Calculating Centre of Mass of a Uniform Object

[BMcC]

Calculate the centre of mass of the object shown below, assuming uniform density

Object: http://i.imgur.com/zuOUyL7.gif



Assume the origin, O, at the lower left, and the positive x-axis pointing to the right.

1) What is the x coordinate of the centre of mass, in terms of a?

2) If the positive y-axis points up along the page, what is the y coordinate of the centre of mass, in terms of a?



So obviously you can break the object into 5 squares, each with a length and width of a and a centre of mass in the middle of the squares. I've done this and tried starting at the origin by adding the masses and positions up according to this formula given by my professor:


X_cm = x₁m₁ + x₂m₂ . . . divided by the sum of the masses, m₁ + m₂ . . .


I'm not sure how to take that gap into account. Should I find the centre of mass of the full rectangle and then subtract the area of the missing square? I don't quite know how to go about doing that.

Thanks in advance!

 

[SteamKing]

Remember, the gap is not there. What does this fact tell you about how to handle it?

You seem to know what the center of gravity of a square is. Can you make the leap and figure out what the center of gravity of a rectangle is?

These are your two hints. Show some work on this problem.

 

[haruspex]

Should I find the centre of mass of the full rectangle and then subtract the area of the missing square? I don't quite know how to go about doing that.

That's certainly the easiest way. Find the moment of the whole rectangle about O, subtract the moment the missing square would have if it were present, then divide by the mass of the object.