyant force, the steps given below have to be followed:

**Find the volume of the object immersed –**Since the object’s volume in the fluid medium is largely influenced by the buoyancy; so we need to find them.**Find the density of the object and the fluid medium –**Density of both the object and the fluid is necessary because density plays a vital role in the exertion of buoyant force.**Find the force of gravity or a downward force –**the buoyant force is an upward force, so it needs to be balanced by a downward force; thus, either gravity or any downward force acting on the object in the medium has to be found out.

**Buoyant force formula**

Since we know that density, volume, and the downward force such as gravity influence the buoyant force directly, using all these entities, the general formula for buoyant force is given by

**F _{b}=V×ρ×g**

**Where; V is the volume of the fluid, ρ is the density of the fluid, and g is the acceleration due to gravity.**

This formula gives the buoyant force of the fluid medium exerted on the object following the Archimedes principle.

**How to find buoyant force with density?**

Density is nothing but the mass per unit volume of the substance, so the density of the fluid is essential to know the entity to calculate buoyant force.

**The density of the fluid is given by the formula,**p=m/v_{f}**Where ρ is the fluid density, m is the mass, and V is the fluid volume.****The formula can calculate the volume of the fluid; V**_{f}=l×w×h; where l is the length, w is the width, and h is the height.

By substituting the value of density and volume, the buoyant force is calculated by the formula, F_{b}=ρgV.

**For example, An object of mass 4kg is submerged in a fluid medium whose volume is 8m ^{3} they how to calculate buoyant force of the fluid? Take acceleration due to gravity as 9.8m/s^{2}.**

Given –the mass of the object, m=4kg

Volume V=8m^{3}

The density of the fluid ρ is given by

p=m/v_{f}

p=4/8

ρ =0.5kg/m^{3}

The buoyant force with density is given by

F_{b}= ρgV

Substituting the value of all the given values, we get

F_{b}=(0.5)(8)(9.8)

F_{b}=39.2N

**How to calculate the buoyant force of a floating object?**

The above calculation involved in the buoyant force gives the amount of force exerted on the object to push it from the submerged state. It is quite easy to calculate, but how to calculate buoyant force for a floating object?

**An object is said to be floating when the buoyancy force is more than the gravitational force. Some extra work has to be done to find the buoyant force on the floating object. The steps to be followed to find the floating buoyancy are given below.****First, calculate the entire buoyant force acting on the object in the fluid, i.e., use the whole volume.****Then find the gravitational force pushing the object downward by the equation, W=mg; where m is the mass of the object and g is the acceleration due to gravity, also known as the object’s weight.****If the force of buoyancy and force of gravity is less than the buoyant force, then the object is floating on the fluid.**

A solved example can be clear to understand the floating buoyant force calculation.

**An object of mass 12kg and density 0.58kg/m ^{3} is dropped in a fluid. How to Calculate buoyant force exerted on the object to float on the fluid?**

Given –the mass of the object m=12kg

The density of the object ρ=0.58kg/m^{3}

The volume of the object can be given as

V=m/p

Substituting the value of m and ρ,

V=12/0.63

V=20.68m^{3}

The buoyant force acting is given by

F_{b}=ρgV

F_{b}=(0.58)(9.8)(20.68)

F_{b}=127.67N

The weight of the object is given by

W=mg

W=(12)(9.8)

W=117.6N

Since the value of buoyant force is more than the value of the force of gravity, the object floats on the fluid.

**How to calculate buoyant force in air?**

The buoyant force in the air is associated with air displacement with the object. The calculation of buoyant force in the air is quite different from the general buoyant force calculation because the air density is much smaller.

**The calculated air density is approximately equal to 1.3×10**^{-3}kg/m^{3}.**Then calculate the density of the object floating in the air medium.**

We know that when the force of buoyancy acts in the upward direction, the force of gravity tries to pull the object downward. The object can float in the air only when the buoyant force is greater.

In the air medium, the buoyant force must be equal to the object’s weight for efficient floating; thus, we can write.

F_{b}=m*g

Where m is the mass of the object and g is the acceleration due to gravity. But in air, the mass can be rewritten as

m=p_{a}/p; where ρ_{a} is the density of the air and ρ is the density of the object.

Substituting the value of mass in the buoyant force equation, we get

F_{b}=p_{a}/p*g

The problem given below helps you to understand better.

**How to Calculate buoyant force on a gold coin of density 19g in the air medium?**

We know that the density of gold coin ρg=19g/cm^{3}

The density of the air ρa=0.0013g/cm^{3}

The buoyant force acting on the gold coin in the air is given by

F_{b}=p_{a}/p*g

F_{b}=0.0013/19*9.8

F_{b}=(6.83×10^{-5})9.8

F_{b}=6.705×10^{-4}N

**How to calculate buoyant force in water?**

Water displaced volume with the density of the water is used to calculate the buoyant force in the water.

**If the object is completely submerged in water, then 100% of the volume should be considered for the calculation.****If the object is partially submerged, 50% of the volume should be considered.****If the object is submerged in only a quarter, then only 25% of the volume should be considered.**

The remaining calculation of the buoyant force is similar to the general buoyancy calculation.

**A body mass of 15kg and density of 0.55kg/m ^{3} is partially submerged in the water. How to Calculate buoyant force acting on the object?**

We know that density of the object ρ=0.55kg/m^{3}

Mass of the object m=15kg

Volume is given by;V=m/p

V=15/0.55

V=27.27m^{3}

Since the object is submerged partially, its volume in the water is equal to half of its total volume; thus, volume V=13.635m^{3}.

The buoyant force can be given by

F_{b}=ρgV

F_{b}=(0.55)(9.8)(13.635)

F_{b}=73.492N.

**How to calculate the buoyant force of a balloon?**

To find the buoyant force of a balloon, we need to know the air-filled volume inside the balloon, which makes the calculation somewhat different.

**The air itself, having low density, still support certain object to float in them. Some balloons, such as helium gas balloons, have less density than the air. So it is easy for balloons to float on them.**

When the balloons are filled with gas, it acquires a shape that resembles a sphere; thus, we need to calculate the volume of the sphere expression as,

V=4/3πr^{3}

Then knowing the density and mass, we can easily calculate the buoyant force.

**A balloon is filled with the air and forms a sphere with a radius of 5cm and is allowed to sail in the air. How to Calculate buoyant force acting on the balloon to lift in the air?**

Given –the radius of the balloon r=5cm.

The volume of the balloon is given by

V=4/3πr^{3}

V=4/3(3.14)*2^{3}

V=33.49m^{3}.

The buoyant force is given by

F_{b}=ρgV

The density of the air is ρ=1.3kg/m^{3}.

Substituting the values, we get

F_{b}=(1.3)(9.8)(33.49)

F_{b}=426.66N.

**How to calculate the buoyant force of a boat?**

The boat always floats on the surface of the water; the buoyancy of the boat should be calculated by considering the entire volume of the boat.

**According to the Archimedes principle, the upward force exerted on the immersed body is equal to the weight of the fluid displacement. And this displacement is also acting in the upward direction toward the center of the mass of fluid displacement.**

The equation of buoyant force is

[latex]F_b=v\times\frac{f}{v}=v\times\frac{mg}{v}[/latex]

But [latex]\frac{m}{v}=\rho[/latex];

F_{b}=ρgV

For a boat sailing on the water, the buoyant force is given by

[latex]F_b=\frac{W}{W_a}[/latex]

Where; W is the object’s weight in the water, W_{a} is the weight of the object in the air.

**How to calculate buoyant force on a submerged object?**

For a submerged object, the volume is the same as the displaced volume in the fluid, and hence we can easily find the buoyant force of a submerged object with the same equation.

F_{b}=ρgV

For a submerged object, the weight of the fluid displacement is to be found. The weight of the displaced fluid is given by

W_{f}=ρ×V

**How to calculate magnitude of buoyant force?**

The magnitude of the buoyant force is always equal to the magnitude of its weight. This is true only when the object floats. Let us understand this concept by considering an example of a slab of thickness t and density ρs floating on the water’s surface with mass m.

**Since the magnitude of the buoyant force is equal to the magnitude of the weight of water given by**

**W _{w}=ρ_{w}Atg; where A is the area of the slab.**

**The magnitude of the weight of the slab is given by,**

**W _{s}=ρ_{s}Atg+mg**

But According to Archimedes’ principle

W_{s}=W_{w}

ρ_{w}Atg= ρ_{s}Atg+mg

mg= ρ_{w}Atg- ρ_{s}Atg

m=ρ_{w}At-ρ_{s}At

m=A(ρ_{w}t-ρ_{s}t)

[latex]A=\frac{m}{\rho_wt-\rho_st}[/latex]

**How to calculate buoyant force on a cube?**

When a cube is submerged in the fluid, its volume is equal to the cubical value of every side. Using this as a reference, we can calculate the buoyant force on the cube.

**For example, a cube of side length 2cm is immersed in an oil of density 800kg/m ^{3}. Calculate the buoyant force acting on the cube.**

The side length of the cube l=2cm=0.2m.

The volume of the cube can be calculated as

V=l^{3}=(0.2)^{3}=0.008m^{3}

The buoyant force F_{b}=ρgV

Let us take g=9.8m/s^{2}.

Substituting the values in the above equation,

F_{b}=(800)(9.8)(0.008)

F_{b}=62.72N.

**Some more solved problems**

**A body of mass 0.56kg is submerged in a fluid of density 910kg/m**^{3}. Calculate the buoyant force acting on the body in that fluid. And hence calculate the weight of the fluid displacement.

^{3}. Calculate the buoyant force acting on the body in that fluid. And hence calculate the weight of the fluid displacement.

**Solution:**

Given –the mass of the body, m=0.56kg

The density of the fluid medium ρ=910kg/m^{3}

Acceleration due to gravity g=9.8m/s^{2}.

The volume of the body in the fluid is

[latex]V=\frac{m}{\rho}[/latex]

[latex]V=\frac{0.56}{910}[/latex]

V=6.153×10^{-4}m^{3}

The buoyant force exerted on the body in the fluid

F_{b}=ρgV

F_{b}=(910)(9.8)(6.153×10^{-4})

F_{b}=5.488N.

**A slab with a length of 20cm, a width of 9cm, and a height of 0.88cm are floating in a fluid of density of 998kg/m**^{3}. Calculate the buoyant force exerted on the object and hence calculate the weight of the object to float on the fluid. (Take acceleration due to gravity g=10m/s^{2})

^{3}. Calculate the buoyant force exerted on the object and hence calculate the weight of the object to float on the fluid. (Take acceleration due to gravity g=10m/s

^{2})

**Solution:**

Given –length of the given slab, m=20cm=0.2m

Width of the slab w=9cm=0.09m

Height of the slab h=0.88cm=0.0088m

Density of the fluid medium ρ=998kg/m^{3}

First, we have to find the volume of the slab

V=lwh=(0.2)(0.09)(0.0088)

V=1.58×10^{-4}m^{3}

The buoyant force acting on the slab in the fluid medium is

F_{b}=ρgV

F_{b}=(998)(10)(1.58×10^{-4})

F_{b}=1.580N.

**A balloon is blown with a gas of density 0.89kg/m**^{3} and allowed to float in the air of density 1.22kg/m^{3}. The balloon formed a sphere-like structure of radius 0.32m. Calculate the buoyancy applied to the balloon and the volume of the balloon.

^{3}and allowed to float in the air of density 1.22kg/m

^{3}. The balloon formed a sphere-like structure of radius 0.32m. Calculate the buoyancy applied to the balloon and the volume of the balloon.

**Solution:**

Given –the density of the balloon filled with the gas ρb=0.89kg/m^{3}

The density of the air ρa=1.22kg/m^{3}

The radius of the balloon r=0.32m.

The buoyant force is calculated as

[latex]F_b=\frac{\rho_a}{\rho_b}g[/latex]

[latex]F_b=\frac{1.22}{0.89}g[/latex]

F_{b}=1.32N.

The volume of the balloon filled with gas is given by

[latex]V=\frac{4}{3}\pi r^3[/latex]

[latex]V=\frac{4}{3}(3.14)(0.32^3)[/latex]

V=0.137m^{3}.

**Check whether the given body sinks or floats on the fluid density of 1025kg/m**^{3}. Given that mass of the object is 46kg.

^{3}. Given that mass of the object is 46kg.

**Solution:**

Given –the density of the fluid ρ=1025kg/m^{3}

The mass of the given body m=46kg.

Acceleration due to gravity g=9.8m/s^{2}.

The volume of the body in the fluid v is given by

[latex]V=\frac{46}{1025}[/latex]

V=0.044m^{3}

The buoyant force acting on the body in the fluid

F_{b}=ρgV

F_{b}=(1025)(9.8)(0.044)

F_{b}=450.8N

The force of gravity acting on the body W=mg

W=(46)(9.8)

W=450.8

Since the weight of the body and the buoyant force exerted on the body are equal; hence the body is under neutral buoyancy condition. The body neither sinks in the fluid, nor rises in the fluid.