conservation-lawsforcesnewtonian-mechanicspotential energywork
I know that the amount of work does not depend on the followed path but on the position when speaking about conservative forces. But could someone please prove/explain why exactly those two are conservative forces?
Has it something to do with the potential energy?
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:
Mechanical energy (ME) is the ability of a body to do [mechanical work]. A body has ME because of:
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?
Why do some forces' work depends on path and others not.
Nonconservative forces cause energy loss during displacement. For example, friction when an objects moves over a surface converts the stored energy to heat that disappears and is wasted. Therefore the final state depends on how long the path was, because that determines how much energy is lost along the way.
Conservative forces cause no loss in energy. Therefore, energy associated with such forces can only be converted into other stored forms in the object (kinetic energy) or system (potential energy). In fact the work done by a conservative force is what we describe as potential energy. The word "potential" gives the feeling that it is stored; it is merely a name for the work that the conservative force will do when released. And when released, that potential energy will be work done on the object and it turns into kinetic energy, which is still stored in the body. If you are told what the start and end speeds are, you therefore know that the difference in kinetic energy must be stored. Regardless of the path.
We can consider the conservation in terms of energy like here or entropy and maybe others as well. I personally find the energy approach the most intuitive.
When I hold a thing in my hand and make it follow a short and long random path in different cases and come to the some position 'x' , I feel I have done different amount of work in both cases. But gravitation being a conservative force says I have done equal amount of work in both cases. Where am I mistaken?
Gravity might be a conservative force, but the force you exert on the object is not.
Also why do 2 forces exist? Forces are forces they must be same nature.
Which two are you thinking of?
In any case, yes, forces are "the same thing" so to speak. It doesn't matter what "kind" of force or what created the force - forces are forces and they can be added, for example in Newton's laws, where we don't care about the "type" of force.
Lastly, -can I state all unidirectional forces are conservative?
What do you mean by unidirectional force?
If gravity pulls downwards, so a box slides down an incline, there can still be a friction in only one direction on the incline. The directionality is not a measure of if a force is conservative or not.
Instead think about what kind of energy that force causes. Is it potential or kinetic, then the force is conservative. Is it heat or alike, then not.
-Are there other classifications of forces?
There are many "types" of forces: Electric, magnetic, chemical, gravitational, elastic etc. Those are just names that tell us the origin of them. As stated above, the "type" or origin is of no importance; all forces can cause acceleration in the same manner.
Best Answer
A conservative force has the property that the work done in moving a particle between two points is independent of the path taken. It implies that the force is dependent only on the position of the particle. Now we can use this idea to define a function called the potential energy.
It is a conservative force that gives rise to the concept of potential energy and not the other way round. If the force were non conservative, the force would not be dependent on position only and thus we could not have defined a potential energy function.
A simpler way to find out whether a force is conservative or not is to find out the closed line integral of force, i.e $\oint dr F$ and convert it into the area integral of the curl of the force by using Green's theorem, i.e $$ \oint F dr = \int_A (\nabla \times F) da$$. Thus if the curl of the force is zero, it automatically means that the force is zero. It is now trivial to see that the gravitational as well as the spring force are conservative as the curl of both forces vanish.