Deriving the formula for energy stored in a spring without using geometry (determining the area under a curve)?

Using Hooke's Law, we know that the force applied is proportional to the extension of the spring. Therefore by plotting a graph of force against extension, through the area under the curve we are able to calculate the elastic potential energy stored in the spring, i.e. the work done.

As there is a proportional relationship between force and extension, the triangular area, $\tfrac{1}{2}bh$, gives the $E_p$ stored as $\tfrac{1}{2}F\Delta L$.

Therefore this is different from normal energy transferred, force×distance. However how would one be able to derive this equation without using the area under the curve?

• It is actually still force times distance but since the force is not constant when you stretch the spring you need to integrate the force over the distance. Mar 3 '16 at 14:45
• Maybe it's just me, but it's kind of weird to see $E_p$ used to mean potential energy. (Not that there's anything wrong with it, really, but I guess it would help the question to explicitly say that's what the symbol means.) Mar 3 '16 at 15:02
• hint- calculate the work done by the force proportional to length l in extension for an element dl and integrate the expression between limits L to L+ delta L-like Jan said above. Mar 3 '16 at 15:22

But quickly, you should take the integral $\int_0^x kx'\,\mathrm{d}x'$. This gives you $U=\frac{1}{2} kx^2$.