Question

You’re an engineer for a company that makes bungee-jump cords and you’re asked to develop a formula for the work involved in stretching cords to double their length. Your cords have force-distance relations described by F = −( kx + bx2 + cx3 + dx4) where k, b,c, and d are constants.
a) How will you model this problem?
b. Given a cord with un-stretched length 4, what is the formula for the work done?
c. Evaluate the work done in doubling the stretch of an 8 m cord with k= 420N/m, b = -86N/m2, c = 12N/m3, and d=-0.50N/m4

Answer 1

Answer:

a) Definition of work, definite integration, b) $W = -\frac{k}{2} \cdot x^{2} - \frac{b}{3} \cdot x^{3} - \frac{c}{4}\cdot x^{4}-\frac{d}{5}\cdot x^{5}$, c) $W = -7773.696\,J.$

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.

$W = \int\limits^{x_{f}}_{x_{o}} F(x) \, dx$

$W = -\int\limits^{x_{f}}_{x_{o}} (k\cdot x + b\cdot x^{2}+c\cdot x^{3}+d \cdot x^{4}) \, dx$

b) The formula for the work done is:

$W = -k\int\limits^{x}_{0} {x}\, dx - b\int\limits^{x}_{0} {x^{2}} \, dx - c\int\limits^{x}_{0} {x^{3}} \, dx -d\int\limits^{x}_{0} {x^{4}} \, dx$

$W = -\frac{k}{2} \cdot x^{2} - \frac{b}{3} \cdot x^{3} - \frac{c}{4}\cdot x^{4}-\frac{d}{5}\cdot x^{5}$

Where $x = x'-4$.

c) The value for x is:

$x = 16\,m - 8\,m$

$x = 8\,m$

The work done to stretch the cord is:

$W = -\left(210\,\frac{N}{m}\right)\cdot (8\,m)^{2}+\left(28.667\,\frac{N}{m^{2}} \right)\cdot (8\,m)^{3} -\left(3\,\frac{N}{m^{3}} \right)\cdot (8\,m)^{4}+\left(0.1\,\frac{N}{m^{4}}\right)\cdot (8\,m)^{5}$

$W = -7773.696\,J.$

The negative sign is because the reactive nature of the elastic forces, which is antiparallel to displacement direction.