# What is the form of energy conservation law when non-conservative force is included?

Does the law of energy conservation hold when a non-conservative force is included? I mean if a ball or point like object goes down a circular ramp and friction is present. Then due to friction being non conservative it is path dependent . So can I use energy conservation law to find the velocity of that object for any moment? I have seen many physics problems where friction is included in a straight path and they use a modified form of that law such as $$∆K+∆P=W$$ (here $$W$$ is the work done by friction) But what will be the modified form in case of such a circular path? I thought that we might calculate the work done by varying friction force along the path but it seems to me now that the previously mentioned formula was derived as if object moves in a straight path. So you know my main focus is to get a formula of energy conservation for any path including non-conservative force (circular or any arbitrary path).

Does the law of energy conservation hold when a non-conservative force is included?

Overall, energy conservation holds whether the force doing work is conservative or non-conservative. Energy is never "lost". It is only converted to different forms. Mechanical energy only encompasses the sum of the macroscopic kinetic and potential energies of a system, that is, the energy of motion or position of the system as a whole. It is only conserved if no energy is "lost" due to non-conservative forces such as kinetic friction.

Note however that the energy is not truly lost. It is just converted to a different form of energy. In the case of kinetic friction, the loss of mechanical energy equals the increase in the internal (molecular kinetic) energy (and thus temperature) of the materials of the sliding surfaces.

I mean if a ball or point like object goes down a circular ramp and friction is present.

It depends on whether or not the object slides on a surface with friction.

If the object slides on a surface with friction it will lose mechanical kinetic energy as heat due to kinetic friction. In order for a ball to roll down the ramp without sliding, static friction is needed. But static friction does not dissipate heat, and therefore mechanical energy will be conserved provided no slipping occurs.

I have seen many physics problems where friction is included in a straight path and they use a modified form of that law such as $$∆K+∆P=W$$ (here $$W$$ is the work done by friction) But what will be the modified form in case of such a circular path?

The friction they are referring to is kinetic friction. Kinetic friction is a dissipative force that transforms macroscopic kinetic energy of the system into microscopic (molecular) kinetic energy, eventually dissipated as heat. In the absence of kinetic friction $$\Delta K+\Delta P=0$$ for an isolated system.

So you know my main focus is to get a formula of energy conservation for any path including non-conservative force (circular or any arbitrary path).

The total energy of a system is the sum of its kinetic energy (KE) and potential energy (PE) at both the macroscopic and microscopic (atomic and molecular) levels. Conservation of mechanical energy only encompasses the macroscopic and involves conservative forces.

But if you include the KE and PE of the system at the molecular level, which in thermodynamics is called its "internal energy, $$U$$" then it doesn't matter what types of forces are involved or the paths over which the forces do work. Overall, energy is conserved. When friction is involved, mechanical energy is transformed into internal energy.

The general form of the first law of thermodynamics (conservation of energy) equation for a closed system (no loss or gain of mass) includes both the mechanical energy of the system and its internal energy, and energy transfer due to heat and work.

Hope this helps.