Center of mass x-coordinate of a metal plate

[cubejunkies]

A uniform flat plate of metal with dimensions 10x 12 is situated in a reference plane with its bottom left hand corner at (-2, -6) and its upper right hand corner at (8, 6). The plate has a circular hole cut out of it centered about (4,0) with a radius of 2. Find the x coordinate of the center of mass of the plate.

First off, I know that center of mass between multiple particles is given by the summation of the mass times the relative location of each respective mass all divided by the sum of the masses of each of the particles, however, I have no idea how to find the center of mass of a single system, notably one that has a hole in it or is inconsistent in its mass in one dimension in some way.

I don't know how or if you can use/adapt the equation to determine center of mass of a system of particles to a single system, but if you can and this is how to find the solution, any help to let me understand how to get there would be tremendously helpful.

Thank you!
Anthony

 

[vela]

Hint: I'll assume the plate is of uniform density. If you have that plate plus the circle, you'd have a solid plate, for which you can easily state the location of its center of mass.

 

[cubejunkies]

Once I do that, what do I do about the circle though? I still am completely lost and don't know what to do :/

 

[vela]

Express the center of mass of the solid plate in terms of the centers of mass of the original plate and the circle.

 

[cubejunkies]

Thank you so much! I got it right!

I tried the problem two other ways before the attempt shown below getting it wrong, but when I finally did:

((10*12*3)-(4*4*pi))/((10*12)-4*pi) which yielded 2.88303, I was right!

Thank you so much! :)