newtonian mechanics - Relation between potential energy and forces

Can you also explain why can't we define potential energy corresponding to a non-conservative internal force? 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?

Answer

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:

• 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:

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.