An interesting application of the above formulas is in
calculating the gravitational potential energy of a body. Suppose
you have a turnip and wish to calculate its potential
energy. Well for every atom i of mass 
in the turnip,
the potential energy, is just 
times the vertical
position of the atom 
. To get the total potential
energy, we sum over all atoms:
By some fancy foot work we see this is
But 
is just the total mass 
, and
the term in parenthesis is just the definition of
the center of mass in the z-direction, 
. So finally
we obtain the simple formula:
In other words, to compute the potential energy of a turnip, we just
have to find its center of mass, and its total mass. If we
replace the turnip with a point of mass located
at the center of mass, it will have the same potential energy.
This gives us a nifty way to experimentally determine the center of mass. Take the right icoscoles triangle we looked at earlier, see 1.2.1.
If we hang it from the right angled corner, the hypotenuse will lie horizontally.
{
Why? We can replace the triangle by the center of mass (the red spot), and we're hanging it from the top (the + symbol). A single mass will always dangle vertically (dashed purple line) from where it's being held.
We could also hang it from another corner and the same thing should happen. The red spot should dangle vertically from the point its being held.
{
If we didn't know where the center of mass was, this method would allow us to determine its position. It's just the intersection of the two purple lines
{
This is a neat way of determining the center of mass of a body.
