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Math Help - Not sure about Hooke's Law...?

  1. #1
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    Not sure about Hooke's Law...?

    This is part of my Calc2 homework that the professor hasn't discussed.

    "If a spring needs 10lbs of work in order to keep it stretched 4 inches from it's natural length, how much work is need it to stretch it to 6 inches from it's natural length?"

    According to the book, the best method is to use Hooke's Law ( F = -kx), but I'm having some trouble applying it.

    I'm going to assume that the natural length is x and stretching it 4 inches would give me x + 4. The force needed to keep it here is denoted by x + 4 = 10, having 10=Force.

    So, would I have: 10=-k(x+4), so that k  =-\frac{10}{x+4} ?

    If that's correct, then all that's left is the integral: 10\int_{4}^{6}\frac{1}{x+4}dx

    Thank you for your help!
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  2. #2
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    "If a spring needs 10lbs of work in order to keep it stretched 4 inches from it's natural length, how much work is need it to stretch it to 6 inches from it's natural length?"
    I think it should say, "10lbs of force."

    In Hooke's law, you don't count the natural length, so 4 inches is the displacement of the spring's end from its equilibrium position. Thus, 10 = 4k, from where k can be found.

    The work needed to stretch the spring to 6 inches from it's natural length is (up to the sign) \int_0^6 F(x)\,dx=\int_0^6 kx\,dx.
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