Answer:
a) Definition of work, definite integration, b) [tex]W = -\frac{k}{2} \cdot x^{2} - \frac{b}{3} \cdot x^{3} - \frac{c}{4}\cdot x^{4}-\frac{d}{5}\cdot x^{5}[/tex], c) [tex]W = -7773.696\,J.[/tex]
Explanation:
a) The bungee-jump cord is modelled by using the integral form of the physical definition of works, as the force varies with the elongation. A definite integral is used because initial and final lengths are supposed to be known.
[tex]W = \int\limits^{x_{f}}_{x_{o}} F(x) \, dx[/tex]
[tex]W = -\int\limits^{x_{f}}_{x_{o}} (k\cdot x + b\cdot x^{2}+c\cdot x^{3}+d \cdot x^{4}) \, dx[/tex]
b) The formula for the work done is:
[tex]W = -k\int\limits^{x}_{0} {x}\, dx - b\int\limits^{x}_{0} {x^{2}} \, dx - c\int\limits^{x}_{0} {x^{3}} \, dx -d\int\limits^{x}_{0} {x^{4}} \, dx[/tex]
[tex]W = -\frac{k}{2} \cdot x^{2} - \frac{b}{3} \cdot x^{3} - \frac{c}{4}\cdot x^{4}-\frac{d}{5}\cdot x^{5}[/tex]
Where [tex]x = x'-4[/tex].
c) The value for x is:
[tex]x = 16\,m - 8\,m[/tex]
[tex]x = 8\,m[/tex]
The work done to stretch the cord is:
[tex]W = -\left(210\,\frac{N}{m}\right)\cdot (8\,m)^{2}+\left(28.667\,\frac{N}{m^{2}} \right)\cdot (8\,m)^{3} -\left(3\,\frac{N}{m^{3}} \right)\cdot (8\,m)^{4}+\left(0.1\,\frac{N}{m^{4}}\right)\cdot (8\,m)^{5}[/tex]
[tex]W = -7773.696\,J.[/tex]
The negative sign is because the reactive nature of the elastic forces, which is antiparallel to displacement direction.
