Buoyant Force 

The buoyant force is a fundamental term in physics that explains why the floating and sinking of objects submerged in fluids take place. This upward force is exerted by the fluid, which might be water or air on an object partially or fully immersed in it. It comes due to the pressure variation of the fluid at different points in its depth.

1.0Insight into Buoyant Force 

Definition 

The buoyant force is the upward force exerted by any fluid (liquid or gas) on an object that is submerged in it or floating on its surface. It is responsible for making objects float or rise in the fluid. For example: When a boat floats on water, a helium balloon rises in the air or an object is submerged in water and experiences an upward force. 

In maths, the same can expressed as: 

$F_b={\rho }_{f}gV$

Here, 

• F_b​ = Buoyant force
• ρ_f = Density of the fluid
• g = Acceleration due to gravity
• V = Volume of the displaced fluid

Archimedes’ Principle and Buoyant Force 

Archimedes' Principle and buoyant force are two related but quite different concepts. Archimedes' Principle is as follows: Any object placed in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. 

The buoyant force is the name of this upward force. While Archimedes' Principle explains why buoyant force occurs, buoyant force is the actual physical force experienced by an object in a fluid. The two concept  ,s give rise to another important concept that is: 

Principle of Floatation: It states that if the buoyant force acting on the object is equal to its weight, then the object will float in fluid. When the buoyant force is less than the weight of the object, then the object will sink. This principle helps in explaining why objects with different densities behave differently in fluids.

Both concepts are essential for understanding why objects float or sink depending on their displacement and density.

Key Takeaways

1. Direction of Force: Buoyant force is always in a vertical upward direction and opposite to the gravity direction.
2. Dependency on Fluid Density: Buoyant force is directly proportional to the fluid density in which the object is submerged. The more the fluid density, the more the buoyant force.
3. Dependence on the volume of the object: It also depends upon the volume of the fluid displaced by the object. The larger volume displaces more fluid, which in turn causes a larger buoyant force.
4. Objects which Float and Sink:

• When the weight of the buoyant force is equal to the weight of the object, the object floats
• When the weight of the buoyant force is less than the weight of the object, the object sinks.
• A body will float in a liquid if the buoyant force is greater than the weight of the object.

2.0Buoyancy and Buoyant Force 

1. Buoyancy: Buoyancy is referred to as the ability of an object to float in fluid. It is due to the buoyant force acting on the object. An object floats if the buoyant force is equal to or greater than its weight.
2. Buoyant Force in Various Fluids:

• In Water: The buoyant force is significant because water has a relatively high density.

• In Air: The buoyant force is still in place, but it's very small since air has a much lower density than water. For instance, balloons rise in the air because the buoyant force overwhelms their weight.

3.0Solved Examples 

Example Problem 1: A metal block of volume 0.5 m³ is submerged in water. The density of water is to be taken as 1000 kg/m³. Find the buoyant force acting on the block.
Solution: Using the formula F_b=ρ_f g V
F_b=1000 kg/m³×9.8 m/s²×0.5 m³

F_b​=4900N
Hence, the buoyant force on the metal block is 4900 N.

Example Problem 2: A body weighing 100 N is floating on water. The volume of the displaced water is 0.1 m³, and the density of water is to be taken as 1000 kg/m³. Is the body floating or sinking?
Solution: Buoyant force F_b=ρ_f g V
Fb=1000×9.8×0.1= 980 N

Since the buoyant force (980 N) is more than the weight of the body (100 N), the body will float.

Problem 3: A cylindrical metal rod of length 2 m and radius 0.1 m is submerged vertically in water. The rod has a density equal to 8000 kg/m³. Calculate The buoyant force acting on the rod when it is completely submerged and The apparent weight of the rod when it is fully submerged in water.

Solution: 

1. The Buoyant force is the force equal to the weight of the displaced fluid. 

$F_B={\rho }_{fluid}.V_{displaced}.g$

$fluid=1000kg/m^3$ density of water

V_displaced is the volume of the submerged part of the rod. Since the rod is completely immersed, the volume of the water displaced would be equal to that of the rod.

Volume of cylinder = $\pi r^{2}h=3.14\times (0.1{)}^2\times 2=0.0628m^3$

Now, the buoyant force: 

$F_B=1000\times 0.0628\times 9.8=616.24N$

2. Apparent weight of the Rod when fully submerged

$W_{apparent}=W_{actual}–FB$

$W_{actual}=Massoftherod\times accelerationduetogravity=m\times g$

$Massoftherod={\rho }_{rod}\times V_{rod}$

$m=8000\times 0.0628=502.4kg$

$W_{actual}=502.4\times 9.8=4923.52N$

$W_{apparent}=4923.52–616.24=4307.28N.$