physics help?

Question

A semicircle of uniform sheet metal has radius R . Find the center

of mass

do we take the sheet as a semi circle and ignore thickness in our calculation?

Answer 1

Yes that's the point, we don't care about thickness (besides, we know COM is going to be in the center between two flat surfaces)

To find COM, we want to divide the circle into many strips parallel to the flat edge and add up the area times the distance from the edge for each strip.

Best to orient the flat edge along x-axis with center at (0,0) and integrate with respect to y .

If the curved part faces the positive y-axis, then a strip at y has length 2sqrt(R^2 - y^2)

int[0 to R] 2y sqrt(R^2 - y^2) dy = 2R/3

Dividing with pi R^2 / 2 gives x_COM = 4R / (3pi)

y_COM = 0

Answer 2

along the +y axis and far R*0.4040 from the center (0 ; 0)