Question

For a particular nonlinear spring, the relationship between the magnitude of the applied force F and the resultant displacement x from equilibrium is given by the equation: F = Kx^2. What is the amount of work done by stretching the spring a distance xo?

Answer 1

Answer:

$F = \frac{k\cdot x_{o}^{3}}{3}$

Explanation:

Given that force changes with distance. From Physics we remember that work can be defined by the following integral equation:

$W = \int\limits^{x_{b}}_{x_{a}} {F} \, dx$ (1)

Where:

$F$ - Force.

$W$ - Work.

$x$ - Distance.

If we know that $F = k\cdot x^{2}$, then the work of the nonlinear spring is:

$W = k\cdot \int\limits^{x_{b}}_{x_{a}} {x^{2}} \, dx$

$W = k\cdot \left(\frac{x_{b}^{2}-x_{a}^{2}}{3} \right)$

$W = \frac{k}{3}\cdot (x_{b}^{3}-x_{a}^{3})$ (2)

If we know that $x_{a} = 0$ and $x_{b} = x_{o}$, then the amount of work done by stretching the spring a distance $x_{o}$ is:

$F = \frac{k\cdot x_{o}^{3}}{3}$