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© The scientific sentence. 2010

Buoyant Forces



1.Pressure


Pressure is the force that is exerted on a unit surface area. It is defined as the dot product of the vector force and the vector normal to the surface. P = F . n = |F| |n| cosθ = |F| cos θ.
θ is the angle between the vector n and F. Since |n| = 1, we have then:
P = |F| cos θ.

Notice that only the component of the force , that is perpendicular to the surface exerts a pressure on this surface. Only normal force exerts a pressure.

Pressure is a scalar quantity, so it has no direction. Its SI unit is the Pa (Pascal) = 1 Newton/m2(N/m2).

The absolute pressure is measured from zero. That is, if we press something, we just add a pressure above the already existing pressure called the atmospheric pressure. The added pressure is called gauge pressure. This added pressure is relative to the ambient atmospheric pressure.



2. Pascal's Principle

It states:

When a pressure is applied to an enclosed incompressible fluid, this pressure is fully transmitted to every point of the fluid, including the walls of the container where the fluid is confined.



3.Archimedes' principle

It states:

When an object is immersed in a fluid, the object is buoyed up by a force that is equal to the weight of the fluid displaced by this object.



4. Column height pressure



At the top, the absolute pressure P1 is equal to the atmospheric pressure Patm.
At the level h, the absolute pressure P2 is equal to the pressure Patm plus the contribution P of the column fluid of height h. Therefore: P2 = Pa + P
Where:
P is the pressure of the weight of liquid corresponding to the height h; that is P = F/S = mg/S = ρ V g/S = ρ gh


P = ρgh



5. Buoyant force


5.1. Archimedes' law

The buoyant force replaces the normal force of contact if an object was at rest on a rigid support. Here, the rigid support is replaced by an incompressible fluid.

The buoyant force is due to the difference between the pressure at the bottom surface and the pressure at the top surface. Thus:

Fb/So = P2 - P1 = P = ρf g h  (1)

Where
So is the area of the object at the bottom,
ρf is the density of the fluid.

We have:
Vf = h SoVf the volume of the displaced fluid.

If mf is the mass of fluid displaced, then:
mf = ρf Vf

Therefore:
mf = ρf h So

If Wf = is the weight of the displaced fluid, then:
Wf = mf g = ρf g Vf = ρf g So h = Fb Then: Fb = Wf = ρf g h So = ρf g Vf

The buoyant force is equal to the weight of the fluid's displaced volume.




The buoyant force is equal to the weight of the fluid's displaced volume
Fb = Wf = ρf g h So = ρf g Vf



5.1. Immersed object

Let's denote by ho, Vo, mo, and &pho;o the height, the volume, the mass, and the density of the object respectively, we have:

Fb = So ρf g ho
Vo ρf g
mo = ρo Vo

The object is at rest. According to the Newton's first law, we have:
Fb = mo g

According to (1)
Fb = ρf g h So

Therefore:
mo g = ρf g h So
That is:
ρo Vo g = ρf g Vf

Hence:
ρo Vo = ρf Vf


ρoVo = ρfVf



5.2. Wholly immersed = totally submerged body

For this case, the volume of fluid displaced is equal to the entire volume of the object.
ρo Vo = ρf Vo
or
Voo - ρf ) = 0

5.3. Does the object sink or float ?

At a certain level, when the object is wholly immersed (Vo = Vf), if we release the object we will have two cases:

1. Fb > mg:

ρf Vf > ρo Vo That is with Vf = Vo:
ρf > ρo
So, the object raise to the surface and floats.


ρf > ρo
The object floats



2. Fb < mg:

ρf < ρo
The object falls to the bottom and sinks.


ρf < ρo
The object sinks



  


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