# Elastic Potential Energy of spring

Suppose I stretch an ideal spring by distance $$x$$ then we know that the potential energy stored in the spring during elongation is $$kx^2/2$$. Now if I leave the spring it returns back to its natural length instantly and its Potential energy becomes 0. But if no body is attached to the spring where was this Potential energy transferred?

The idealization of zero mass often presents a contradiction when one looks at the system energy, as the kinetic energy is always identically zero, rendering many work–energy analyses invalid and ineffective.

The idealized spring has no mass and extends and retracts instantly, among other unrealistic qualities. The real spring has mass, extends and retracts over finite time, and snaps back and bounces with internal friction when suddenly released, ultimately dissipating stored strain energy as heat.

But if no body is attached to the spring where was this Potential energy transferred ?

A released spring oscillates, with the energy converting back and forth between potential and kinetic energy. Since this is an ideal spring the oscillation frequency is infinite and it is undamped so it oscillates at that infinite frequency forever.

it returns back to its natural length instantly and its Potential energy becomes 0

Indeed, and it then compresses instantly, returns to its natural length instantly, extends instantly, and so on. It is not a realistic motion because ideal springs are not realistic objects.

If there were an object of mass $$m$$ attached to the end of the ideal spring then the object would execute SHM with amplitude $$x$$ and period $$T = 2 \pi \sqrt {\frac m k}$$. During SHM energy is transferred from potential energy to kinetic energy and back again, so that after time $$T/4$$ the potential energy in the spring is zero and the kinetic energy of the object is at a maximum.

The motion when no object is attached to the spring can be found by letting $$m$$ tend to zero. You can see that the period of the SHM would also tend to zero. In the (unrealistic) limit of an ideal spring with zero mass attached to it, we have SHM with amplitude $$x$$ but period $$0$$ i.e. SHM with "infinite" frequency.

After considering the prior answers, I don’t even think it would retract, since that would violated conservation of energy. If it was released at $$+x_0$$, then at any value of with $$|x| there would be missing potential energy, with no way to have kinetic energy.

So, it would just oscillate between $$\pm x_0$$ discretely at infinite frequency.

Of course that doesn’t make sense even by the low standards of Newtonian physics, so ideal springs are just abstractions.

It raise the general question of where our abstractions breakdown. For instance: elastic collisions between macroscopic classical objects… which are simply not allowed by special relativity. However, quantum particles are fine with elastic scattering.