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I have learned that ideal spring has no mass. Suppose, I attach a ideal spring (spring constant $K$) to a wall and pull it a distance $x$ it will have a potential energy $U = \frac{1}{2}Kx^2$ and if I release it then what happens it has no mass so it can't have kinetic energy. What will happen to it, will it be at rest? If yes then how can it be possible that a spring extended a distance $x$ (force $Kx$ towards wall is acting on it) is at rest. If it will come to its original position then what happens to the potential energy? Where does it go?

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There are no ideal springs. Therefore the paradox with the infinite acceleration is not a physical one, but an artefact of the mathematical modelling. Conservation of energy does obviously not hold, when there are objects of mass zero in a system (because the kinetic energy will always be 0). So your setting simply does not fulfil the requirements for energy conservation.

It is not uncommon for systems not to conserve energy. Mechanic problems involving friction or time dependent constraints, for example, do not conserve mechanic energy either. It is all a question of modelling. If you model something completely, then energy will be conserved. But if you let masses go to zero or infinity or simply not include degrees of freedom (as when ignoring the heat generated by friction), energy need not be conserved. The modelling can nevertheless be useful or give meaningful results. In the case of the ideal spring without any connected masses it does not.

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Normally it is said that the spring has no mass, but in problems where the spring is attached to bodies with larger mass compared to the spring mass. So you pull the string from the side where you have a body of mass $m$ (tha other side is attached to the wall). Then when you release this body with potential $\frac{1}{2} K x^2$ it will be converted in kinetic energy (maximal at $x=0$).

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  • $\begingroup$ there is no mass attached to spring in my question so potential energy can't convert into kinetic energy . I am saying consider a massless spring and answer this question.I know this is not practical but just theoretically ,can't you answer this question. $\endgroup$ – Dhruva Mehrotra Nov 19 '15 at 13:51
  • $\begingroup$ @DhruvaMehrotra, that's correct. The theory doesn't work---equations can't be solved---if there's no mass. But that's not a problem because an ideal spring can not actually exist. $\endgroup$ – Solomon Slow Nov 19 '15 at 14:02
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where spring's energy goes

It does not go.

From the conservation of energy, $\frac12Kx_\text{max}^2=\frac12mv_\text{max}^2$, and $m = 0$ you get $v_\text{max}=\infty$.

In conclusion, your spring will oscillate with $\text{Amplitude} = x_\text{max}\;,v_\text{max}=\infty \; \&\; \omega = \infty\; .$

For more information see: https://en.wikipedia.org/wiki/Harmonic_oscillator

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  • $\begingroup$ $v_{max}$ can't be infinite as max velocity that can be achived is c=speed of light $\endgroup$ – Dhruva Mehrotra Nov 19 '15 at 14:34
  • $\begingroup$ I believe I have answered your question in an elegant way using pure Newtonian mechanics. If you ask for impossibilities the answer is an impossibility. $\endgroup$ – PeterS Nov 19 '15 at 14:50
  • $\begingroup$ @PeterS: An apt & smart reply:) $\endgroup$ – user36790 Nov 19 '15 at 14:51

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