# Elastic force expression

I would like to know how to properly write the elastic force in the classic x and y axes. I know that the global formula is $$\vec{F_{el}}= -k\vec{x}$$, but sometimes the situation is little bit difficult. Here is an example of a situation where I struggle :

Considering here two elastic forces (springs of k coefficient and $$L/2$$ undeformed length) with the origin $$O$$ in the middle (A and B slide), for the "left" elastic force I write it this way : $$\vec{F_{el_A}}= -klsin(\phi)\vec{i}$$ for the "right" one my guess is $$\vec{F_{el_B}}= -klsin(\phi)\vec{i}$$ too. I do not know if I am right but the elastic energy says that $$V_A(\phi)=\frac{1}{2}k(lsin\phi)^2$$ but $$V_B(\phi)=\frac{1}{2}k(-lsin\phi)^2$$ which makes me think I was wrong for the forces.

In general, how to write properly the elastic force and eventually elastic energy ?

• There might be something with the sign of the force analysis. If the object $m$ is stationary (not moving), then the spring forces must be opposite and then they must have opposite signs. Jan 6, 2021 at 20:24
• The object m moves freely $\implies \phi \in [0,2\pi]$ and the extremities A and B slide on the walls so the springs are always horizontal $\implies$ the elastic forces are always horizontal too, I would like to know if the elastic forces are equal in this case Jan 6, 2021 at 20:27
• The forces are in the same direction. For the energy, the direction does not matter.
– nasu
Jan 6, 2021 at 21:40