Does potential energy only happen when the work done is by a conservative force? Or does work done by non-conservative forces also create potential energy?
Newtonian Mechanics – Relation Between Potential Energy and Conservative Force
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Related Solutions
Relation between Forces and Potential Energy
Can you explain why can't we define potential energy corresponding to a non-conservative internal force?
In order to examine the relation between two terms we must consider the definitions of each term:
a) forces:
- 1) internal forces are those that act inside a body (note that in engineering also a structure is considered a body), they interact between the parts of a body and keep it together. If a body is elastic it can have an internal force when it is compressed or stretched beyond the line of natural equilibrium.
- 2) contact (or applied) forces are those that act from outside and are in contact with a part of the body. A push or a pull do positive work, while friction and drag do negative work on a body
- 3) non-contact forces (gravity, electric and magnetic) can accelerate a body without any contact. We can consider these forces as elastic if we connect, for example, B and the ground with an ideal spring that stretches out when we separate them as in the bottom sketch:
b) Potential Energy
Mechanical energy (ME) is the ability of a body to do [mechanical work]. A body has ME because of:
- 1) motion: kinetic energy is defined as the ability of a body to do work. If a massive body A impacts on another body B it will give some KE to B and do work. If KE is lost by a body B because of a conservative force (2,3) it is conserved PE
- 2) position: if a body is distant from the source of non-contact force has PE and it will acquire the KE lost
- 3) condition, (compressed/stretched): if a body is elastic it can have PE, it will tend to reach the position of natural equilibrium and do work on another body
Potential energy is associated only with elastic, conservative forces, that act on a body in a way that depends only on the body's position in space. These forces can be represented by a vector at every point in space forming what is known as a vector field or force field
An elastic force is conservative because it conserves the KE it subtracts to a body as potential energy. In the bottom sketch when body B is shot up in the air, it has PE = 0 and KE (mgh) = 10 (mg) * h, when it reaches h/2 has KE = 5 * h and PE = 5 *h, and at height h has KE = 0 and PE = 10 * h: ME is costant = mgh.
mechanical energy is the sum of potential energy and kinetic energy ($ME = KE +PE$. It is the energy associated with the motion and position of an object. The principle of conservation of mechanical energy states that in an isolated system that is only subject to conservative forces the mechanical energy is constant. If an object is moved in the opposite direction of a conservative net force, the potential energy will increase.
This you have learned in another answer. You ask now:
Non-conservative forces are those which don't depend on the initial and final states but on the path taken. If such a force act as in a system as internal force why can't we define potential energy?
- Probably you realize by now that PE cannot be associated to a non-conservative force, it would be a contradiction in terms, since PE is the conserved energy
- Besides that, no internal force is known apart from the spring force. If other non conservative force exist or existed inside a body we could never define a PE associated to them. The only forces associated with PE are the non-contact forces and the internal spring force. If you are interested you can find here details on how PE is stored in a spring.
Your question seems to arise from a problem in which there is both a conservative and a non-conservative force. When you say "PE" you must be referring to the PE of the conservative force (by definition there is no PE of a non-conservative force).
The work done by the conservative force does not depend on the path. Therefore you can define the potential as
$$\phi(x_0) - \phi(x) \equiv W_{x_0\to x}$$
Notice that:
- The potential is defined up to a global offset: you can arbitrarily choose the value $\phi(x_0)$ but afterwards any value of $\phi(x)$ is defined.
- This is a well posed definition just because $W_{x_0\to x}$ is a well defined quantity (depends only on $x_0$ and $x$, by definition of conservative force). This is not the case for the non conservative force.
Best Answer
Forces can be conservative or non-conservative. But conservative forces do work where this work is equal to the change in potential energy. Conservative forces are also characterized by the fact that the work done by the force that moves an object from one point to another is independent of the path taken between these points (and the total work done will be zero when the path forms a closed loop).
However, a non-conservative force is one where the work done will indeed depend on the path. A good example of a non-conservative force is friction. The work done against a frictional force will depend on the length of the path between the two points, and due to this path dependence, there will be no potential energy we can associate with this force, and indeed the same is true for all non-conservative forces.
Non-conservative forces will either add or remove mechanical energy from a system. Friction, energy dissipation in the form of heat, removes energy from a system which cannot be fully converted back to work.