# Elastic potential energy

Elastic potential energy derivation

For a string If $$F_1=0$$ then $$x_1=0$$ , $$F_2=F$$ then $$x_2=x$$ Elastic potential energy = work done = $$\frac{F_1+F_2} {2} \cdot x = (0+F)2 \cdot x= \frac{Fx} {2} = -\frac{1} {2} Kx^2$$ In this derivation why should we take average force $$\frac{F_1+F_2} {2}$$ and why we are not taking average displacement of wire $$\frac{x_1 +x_2} {2}$$

• Welcome to Physics SE! You're highly encouraged to use LaTeX in your questions. – Mauro Giliberti Jun 13 '19 at 5:58

You can take the average position as well, the result don't change. Be aware that in those formulas what's important isn't really the displacement $$x$$, but the change in displacement $$\Delta x$$. In this case they are the same only because $$x_1=0$$.
$$\frac{x_1+x_2}{2} \cdot F =\frac{0+x}{2} \cdot F =\frac{Fx}{2}$$