# Conservative force for an elastic force?

In summary, elastic force is a type of force that is linear to displacement, such as a spring obeying Hooke's Law. This type of force is conservative, meaning that the work done against it does not depend on the path taken, only the start and end point. Any force with a potential is considered conservative, while the reverse is not necessarily true. Nonlinear elasticity theories, like the Neo-Hookean model, also exist and can accurately describe large deformations. However, when considering frictional dissipation of energy, the force may not be conservative.
Hi ,

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

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.

What's an "elastic force"?

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!

Dale
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).

vanhees71

## What is a conservative force?

A conservative force is a type of force that conserves mechanical energy. This means that the work done by the force is independent of the path taken by the object, only the initial and final positions matter.

## What is an elastic force?

An elastic force is a type of force that causes an object to return to its original shape after being deformed. This force is a result of the object's elasticity, which is its ability to stretch or compress and then return to its original state.

## What is the relationship between conservative force and elastic force?

Conservative forces and elastic forces are related in that elastic forces are a type of conservative force. This means that the work done by an elastic force is independent of the path taken by the object, as long as the initial and final positions are the same.

## How is the potential energy of an object related to conservative and elastic forces?

The potential energy of an object is a result of conservative forces, including elastic forces. When an object is subjected to an elastic force, its potential energy increases as it is stretched or compressed. As the object returns to its original shape, the potential energy is converted back into kinetic energy.

## What are some examples of conservative forces for elastic forces?

Some examples of conservative forces for elastic forces include spring forces, elastic band forces, and rubber band forces. These forces all have the ability to store potential energy in an object through stretching or compressing, and then release that energy as the object returns to its original shape.

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