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Work

10 November, 2015 - 14:44

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 F acts on an object that moves in the direction of the force by an infinitesimal distance dx, the infinitesimal work done is dW=F dx. Integrating both sides, we have W=\int_{a}^{b}F dx, where the force may depend on x, and a and b represent the initial and final positions of the object.

Example

A spring compressed by an amount x relative to its relaxed length provides a force F=kx . Find the amount of work that must be done in order to compress the spring from x=0 to x=a. (This is the amount of energy stored in the spring, and that energy will later be released into the toy bullet.)

\begin{align*} W &=\int_{0}^{a}Fdx \\ &=\int_{0}^{a}kxdx \\ &=\frac{1}{2}kx^2\mid ^a_0 \\ &= \frac{1}{2}ka^2 \end{align*}

The reason W grows like a^2 , not just like a, is that as the spring is com- pressed more, more and more effort is required in order to compress it.