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# principle of moments

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The principle that, when a system of coplanar forces acts on a body and produces a state of equilibrium, the sum of the moments of the forces about any point in the plane is zero.

Suppose, for example, that a light rod is supported by a pivot and that a particle of mass m1 is suspended from the rod at a distance d1 to the right of the pivot, and another of mass m2 is suspended from the rod at a distance d2 to the left of the pivot. The force due to gravity on the first particle has magnitude m1g, and its moment about the pivot is equal to (m1g)d1 clockwise. Corresponding to the second particle, there is a moment (m2g)d2 anticlockwise. By the principle of moments, there is equilibrium if (m1g)d1=(m2g)d2.

To use the vector definition of moment in this example, take the pivot as the origin O, the x-axis along the rod with the positive direction to the right and the y-axis vertical with the positive direction upwards. The force due to gravity on the first particle is −m1gj, and the moment of the force about O equals (d1i)×(−m1gj)=−(m1g)d1k. Corresponding to the second particle, the moment equals (−d2i)×(−m2gj)=(m2g)d2k. The principle of moments gives the same condition for equilibrium as before.

Subjects: Mathematics.

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