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

• Mar 29th 2011, 06:34 AM
Polyxendi
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 ($\displaystyle F = -kx$), but I'm having some trouble applying it.

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

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

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

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