Formula for Work & Energy
Energy is a paper-and-pencil (mathematical) construct instead of a count of something physical. It has the property of being conserved in any closed system. It can convert into different forms, all represented formulaically, but it cannot be created or destroyed. Work is one manifestation of energy. Its paper-and-pencil representation is the force (in the load's direction of motion) applied to a load times the distance the load is moved by that force.
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Energy Formulas
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Energy can "hide" in many different forms: gravitational (mgh), kinetic (0.5mv^2), relativistic (mc^2), electrical (eV), heat (3/2kT), electromagnetic, elastic (0.5kx^2), and radiant energy (not a complete list). When one of these equations is found to drop in value, others are found to have increased in total by the same amount.
Work
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Of the various conversions that energy goes through, the final form of interest to us is work. Work is a form of energy as well. It is defined mathematically as the product of the distance over which a load is moved times the tangential force applied to make it move. Work = Force --- Distance. Notice the word "tangential." A sled may be pulled 10 meters with a force of 10 Newtons, but if the rope used for pulling is at an angle so that the forward component of the force is only 5 Newtons, then the work done is 50 Joules, not 100 (1J = 1N --- 1m).
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Stasis
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Notice also the word "distance" in the definition. If the sled doesn't move, then no work is done. Chemical energy stored in glucose may convert into another form of energy (heat, e.g.) but not work. The definition of work is therefore made quite carefully, so that the conversion of bodily energy into another form of energy does not get double-counted.
Vector Representation of Work
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When force is represented as a vector, its dot product with the distance vector (pointing in the direction of motion) gives the magnitude of work done. If the path is crooked, the distance vector can be made small and the sum F(dot)"s for each straight segment of path s can be summed. If "s is very small, then the summation is called integration, represented as W = ∫F(dot)ds.
Conservative Forces
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If the path over which a sled is pulled between two points is varied, naturally the work done will be different for each path, being a function of the friction being pulled against and the length of path. If the work done between two points is independent of the path, the force is called "conservative." An example is a gravitational field in which the load does not experience friction. If the path loops back on itself, then whatever work was performed against the field has been returned back to stored gravitational potential energy, and W=0.
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