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This is called the actual energy of the motion of the body, and is half the quantity which in some treatises is called vis viva.
The energy stored or restored, as the case may be, by the deviations of velocity of a body or a system of bodies, is the amount by which the actual energy is increased or diminished.
§ 119. Principle of the Conservation of Energy in Machines.—The following principle, expressing the general law of the action of machines with a velocity uniform or varying, includes the law of the equality of energy and work stated in § 89 for machines of uniform speed.
In any given interval during the working of a machine, the energy exerted added to the energy restored is equal to the energy stored added to the work performed.
§ 120. Actual Energy of Circular Translation—Moment of Inertia.—Let a small body of the weight w undergo translation in a circular path of the radius ρ, with the angular velocity of deflexion α, so that the common linear velocity of all its particles is v = αρ. Then the actual energy of that body is
wv2 / 2g = wa2ρ2 / 2g.
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By comparing this with the expression for the centrifugal force (wa2ρ/g), it appears that the actual energy of a revolving body is equal to the potential energy Fρ/2 due to the action of the deflecting force along one-half of the radius of curvature of the path of the body.
The product wρ2/g, by which the half-square of the angular velocity is multiplied, is called the moment of inertia of the revolving body.