# Difference between force and energy equilibrium

I want to find out how far a spring stretches.

1. If I use the force equilibrium, I get $kx=mg \to x=mg/k$.

2. If I use the conservation of energy, I get $0.5kx^2=mgx \to x=2mg/k$.

How can I choose whether to use force or energy equilibrium?

• Can you give some details on how you find x using the energy ? The two results should be the same. – Cabirto Mar 6 '18 at 13:29
• $x$ is not a height from the ground where you use in potential energy ...The force term is correct... – Nehal Samee Mar 6 '18 at 13:53
• hint ; why not do the experiment..hanging a mass from the lower end of spring....observe carefully the mass till it reaches an equilibrium position.then you can yourself decide to use the energy equation or the equilibrium condition....the later is safe as in the former some kinetic part is lost. – drvrm Mar 6 '18 at 14:51
• You don't need to choose between force and energy equilibrium; they give different results for different situations. Force balancing tells you at what height the net force on the hanging object would be $0$ (or, it would be at rest). Energy balancing tells you where the instantaneous velocity (or kinetic energy) vanishes and completely gets converted into the potential energy of the system. – PhysicsMonster Aug 16 '20 at 5:19

## 1 Answer

When using Newton's law you made $\sum{\vec{F}}=0$, that is $mg-kx=0$ and that is the correct equation when there is no acceleration.
The correct energy conservation of spring would be $$GPE + KE + EPE = 0$$
That is, gravitational potential energy + kinetic energy + elastic potential energy $=0$. If we are talking about a vertical spring, then the only difference is the equilibrium position (everything else is the same as a normal horizontal spring-mass system).

Explanation: http://hirophysics.com/Study/vertical-spring.pdf

Energy conservation, for me, is always the easier way of calculation, whether by Energy Conservation or it's cousin, Work (energy difference between two points).