Kinetic Energy of The Moving Parts of A Machine
The kinetic energy of a machine is the sum of the kinetic energies of its individual moving parts. A machine with moving parts can, mathematically, be treated as a connected system of bodies, whose kinetic energies are simply summed. The individual kinetic energies are determined from the kinetic energies of the moving parts' translations and rotations about their axes.
The kinetic energy of rotation of the moving parts can be determined by noting that every such system of moving parts can be reduced to a collection of connected bodies rotating about an instantaneous axis, which form either a ring or a portion of an ideal ring, of radius rotating at revolutions per second. This ideal ring is known as the equivalent fly wheel, whose radius is the radius of gyration. The integral of the squares of the radii all the portions of the ring with respect to their mass, also expressible if the ring is modelled as a collection of discrete particles as the sum of the products of those mass and the squares of their radii is the ring's moment of inertia, denoted . The rotational kinetic energy of the whole system of moving parts is, where is the angular velocity of the moving parts about the same axis as the moment of inertia.
The kinetic energy of translation of the moving parts is, where is the total mass and is the magnitude of the velocity. This gives the formula for the total kinetic energy of the moving parts of a machine as .
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