What are Conservative Forces?

This is a topic which was pretty confusing according to me. Most people won’t find it confusing because they are satisfied with the general definition provided for conservative forces.As per LibreTexts, Physics the definition of Conservative forces is as follows:

“The work done by Conservative forces is independent of the path taken i.e. the work done by a Conservative force is same between two points A and B for any path taken.”

Nonconservative forces are dissipative forces such as friction. These forces take energy away from the system during a process and hence can’t be gotten back. Accordingly, the definition is given as:

“The work done by a Nonconservative force depends on the path taken.”

Equivalently, it can be said that the work down by a Conservative force along a closed path is zero.

The last two definitions or points are interesting and most overlooked by students and teachers alike.But first I need to answer a few questions that might arise in the mind of the reader.Q.1: What is the reason behind my confusion for a simple definition?Q.2: What do we mean by heat being taken away from the system by dissipative forces?I’ll try to answer the second question first. I must inform that my answer is based on my current knowledge of the subject and my best understanding. This theory is open to discussion. Let’s look in terms of friction.When an object slides on a rough floor, we say that work is being done against friction. The question arises, where does that energy go. Bang! Comes the answer, it is dissipated as heat. Ignorance is bliss. Most people are content with this answer and they move on.I couldn’t. As far as I can understand, when an object slides on a floor, whole of its surface isn’t in contact. What seems like a plane surface at macroscopic level is actually quite uneven at a microscopic level.The following image from mechdesigner.support explains what happens when two surfaces come in contact.

Thus, even though the force applied may not seem very large but the points in contact experience immense pressure due to their microscopic areas of cross-section and get “cold welded” to each other. This leads to formation of temporary bonds which I believe, release some energy in form of heat and radiation. As one tries to slide the object ahead, energy has to be supplied to break these cold welded points which can be termed as the work done against friction. As these joints are broken, material transfer also takes place between the surfaces.Recently, I came across realisation that the work done against friction does not really depend on the path taken. The mathematical derivation for my statement is given as follows:We can assume an arbitrary path AB with constant coefficient of friction as μ.

(I apologize in advance for the messy diagrams)

An object is made to slide from A to B with uniform speed. Let us consider a small section of horizontal length dx. Magnified image of the above mentioned section is given below.

dx is small such that the path is a straight incline with inclination θ. The free body diagram of the object is also given along with and the forces acting on the object can be seen.

So, F = f + mg sin θ

As the block slides from P to Q, the work done against friction is given by,

dW = μmg(cos θ) x (dx/cosθ)

dW = μmg(dx)

Hence, integrating over the whole path length AB

Thus, we can see that work done against friction depends on the horizontal path length and independent of the vertical distance covered.

This might give rise to the misconception that it work done by friction is independent of the path taken.

So, how is it different different from the work done by conservative forces?

As it has been defined before the work done by conservative forces over a closed path is always zero. However in case of friction the energy spent over a closed path cannot be zero.

Let’s say amount of work done against friction over path AB is U. The amount of work done against friction over path BA will also be U (not -U).

The total work done over path A to B to A is 2U which shows work done against friction may not depend on length of the path but it doesn’t make friction a conservative force.