Center of Mass?

Question

Assume that we have a uniform, square metal plate with side L = 4.30 cm and mass 0.205 kg. The plate is located at (x,y) = (0,0) We remove a square from of the plate with side L/4 with lower left edge located at (x,y) =(0,0). What is the distance of the center of mass of the remaining plate from the origin?

I tried using this:
X (whole)=L/2; X (piece)=L/8
M (whole) is given; M (piece)=M (whole)/16;
Then, I used the formula X=( X whole*M whole-X piece*M piece)/(M*whole-M piece).

I believe that theY coordinate is equal to X because the sides of the
piece are the same.

Then, I took sqrt(X^2 + Y^2) to get the answer, but It is not correct.

Can anyone help? Thanks.

Answer 1

The center of mass is
R=1/M*sum(mi*ri)

The original mass has center of mass equal to

1/M*(M*sqrt(2*L^2/4))
or L*sqrt(2)/2

The new shape subtracts a shape that has
r=L*sqrt(2)/8

and mass equal to

d*L^2/16

where d is the density of the material

The new mass of the system is
d*(L^2-L^2/16)
or
d*15*L^2/16

the center of mass is located at

(16/(d*15*L^2))*(d*L^3*sqrt(2/2)*(7/8)

simplify
7*sqrt(2)*L/15
this is a radial from the origin.
The x and y coordinates are equal
and can be computed as
(sqrt(2)/2)*(7*sqrt(2)*L/15)
7*L/15


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