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Math Help - Work Integration

  1. #1
    Member
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    Alberta
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    Work Integration

    Hey folks, I've got a problem here that I can't figure out how to start. Just a pointer in the right direction would be great.

    What is the natural length of a spring (in cm) which needs 6 joules of work to stretch it to a length of 10 cm to 12 cm, and another 10 joules to stretch it from 12 cm to 14 cm
    Havn't yet hit a problem where I need to work backwards to find this. I'm not sure how to start, because even if I do find the spring constant and the force, I can only find the displacement from the natural length, not the natural length itself. What am I missing?

    Thanks!
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  2. #2
    Member Black's Avatar
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    Let \ell be the natural length of the spring (in cm) and k be the spring constant. Then we have the following equations:

    \int_{(10-\ell)\times 10^{-2}}^{(12-\ell)\times 10^{-2}}kxdx=\frac{k}{2\cdot10^4}[(12-\ell)^2-(10-\ell)^2 ]=6

    \int_{(12-\ell)\times 10^{-2}}^{(14-\ell)\times 10^{-2}}kxdx=\frac{k}{2\cdot10^4}[(14-\ell)^2-(12-\ell)^2 ]=10.

    Divide the two equations to get

    \frac{(12-\ell)^2-(10-\ell)^2}{(14-\ell)^2-(12-\ell)^2}=\frac{3}{5}.

    Solve for \ell.
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  3. #3
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    Alberta
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    Was just looking for a push, thank you very much. Makes sense to include the natural length in the limits rather than trying to integrate and solve for it after. Thanks!
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