Suppose a box (which I assume to be a rigid body) with an initial velocity that starts to slide on a level surface with friction. Imagine this experiment is done in vacuum, so there is no air drag or dissipation of energy as sound waves. What I've learned is that as the box slides on the surface, the work done by friction decreases the kinetic energy of the box, and part of this energy is converted into the thermal energy of the box itself (it heats up) and part of it exits the box as other forms of energy (the floor heats up).
My textbook suggests the following formula for conservation of energy in a system:
$$W_{external}=\Delta E_{mechanical} + \Delta E_{thermal} + \Delta E_{internal}$$
Where $\Delta E_{internal}$ represents the change in forms of internal energy other than thermal energy (that is accounted for separately). I deliberately took the box as the system and didn't include the floor/ground, etc. There is no potential energy, and we neglect other forms of internal energy than thermal energy. So for the "box" system, the equation above can be rewritten as:
$$W_{external}=\Delta K + \Delta E_{thermal,box} ~~~~~~~~~~(1)$$
Whereas the work-energy theorem states that:
$$W_{net}=\Delta K ~~~~~~~~~~(2)$$
The box is our system, and the only force that is exerted on the box and does work, is friction. Thus $W_{net}=W_{friction}=W_{external}$ (is this correct?). But if we apply this latter result on equations (1) and (2), it gives that $\Delta E_{thermal,box}=0$, meaning that the box has not heated up, which is of course, incorrect.
The only thing I could think of is that the $\Delta K$ in the work-energy theorem also includes the change in thermal energy, as itself is the kinetic energy of the molecules and atoms vibrating (or not?), whereas the $\Delta K$ in equation (1) only accounts for the change in kinetic energy of the center of mass of the box, and the change in thermal energy of the box is written as a separate term, $\Delta E_{thermal,box}$.
This is very confusing. Is there any assumption behind these formulas that I overlooked? What am I missing?
Best Answer
In physics mechanics texts a typical assumption is that we are dealing with rigid bodies which have constant internal energy (no "heating" effects), and work is defined as the change in the kinetic energy (work by a conservative force is equivalently the negative of the change in potential energy).
In thermodynamics, the definition of work is expanded to be "energy that crosses a boundary between a system and its surroundings, without mass transfer, due to an intensive property difference other than temperature". The first law of thermodynamics allows work (and heat) to change internal energy.
The mechanics definition of work is always applicable to the center of mass (CM). Some call this work "pseudo work" to distinguish it from the broader thermodynamics definition of work.
See other questions on this exchange regarding work