# find the center of mass

label Calculus
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schedule 1 Day
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May 25th, 2015

According to given question

he plate has a constant density and thickness so the mass is evenly distributed. You only have to find the gravitational center of the figure.

You see instantly that the figure is mirror-symmetric around the x-axis, so the gravitational centre must be on the y-axis.

The y-coordinate of the gravity centre is difficult to calculate (using arccos) over the y-axis. But when we look at the surface of the figure you see 3 distinct areas. make (o,g) the gravitational point. and h the point where y = g = 2cos(h) . Then we integrate over the x-axis
The integral from x = 0 to h must be equal to 1/2 (half the total area) - g.h
1 - 2sin(h) = 1 - g.h or 2sin(h) = 1 + 2cos(h).h
I can't see how to calculate this in an algebraic form. you can approach it by interpolation.
f(h) = 2sin(h) - 1 - 2cos(h).h
gives h = 1.202 g = 0,36

May 25th, 2015

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May 25th, 2015
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May 25th, 2015
Nov 21st, 2017
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