# Generalized version of work-energy theorem [closed]

I know that for rigid bodies only the work-energy theorem states that the net work done on the body equals the change in kinetic energy of the body since a rigid body has no internal degrees of freedom and hence no other forms of energy such as potential energy. Is there a most generalized form of work energy theorem that is valid for rigid as well as non rigid bodies and for conservative as well as non-conservative force? I would like a work-energy equation that would be valid for point particles, rigid bodies and non-rigid bodies.

• What do you mean? The work-energy theorem for $N$ particles follows directly from Newton's 2nd law. The net work is always equal to the change in kinetic energy. Oct 13, 2016 at 11:03
• I know but that is valid only for rigid bodies. I am talking about a generalized form of it which is valid for all types of bodies and forces.
– MrAP
Oct 13, 2016 at 11:07
• It is also valid for non-rigid, non-isolated bodies and non-conservative forces. Oct 13, 2016 at 11:15
• How?There is no account of change in potential energy and other forms of energy in the equation.
– MrAP
Oct 13, 2016 at 11:33
• $W = \Delta K + \Delta U -Q$? As far as I know, the conventional work-kinetic energy theorem is valid only for isolated point particles. I'm really not sure what you are asking. Oct 13, 2016 at 11:53

Good question! In my opinion as a physics teacher, the work-kinetic energy theorem is one of the worst things perpetrated on beginning physics students, and its ill-effects stay with us and cloud things up for a long time

Work is nothing but energy transfer. A change in kinetic energy for an object is a change in one of its properties. So the Work-Kinetic Energy Theorem:

$$W = \mathbf F\cdot\mathbf d = \Delta KE$$

says that a transfer of energy, resulting from a force exerted through a distance, results in a change in this particular property of an object. Elevating this fact to a special theorem tends to obscure in our minds the useful fact that it is true for all changes in properties that (and here's your generalization, which I call the Law of Change):

Any change in any system's properties is always associated with a transfer of energy into or out of it, which is always associated with force exerted through distance, which, in turn, is always associated with a change in motion or position.

• I've never understood the emphasis placed on the theorem. It is of such limited applicability. Watch out for your "always"; and be more clear what forces you are talking about. Are you ignoring heat? radiation? internal processes? All of these can change a system's properties while doing no work using the conventional meaning of work. Oct 13, 2016 at 12:33
• "Always" is what makes it a law. And it does apply to heating, radiating and internal processes. If we "zoom in" we find that all property changes occur through a mechanism, and that mechanism always involves energy transfer between systems in association with force exerted through distance. Good subject for a discussion in another place, but probably not here. Oct 13, 2016 at 14:57
• But the equation W=F.d=deltaK.E. does not account for change in other forms of energy.
– MrAP
Oct 14, 2016 at 6:36
• @MrAp, it appears to me that if you turned that into a question, it would be answered above. If I'm wrong, I'd encourage you to ask another question. Oct 14, 2016 at 11:37
• I want an equation that would include changes in all forms of energy.
– MrAP
Oct 14, 2016 at 21:33

The "generalized work energy theorem" is extremely simple:

The work done on or performed by the system equals the difference in energy of the system. The energy of the system is the sum of all different kinds of energies of your system: kinetic, potential, chemical, etc.

How does the work split across different types of energies? I can't tell you, that really depends on the body in question.

Ultimately, you only have two "energies": potential energy of conservative forces and kinetic energy. All other energies, such as heat or chemical energy or the like constitute an effective description of something that ultimately can be described by kinetic and potential energy only. Stored heat energy is undirected kinetic energy of internal degrees of freedom (how to model that is one of the questions of statistical mechanics and condensed matter theory). Chemical energy is mostly potential energy of atomic bonds and atoms. Reversible deformation results in potential energy from potential energy in distorted atomic bonds. There is no simple equation to accurately model all of it for every system - otherwise we wouldn't use effective descriptions and physicists would be out of work soon.

• Our generalizations agree where they overlap, but mine also points to inevitable changes in the properties of the system when its energy changes, and also allows for system property changes when the energy of the system doesn't change, such as when all of a stretched spring's elastic energy has transferred to thermal energy after it's released and allowed to vibrate to rest. Oct 14, 2016 at 11:17