Elastic potential

Question

I have a doubt:

elastic potential energy is given by: $U=\frac{k}{2}x^2+K$

but does elastic potential exist? (for example: potential gravitational energy is given by $U=mgz+K$ and gravitational potential is given by $V=-U$)

($K$ is the constant of integration)

Answer 1

Elastic type potential exists for the system in equilibrium. If you have a potential function $U(x)$, you can always write down its Taylor series around the equilibrium point $x_0$ as

$$U(x) = U(x_0) + U'(x_0)(x-x_0) + \frac{1}{2}U''(x_0)(x-x_0)^2 + ... $$

Since the system is in equilibrium at $x_0$, so the net force $F=-U'(x_0)=0$. If we only consider the local region around the equilibrium point and set $x_0=0$, then the result is

$$U(x) = \frac{1}{2}k x^2 + K $$

where $K=U(x_0)$ and $k=U''(x_0)$.

You might argue that it is only approximation, but in reality, we always focus on a small region and wont consider large $x$ because the system is not isolated anymore. Also mathematically this expression becomes true if you take the limit of $x\to 0$.

Spring push objects away when compressed and pull them together when stretched. Therefore, it creates a potential energy function with a minimum and so the elastic potential for Hookes law follows. Note that when you stretch the spring heavily, it starts to distort and the potential is not quadratic anymore as we discussed above.

Answer 2

Yes, I think it makes sense to talk about the potential energy stored in a piece of elastic.

To see this imagine tieing two cannonballs together with a piece of elastic, then start them moving apart. The cannonballs originally have some kinetic energy, and as they are slowed and finally stopped by the elastic the kinetic energy is converted into potential energy. As the elastic pulls the cannonballs back together again the potential energy is converted back into kinetic energy.

I suppose you could argue about the terminology, and claim that the energy stored in the elastic band is chemical energy (the conformation of polymer chains) and it shouldn't really be called potential energy. Still, the "two masses + elastic" has the same force/distance curve as a harmonic oscillator, so I think it's fair to call it potential energy.