# I Conservative force for an elastic force?

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1. Dec 28, 2016

Hi ,

I wanted to know how elastic force could be a conservative force ???

2. Dec 28, 2016

### FactChecker

When something is moved against an elastic force, the work done does not depend on the path taken -- only the start and end point. That makes an elastic force a conservative force.

3. Dec 28, 2016

### vanhees71

What's an "elastic force"?

4. Dec 28, 2016

### phinds

5. Dec 28, 2016

### vanhees71

I see, so it's forces that a linear to some displacement, e.g., a spring in the linear realm, where Hook's Law is valid, i.e., for the elongation in $x$ direction, $\vec{F}=-k x \vec{e}_x$. Then it's of course conservative since obviously a potential exists, namely
$$V(x)=\frac{k}{2} x^2 \; \Rightarrow \; \vec{F}=-\vec{\nabla} V.$$
Any force that has a potential is conservative, i.e., the energy-conservation law holds true.

Note: The other direction of this statement is not true. E.g., the magnetic force on a charge hasn't any potential (it's even velocity dependent) but still the energy-conservation law holds true!

6. Dec 28, 2016

### hilbert2

There are also nonlinear elasticity theories that hold reasonably well even for large deformations (outside linear realm). One of these is the Neo-Hookean model. The potential energy of an elastic object is some function of the displacements of its volume elements from their equilibrium positions, and is conservative unless you take in account the frictional dissipation of energy (viscoelasticity).