Question

The force needed to hold a particular spring compressed an amount x from its normal length is given by f = kx + 2ax3 + bx5. how much work must be done to compress it by an amount x, starting from x = 0? (answer in terms of a, b, k, and x, as necessary.)

Answer 1

Given the expression for the force:

`F(x)=kx +2ax^3+bx^5`
The work done to compress the spring of an amount x is equal to the integral of F(x) in dx:

`W= \int {F(x)} \, dx =\int {(kx+2ax^3+bx^5)} \, dx =k (x^2)/(2)+a (x^4)/(2)+b (x^6)/(6)`