Finding Center of mass

In order to find the center of mass of certain objects, we use:

where M is the mass, r is a vector and dm is the what we are integrating over, in this case the mass.  Because M is the mass, we have to integrate dm in order to find out what our mass is. This is shown in the equation:

Next we need to pick a shape.  A simple example would be to pick a rectangle.  This is shown in the following figure:

The Center of mass of a rectangle can be easily determined using logic.  We know that it is the exact middle because of symmetry.  

However, we can determine this using the equations above. We can put our coordinate system anywhere around the rectangle.  In this case, we will put it in the back left corner at the bottom of the rectangle.  

First, we need to find out what dm is.  In this case, dm is represented by:

In our case.  The density is rho and the volume is l * w * h.  After plugging this back into the integral, we figure out that the C.O.M is L/2, w/2, and h/2.  This is what we predicted above, but now you know the reasoning!

We can also do this with more complicated figures.. A cone for example.  

As you can see, this shape is much harder to determine what the center of mass is.  If we use the integral above, we can figure what it is quite easily.  If we plugged in the values of a cone in the above integral, we would find out that the C.O.M = h/4, where h is the height of the cone.