Physics - Classical Mechanics - Archimedes' Principle and Buoyancy

about 2 years ago

Introduction

Hey it's a me again @drifter1!

Today we continue with Physics, and more specifically the branch of "Classical Mechanics", in order to talk about Archimedes' Principle and Buoyancy.

So, without further ado, let's get straight into it!

Buoyant Force

When an object is submerged into a fluid, the fluid exerts an upward force to it. This force is known as buoyant force. Buoyancy (also called upthrust) is the exact reason why some objects float and others do not. So, the buoyant force is an upward force that's exerted on any object in any fluid.

Buoyancy is related to the increase in pressure with depth in a fluid. There is always a difference in pressure between the bottom and the top of a submerged object. The pressure from the top is of course less than the pressure exerted upward from the bottom.

If the force exerted through this pressure is greater than the object's weight the object will rise to the surface of the fluid and float. If it's less the object will start sinking, until it eventually reaches the bottom. So, the buoyant force is always present whether the object floats, sinks or stays suspended at the same depth (the force equals its weight).

Archimedes' Principle

Up to this point we only covered the basic concept of buoyancy. Let's now get into how we determine the exact value of the buoyant force.

The answer is given by Archimedes' principle, which states that the buoyant force on an submerged object is equal to the weight of the fluid it displaces:

Let's try to understand this equation better, by proving it!

Proof

The buoyant force is the net force acting upon the object. The pressure from the sides is cancelled out. And so, the only important forces are a force F_down from the top that pushes the object down, and a force F_up from the bottom which pushes the object up. So, the buoyant force thus equals:

Relating these forces to pressure, yields:

Of course, in this calculations we are again talking about a cylinder. So, the final term can be replaced by the volume V of the cylinder. This volume also equals the volume of the displaced fluid. So, putting this instead yields:

And with that we've proven the principle!

Common Mistakes

There are some common mistakes that are made when applying the principle that need to be mentioned. The term ρ refers to the density of the fluid and not of the submerged object. The volume V refers to the volume of the displaced fluid, which doesn't necessarily have to be equal to the entire volume of the submerged object. Lastly, a common misconception is that the buoyant force changes with depth, which is not true as it's the same at any depth.

Relation with Density

The density of an object also determines whether it floats. If it's less than the surrounding fluid it will float. That's because the fluid with higher density contains more mass and thus more weight in the same volume. The buoyant force will therefore be greater than the weight of the submerged object.

The ratio of the submerged volume to the total volume of the object is defined as the fraction submerged:

Obtaining a relationship between the densities is now a simple substitution:

When floating the masses are equal and can be cancelled out, which yields:

So, if the density of the object is less than that of the fluid, some "part" of it will float. Otherwise, the complete object will be submerged, which basically means that the object will sink.

RESOURCES:

References

Images

https://pxhere.com/en/photo/1045542

Mathematical equations used in this article, where made using quicklatex.

Visualizations were made using draw.io.

Previous articles of the series

Rectlinear motion

Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
Rectlinear motion exercises -> examples and tasks in rectlinear motion

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Projectile motion as a plane motion -> missile/bullet motion as a plane motion
Smooth Circular motion -> smooth circular motion theory
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Mass and Newton's second law -> mass, 2nd law
Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
Contact forces and friction -> contact force, friction
Dynamics of Circular motion -> circular motion dynamics, applications
Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
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Advanced Newton law examples -> advanced (more difficult) exercises

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