Question

A spring does not obey Hooke's law. Rather it follows, $F=kx-b{x}^2+c{x}^3$. If the spring has natural length $l$ and compressed length $l^{\prime }$ then find the work done.

Answer 1

Work done in compression=${}_0^x\int Fdx$

${=}_0^x\int (kx-b{x}^2+c{x}^3)dx$

$=\frac{1}{2}k{x}^2-\frac{1}{3}b{x}^3+\frac{1}{4}c{x}^4$

$=\frac{1}{2}k(l-l^{\prime }{)}^2-\frac{1}{3}b(l-l^{\prime }{)}^3+\frac{1}{4}c(l-l^{\prime }{)}^4$