What is the Natural Length of a Spring Based on Work Required for Stretching?

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The discussion focuses on determining the natural length of a spring using the work-energy principle. The user sets up two equations based on the work done to stretch the spring: 6 J for stretching from 10 cm to 12 cm and 10 J for stretching from 12 cm to 14 cm. The equations are formulated as 6 J = K(10-L)^2/2 and 10 J = K(12-L)^2/2. The user concludes that eliminating the spring constant K from the equations is essential to derive a single equation for the natural length L, ultimately suggesting a calculated length of 8 cm.

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I'm having problems with the work section of my calc book. This is the problem:

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

I set up two equations:

6 J= Kx^2/2 evaluated at 10-L and 12-L
10 J= Kx^2/2 evaluated at 12-L and 14-L

I'm not sure what to do after the mess hell of algebra, I end up with a K and an L. Do I solve for K in one equation and plug it into the other?
 
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I got a length of 8
 
If that's the easiest way to solve the two equations. Any any case, you want to eliminate K from the equations, resulting in single equation for L.
 

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