In physics, work is a measure of the amount of energy transferred by a force; for example, if a horse sets a wagon in motion, the horse’s force on the wagon is putting some energy of motion into the wagon. When a force acts on an object that moves in the direction of the force by an infinitesimal distance , the infinitesimal work done is . Integrating both sides, we have , where the force may depend on , and and represent the initial and final positions of the object.
Example
A spring compressed by an amount relative to its relaxed length provides a force . Find the amount of work that must be done in order to compress the spring from to . (This is the amount of energy stored in the spring, and that energy will later be released into the toy bullet.)
The reason grows like , not just like , is that as the spring is com- pressed more, more and more effort is required in order to compress it.
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