# Work done by conservative and non-conservative forces

Work done by conservative forces changes one form of mechanical energy into another. Is it correct to assume that work done by non-conservative forces changes one form of energy to another, for e.g., from internal energy to a form of mechanical energy or vice versa?

Also, consider the situation where one moves a object up (case 1) and down (case 2) from a table to the floor. Now, what are the roles of the works done by the two forces (conservative (gravity) and non-conservative (us)) in each case?

As far as I understand, conservative forces cannot change the net energy of the system and only the conservative forces can bring change in potential energy of the system, whereas change in kinetic energy can be caused by both conservative and non-conservative forces.

In case 1, where we move the object up, the system (object and earth) gains potential energy and this energy was supplied from us. Now, the work done by us causes the transfer of chemical energy from us to the kinetic energy and the work done by the gravitational force converted this kinetic energy into potential energy such that $\Delta K=0$. Also, the change in chemical energy in us would be greater than the energy transferred to the system since some chemical energy gets converted into heat inside us and this cannot be regained. And in this case, there is no change in internal energy of the system (or is it possible for the work done by us to transfer some of our chemical energy to the internal energy of the system?).

In case 2, where we move the object down, the potential energy decreases and we don't gain that energy. The work done by the gravitational force converts the potential energy into the kinetic energy and the work done by us converts this kinetic energy (and some of our chemical energy) into internal energy of the system (and us) such that $\Delta K=0$. Here, the increase in the internal energy of the system is equal to the decrease in its potential energy (or greater than it, if the work done by us also transfers some of our chemical energy to the internal energy of the system).

Is this right? Nobody explains it in this way. Correct me if I'm wrong.

• It seems like you are asking more about accounting for all of the different types of energy in the system rather than the roles of conservative and non-conservative forces. When we say the change in kinetic energy is equal to the net work done, we are just talking about the work done on the object. In other words, there is a difference between looking just at the work the non-conservative force does on the object and where the energy comes from on a biological level to supply this energy. Of course all energy can be accounted for, but how we do this it seems like is the focus of your question. May 27 '18 at 14:20
• Are these correct? case 1: "the work done by us causes the transfer of chemical energy from us to the kinetic energy and the work done by the gravitational force converts this kinetic energy into potential energy such that K=0" case 2: "The work done by the gravitational force converts the potential energy of the system into its kinetic energy and the work done by us converts this kinetic energy (and some of our chemical energy) into internal energy of the system (and us) such that ΔK=0" May 28 '18 at 3:08
• Can the work done by us also transfer some of our chemical energy to the internal energy (heat) of the system? @Aaron Stevens Is my understanding about the role of the works done by the forces correct? Is my understanding about the changes in the energies correct? May 28 '18 at 3:14
• OP wrote (v5): Only the conservative forces can bring change in potential energy of the system. This is incorrect. Dec 30 '18 at 9:51
• @Qmechanic Can you provide an example? Mar 3 '19 at 21:15

Change in Potential energy of a system is defined as the negative of work done by the internal conservative forces of the system $$dU_{system}=-dW_{int,con}$$

Work-energy theorem states that $$dW_{total}=dK_{system}$$

There may be internal and external forces present in the system,then $$dW_{total}=dW_{int,con}+dW_{int,non-cons}+dW_{external}$$

Simplifying this equation further $$dW_{total}=-dU_{sys}+dW_{int,non-cons}+dW_{external}$$

It immediately follows from the above equation that

$$dW_{int,non-cons}+dW_{external}=dU_{system}+dK_{system}$$

We know that the $$RHS$$ of the above equation is nothing but $$dE_{mechanical}$$,then $$dE_{mechanical}=dW_{int,non-cons}+dW_{external}$$

It is now clear that only the the work done by the internal non conservative and external forces can change the mechanical energy of a system.

As the common phrase goes: "Energy cannot be created or destroyed." So all you can really have is just changing forms of energy. Although this is just limited by the different definitions we can come up with. At the end of the day energy is just the capacity for something to do work, which is the capacity for it to apply some force over some distance. So the "conversion" of energy I would say is more for convenience in our understanding with our different energy classifications rather than having different things at the fundamental level.

As for the box scenario, let's make it simple and say we just have 2 forces, gravity (conservative) and my upwards force (non conservative). Or maybe we drop the box in the presence of large air resistance. The work done by the both forces are found by the same means, we just add up (integrate) all of the $$\mathbf F\cdot \text d\mathbf x$$'s along the path of travel. We only define these forces differently because it's useful to use potential energy. The work done by the non-conservative force just depends on the nature of the force, and it will most likely dissipate energy into, as you said, "internal energy". This can be seen if we lower the box at a constant speed. If this is true, then the net work done is 0, but we are still losing potential energy. So if we want to look at "where the energy went", then yes it is in "internal energy" transferred as heat (or you can get into the biology of muscles doing work). But at the end of the day work done by all forces can be found in the same way, it's just that sometimes we can define things to make life easier on us.