Work Between Colliding Objects
In the previous section we introduced the term thermal energy. We used this phrase as a catch-all to describe the form that energy takes when non-conservative forces internal to the system do work. It was not clear at that time why we had to introduce this element to our model, so let's examine it closer here.
When we first introduced the idea of a system, it was mentioned that while internal forces all come in third law pairs, the work done by these pairs are not generally equal-and-opposite. Applying this fact to a non-conservative force like friction is particularly enlightening. Consider the following physical situation: A book slides off a frictionless horizontal ledge directly onto another book which is at rest on a frictionless horizontal surface. There is a kinetic friction force between the two books, and the rub against each other until they are both moving along together at the same speed (see Figure 3.5.1 below).
Newton's third law assures that both books experience equal friction forces in opposite directions, but the work done by the friction force on book A is greater (in magnitude) than the work done on book B, because the displacement is greater. The friction force opposes the displacement of book A, so negative work is done on that book, while positive work is done on book B. Remember that we are interested in the overall effect of internal forces on the system, which in this case is the two books. We see that in fact in this case there is a net negative work done on the system:
\[ work\;done\;on\;system = -f \Delta x_A + f \Delta x_B = -f \cdot \left(\Delta x_A - \Delta x_B \right) = -f \cdot \left( distance\;top\;book\;slides\;across\;bottom\;book \right) \]
This is an isolated system, which means that the negative work calculated above is not coming from outside. Our conservation models require that the total energy of an isolated system doesn't go up or down, though it can change form. The kinetic energy of this system drops as a result of the books coming in contact (this may not be clear at this point, but we will see this is true in the next chapter, so for now let's just accept it as true), and the energy can't go into potential energy, since the kinetic friction force is non-conservative. We put this energy conversion into the third type of energy in our model – thermal energy – and it is precisely equal to the (negative of the) work computed from the friction force acting through the rubbing distance.
It should also be noted that this analysis is not exclusive to kinetic friction. We would get the same result if the two books compressed a spring, provided we take as the "after" time the moment when the spring is completely compressed. In that case, the magnitude of the work done by the spring force on one book is not equal to the work done by the spring force on the other, but the difference in work (which is again negative) is stored in the potential energy of the spring.
While we will be staying with our macroscopic model of interactions, it helps with our understanding of thermal energy to consider a model of what is happening on smaller scales. We know that a macroscopic amount of matter is comprised of trillions upon trillions of atoms. If this collection of matter is a gas, then the particles don't interact very much with each other, while they do interact (with electromagnetic forces) if the particles make up a liquid or solid. Regardless of the forces acting between them, the particles are able to move, which means that they possess kinetic energy. If they interact electromagnetically, then the presence of this conservative fundamental force means that the particles also possess some potential energy.
So what happens when a system gains thermal energy, like when kinetic friction does net-negative work on the system, as in our example above? This energy is conveyed from the macroscopic to the microscopic modes of kinetic and potential energy. In other words, the energy doesn't mysteriously disappear from the universe, it just disappears from our macroscopic model, and can only truly be taken into account with a microscopic model. But there is something very subtle and fundamental that is going on at the same time.
If the energy that was macroscopically-mechanical (e.g. the kinetic energy of book A before it reaches book B in the example above) simply changes into energy that is microscopically-mechanical (kinetic and potential energy of the atoms in both books), then why refer to thermal energy as being fundamentally different from mechanical energy in our macroscopic model? The reason has to do with the one-way nature of the energy transfer into thermal. The kinetic and potential energies of the atoms are randomly-distributed and randomly-oriented. With the atoms vibrating in random directions (they vibrate because molecular bonds create restoring forces similar to tiny springs), all completely out-of-sync, then to give the kinetic energy back to the macroscopic realm, these trillions of particles would have to somehow coordinate their motions. The probability of the random motions of all the atoms being in the same direction at the same time – which is what needs to happen for the macroscopic object to move faster and get back its kinetic energy – is vanishingly small.
So for our macroscopic model, we separate the kinetic and potential energies that apply to large objects (mechanical energy) from those that apply to their randomly-moving atomic constituents (thermal energy). Both are energy, so both measure the same physical property, but conversions of mechanical energy into thermal energy are a one-way street – it's easy to disorder mechanical energy into thermal energy, but the reverse is too improbable to even consider as a possibility.
What we have been calling "non-conservative forces" are simply forces that can't help but bridge the macroscopic model to the microscopic. It's not a coincidence that when we first discussed friction, we took a brief detour into a microscopic model of irregularities of surfaces. Similarly, our first discussion of air drag included a diversion into a microscopic model of tiny particles bouncing off the affected object. The conservative force of gravity, on the other hand, acts on every particle in a macroscopic object at the same time, introducing no random differences between the fates of individual particles.
As we head back into our macroscopic model, it is useful to revise our notion of what work is. Up to now, we have thought about it as a way of isolating Newton's second law to changes in speed, in terms of the work-energy theorem. But an even more useful way to think about work is as a means by which energy is transferred. This transfer comes in one of three forms:
- Energy is transferred into or out of a system (through an external force)
- Energy is transferred from one mechanical form (KE or PE) into another mechanical form (through an internal conservative force).
- Energy is transferred from a mechanical form into thermal (through an internal non-conservative force).
When it comes to solving problems that involve calculating the energy transferred, in case #1 we generally have no choice but to calculate the work done directly, though a line integral. In case #2, we have the potential energy functions already cataloged, and we don't have to deal with a work calculation at all. But in case #3, there are multiple approaches that might arise. One might be a direct calculation (use the friction force and displacement to find the work done). Another might be simply to find what is "left over" – if the system is closed and the mechanical energy is not conserved, the remaining energy transferred must have gone into thermal energy.