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Calculus/Hooke's Law Problem

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Expert: - 2/6/2008


Question
If 6J of work is needed to stretch a spring from 10cm to 12cm and another 10J is needed to stretch it from 12cm to 14 cm, what is the natural length of the spring?



With this problem, I know I need to find the constant, but I'm just stumped at exactly how to do it. What I've already done is found the force used over the 2 intervals, which is 3N for the first and 6N for the second, but I can't seem to
figure out how to use it to get the constant. If you could even just show me how to do it, and not even solve it, it would be greatly appreciated.

Answer
Try
 the integral from 10-x to 12-x of kx = 6  and
 the integral from 12-x to 14-x of kx = 10,
where x is the springs stationary length.

After integrating, multiply out all of the squares and notice that the x^2 disappears.  Simple subtraction of the equations can solve for k and back substitution of k into the equations can solve for x.  I get two simple integers as the solution.

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