# Buoyancy and Archimedes Principle

Objects in a fluid seem to weigh less than they do because of a concept
called **buoyancy**. Buoyancy occurs because the pressure of a fluid increases with depth. When an object is in a fluid, the force pushing up on the bottom of the object is larger than
the force pushing down from the top. The **buoyant
force** can be calculated by finding the difference between the force pushing on the bottom of the object (F_{2}) and the force pushing on the top of the object
(F_{1}):

**F _{B} = F_{2} –
F_{1}**

****

F_{1}, or the force pushing down on the top of the object, is equal
to the pressure of the fluid multiplied by the area of the top of the object, or:

F_{1} = PA

The pressure of the fluid on the object can be determined by multiplying the density of the fluid by gravity by the height of the fluid from the top of the object to the surface:

P = **r**_{f} g h_{1}

So:

F_{1} =**r**_{f} g h_{1}A

The pressure of the fluid pushing up on the bottom of the object can be
calculated in much the same way, except the height used when calculating the pressure is the distance from the bottom of the object to the surface (h_{2}):

P = **r**_{f} g h_{2}

So:

F_{2} = **r**_{f} g h_{2} A

The buoyant force = F_{B} = F_{2} – F_{1},

so:

F_{B} = **r**_{f} g h_{2} A – **r**_{f} g h_{1} A

We can simplify the equation:

F_{B} = **r**_{f}gh_{2}A – **r**_{f}gh_{1}A = **r**_{f}gA(h_{2}-h_{1}) (h_{2}-h_{1}= h, the height of the
object)

F_{B} = **r**_{f}ghA

And Ah = V, so:

F_{B} = **r**_{f}gV

To continue manipulating the equation, we can use the density equation ρ = m/V to determine that:

F_{B} = **r**_{f}gV = m_{f}g, because **r**V = m

In plain words, this equation says that **the buoyant force on an
object in a fluid is equal to the weight of the fluid that object displaces** (we will define this weight as the weight of the fluid that would occupy the volume of the object submerged in
it)**.** This concept is **Archimedes’
Principle,** and was discovered by the famous scientist when he determined the density of a crown by comparing its weight when submerged to its weight when in the air.

These concepts can be applied when dealing with floating objects as well.
Generally, an object will float on a fluid if the fluid has a greater density than the object. An object floats at **equilibrium** when the buoyant force acting on it equals its weight.
It is this principle that allows a submarine or fish to hover at the same depth in water, or a hot air balloon to float in the atmosphere.

**Examples:**

**A)** What is the buoyant force on a block of gold with a volume of .025m^{3} submerged
in a tank of water (density 1.0 x 10^{3} kg/m^{3})?

*Use the buoyant force equation* F_{B} = **r**_{f}gV

*Plug in the values* F_{B} = (1.0 x 10^{3} kg/m^{3})(9.8
m/s^{2})(.025m^{3})

*Solve for the buoyant force* F_{B} = **245 N**

Often it is helpful to see what others have to say on the subject. Here are a few good sites to check out:

http://www.pbs.org/wgbh/nova/lasalle/buoybasics.html

Buoyancy basics in simple terms

http://www.grc.nasa.gov/WWW/K-12/WindTunnel/Activities/buoy_Archimedes.html

Good overall introduction to Archimedes' Principle, with exercises

http://www.infoplease.com/ce6/sci/A0804583.html

Basic definition of the Principle with relevant links

http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html

All-around site on Archimedes, including biography and the story of the golden crown

The NTNU Virtual Physics Laboratory provides several excellent applets that demonstrate principles of Physics. Click Here for an an interactive introduction to buoyant forces, and here for an applet that shows how buoyant forces work with different densities, volumes, and masses.

For Practice Problems, Try: *Giancoli Multiple Choice Practice Questions (It will be a few lessons before all of this is covered)*