Archive for the ‘Applications of Integration’ Category
April 13th, 2009 | 
Author: In physics there is a useful quantity to calculate which has the dimensions of energy. Work can be thought of as the change in energy required to move a system to a new state. By system, perhaps we refer to some particles that are connected by springs or forces. The equation for work done with a force F is
Example 1. Find the work done by pulling a mass connected to a spring a length L from an equilibrium length of $latex L_0$
Solution 1. According to Hooke’s law the force is just $latex F=-k(x-L_0)$ and directed towards the equilibrium position. This means our work integral is just a polynomial.
In physics there is a sign convention for the work. If it takes some energy to store a final potential energy then the work is positive. If the final potential energy has decreased then the work is negative. You can get into this more if you go on to study physics properly. By the way, the way I remember the sign is to imagine dragging a block along a surface. The frictional force is in the direction opposite how the block travels. The negative force and the negative sign from the integral cancel giving you a positive quantity. It obviously takes work to drag a block against friction. Negative work is when you get energy “for free” like rolling down a hill. The work integral is a very simple application of integration.
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