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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

Nash
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  • Not sure how far you have proceeded. Have you divided the shaded figure into 3 more manageable figures? If you have, labeling them on your drawing would be helpful. – Gummy bears Feb 14 '15 at 12:33
  • Wait, never mind. I think I have understood your divisions. Now let's see. Where exactly is the problem? – Gummy bears Feb 14 '15 at 12:35
  • I read that I am supposed to consider the third small circle on the bottom left (under the X-axis) as negative mass. But when I do that I get zero for the coordinates of center of gravity for the x-axis (for the composite body) – Nash Feb 14 '15 at 12:38
  • Okay, then wait a second. Let me try and solve the question and see where you may have gone wrong. – Gummy bears Feb 14 '15 at 12:40
  • Moreover, upon closer inspection, the formula you have written is wrong. It would be better if you calculate the center of mass of the shape below the x-axis first, rather than attempting to create a single formula. – Gummy bears Feb 14 '15 at 12:44
  • I still get wrong – Nash Feb 14 '15 at 12:48
  • Check my post below. Remember that the area of the shape below is the area of the whole semi-circle minus the area of the small semicircle. – Gummy bears Feb 14 '15 at 12:52

1 Answers1

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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: $$X = \frac{A_3x_3 + Ax}{A_3 + A}$$

Gummy bears
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