finding work done

Question

 A bungee cord exerts a nonlinear elastic force of magnitude F(x) = k₁x + k₂x³, where x is the amount of stretch in the cord (its overall length minus its unstretched length), where k₁ = 237 N/m and k₂ = -0.27 N/m³.

Write an expression for the work done on the cord to stretch it an amount x = s.

What is the work done on the cord, in joules, if it is stretched a distance s = 23.8 m?

Answer 1

Hi Jule!

In order to find the work done on the cord, we want to use W = ∫ F·dx, since the force applied varies as a function of the distance stretched.

We can therefore integrate W = ∫ k₁x + k₂x³dx = .5k₁x² + .25k₂x⁴ +C. I will leave it to you to 1) decide on what the correct bounds of the integral would be, so that we get a general expression for distance s, and 2) plug in the particular value for s.