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"What is the total (1/2)·m·v^2
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summed over all the particles?"
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For that purpose,
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it's convenient to take the following simpler rigid body.
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We take a rigid body by taking a mass m1 at a distance R1,
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connected to some central hub with a massless rod,
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and you take another mass,
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and you take a third mass.
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This is m2, at a distance R2,
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there's m3, at a distance R3, and so on.
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You can make a rigid body
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by modeling it in this simple fashion.