Find the coordinates for the center of mass to the shaded out shape. How does one tackle these problems? I have tried a bunch of stuff... like considering the small halfcircle hole on the left under the x axis as negative area but I did not manage to come to the right answer.
where $X_i,A_i$ indicates the coordinates for the center of mass on the X-Axis and the area respectively.
$X_G = (A_1X_1+A_2X_2-A_3X_3)/(A_1+A_2-A_3)$
$A_1$ - > small halfcircle on the top right above the x axis
$A_2$ - > big halfcircle below the x axis
$A_3$ - > halfcircle on the bottom left below the x axis
Is this right?
Thanks
For the shape below the x-axis, let it be known as $A$ The coordinates of center of mass of this shape be $x$
Now consider a semi-circle that fills the hole. Let this shape be $A_1$ and its center of mass be $x_1$
Now consider the entire semi-circle below the x-axis ($A + A_1$). Let this be $A_2$ Let the center of mass of this be $x_2$
Thus, we have: $$\frac{Ax + A_1x_1}{A + A_1} = x_2$$
Calculating the area of the shapes, and upon rearranging, you will get: $$\frac{4x_2 - x_1}{3} = x$$
That was just a quick calculation, please feel free to correct.
Then find the center of mass of the semi-circle above the x-axis ($A_3,x_3$)
Let $X$ be the center of mass of composite body: $$X = \frac{A_3x_3 + Ax}{A_3 + A}$$