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and the useless part,
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which is trying to pull the rod off the hinges,
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is going to be balanced by a force from the hinges,
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because the rod is a rigid rod,
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it's not going to move.
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But the one perpendicular to the rod can, in fact,
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turn the rod without affecting the rigidity.
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So, we need an extra factor here,
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which is sin(θi),
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where θi is defined as follows.
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If you're at a point Ri,
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and you apply a force F that way,