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Where does the center of gravity of a fine rod 𝐴𝐵 of uniform density lie?
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If we draw a diagram of this rod and divide it up by equally sized segments along its length, if we draw a vertical line through the center of the rod and start to eliminate segments of the rod on either side in pairs, we eventually get to a point where only two segments remain.
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And they’re equally positioned on either side of the center of this rod.
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This means that the point at the very center of the rod through which we’ve drawn our line is at the rod’s midpoint, which is also its center of gravity.
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So halfway from one end of the uniformly dense rod to the other is the rod’s midpoint, which is also its center of gravity.