We can show from Newton's second law
with the work calculated by integrating the force along the path
But as the latest question in previous page has shown, this does not seem to always be correct.
Let us look at that question in more details. In the problem shown below, both block A and block B will speed up (assuming no friction). The system of both blocks will increase their kinetic energy but yet the net force on the system is zero since the two forces F cancels.
If you calculate the work using the net force for this system you would find that in the two blocks systems.
The problem is that the system is not just a single particle but it is instead made of multiple parts. These parts can move independently.
The system (made of both blocks) is "deformable" and energy can be put into the deformation.
As we go beyond the single particle model we will start adding terms to our energy equations.
The form of the equation is always as
"energy transfers" = change of energy of system
From Newton's second law applied to a single particle we get
but for systems with multiple parts, we will need new forms of energy beside kinetic. The new energy equations cannot be easily derived from Newton's second law (it can be done in principle depending on the number of parts, just not easily).
Potential energy is the subject of next week so we will come back to it. In a nutshell, potential energy is the energy related to the relative position of "parts" of a system that can be stored and retrieved.
If in the image above, there was a spring between block A and block B, the energy from the work done on the two blocks could be stored in the spring and released when the spring snaps back at some later time. The system would have "potential" energy that we can retrieve when we want it.
Another form of energy is thermal energy which is the sum of all the random kinetic and potential energies of the atoms in an object.
Kinetic (and rolling) friction are good examples where the naïve use of
It is never correct to say that friction does work since work is a transfer of energy to the system. When you rub your two hands together, both hands get warmer. Friction does not just transfer energy to the system on which it is applied. Both the objects where friction is applied and the agent that create the friction get warmer.
Because friction is a force applied on a surface with many particles, the derivation of work from Newton's second law does not apply to it. Kinetic and rolling friction involve multiple deformation, breakage that are hard to fully account for.
How can we use energy method in the presence of friction? If you define the system to include the object and the surface on which it slides, then the friction is an internal force to the system. It does not do work but it will increase the thermal energy of the system. As shown in your book, it is simple to prove that the amount of thermal energy increase is
replacing
There are many important points to remember here
Static friction never does work or increase thermal energy. It does neither. For a force to do work, the point of contact of the force needs to move and static friction is static so there is no motion where it is applied. A car that propel itself with static friction is using the energy from the gasoline or battery, not from the road.
The choice of system is important. If you choose a system with a dissipative force (like friction) as an external force, you will not be able to (at least easily) apply conservation of energy since it is hard to know how much of the dissipated energy goes in the system versus how much goes out. Just choose a different system! Always put friction forces (kinetic or rolling) inside the system for which you will do conservation of energy.
The formula for