Calculate the center of mass of a semi-circular metal plate

[mit_hacker]

Homework Statement

(Q) Calculate the center of mass of a semi-circular metal plate of uniform density ρ and thickness t. Let the radius of the plate be a. The mass of the plate is thus 1/2 (ρπat2). In your co-ordinate system, you must consider the x-axis passing through the bottom of the plate and the y-axis to be bisecting the metal plate.

Homework Equations

xcm = 1/M int [xdm] and ycm = 1/M int[ydm]

The Attempt at a Solution

If we take a small change in length dx, the area of the rectangle formed will be adx. The volume therefore will be atdx. The change in mass dm = ρatdx.

Substituting and integrating gives us 2a/π which is definitely wrong since under the is co-ordinate system, x-value should come to 0 as it is a uniform object. Am I right? Please help me!

 

[learningphysics]

The height of the rectangle is not a... it is \sqrt{a^2 - x^2}.