xlabel(’U/Uˆ{\ast}’), ylabel(’x/b’) subplot(2,1,2); plot(x,t,’-b’,x[i],t[i],’o’); ylabel(’t/tˆ{\ast}’), xlabel(’x/b’)

11.4 The Energy Principle

So far we have addressed processes where the work is done by conservative forces only. Let us now use what we have learned about potential energy to also discuss processes with non-conservative forces. Non-conservative forces are typically forces that not only depend on position, but for example also on velocity, such as air resistance or friction. Let us examine the motion of an object subject to both conservative and non-conservative forces.

We decompose the net force acting on an object into conservative forces, F j , and a non-conservative force, f :

j

If the object moves from r(t0) = r0 at t0 to r(t1) = r1 at t1, the net work is:

j

The work-energy theorem relates the net work to the change in kinetic energy for the object:

We group terms related to position 1 on the left hand side:

If we introduce the term E1 = U1 + K1 for the total (mechanical) energy of the object, we get:

The change in the (mechanical) energy of the object is equal to the work done by the non-conservative force. This law is a completely general law of nature usually called the second law of thermodynamics or the energy principle:

The change in (mechanical) energy is equal to the work done by nonconservative forces (W ).

We will later see how we can relate this energy principle for a single object to a more general principle for a system of several objects interacting with its environment through work and heat (thermal energy transfer).

11.4.1 Example: Lift and Release

Problem: If you lift a book from the floor to a height h using a constant force F , and release it, you have increased the total energy of the book, but the book was affected by a constant force. How can we explain this using the energy principle?

Solution: When you are not holding the book, and the book is falling, it is affected by the gravitational force alone. The total mechanical energy of the book (when falling) is therefore:

Initally, the book was lying on the floor with zero velocity. In this case, the total mechanical energy is:

If we choose the potential energy to be zero when the book lies on the floor, that is, when y = 0 m and v0 = 0 m/s. When the book is at rest (v1 = 0 m/s) at y1 = h, the total mechanical energy is:

The change in mechanical energy is therefore:

where did this energy come from? It came from the work done by the constant force F lifting the book to the height h.

Because F is a non-conservative force, the energy principle gives that:

The work done by F corresponds to the work done by a constant force mg. But, you protest, the force F is a conservative force, since it is a constant. Something must be wrong with the energy principle!

Not so fast. The force F was constant only when you lifted the book. Afterwards, you released the book, and this applied force therefore became zero. The force F is therefore not that constant after all. This is indeed how we should regard such a force: The force F lifting the book is a time-dependent force. It only acted when the book was lifted. It does not act when the book is released afterwards. Then it is only affected by gravity. This is why we call it a non-conservative force. We could look at it in a different way: The force F does not only depend on the position y of the book, because when we lift the book, it is affected by the lifting force, but afterwards, when the book is released, it may be falling through the same position without being affected by the lifting force. Thus the lifting force is not a conservative force, and we must use the energy principle to understand the change in mechanical energy as caused by the work done by non-conservative forces.

11.4.2 Example: Sliding Block

Problem: A block is sliding down an inclined plane forming an angle α with the horizontal. The kinetic coefficient of friction for the contact between the block and the plane is µ. What is the velocity of the block as a function of the length L it has slid?

Identify: We use a coordinate system oriented along the plane, as illustrated in Fig. 11.15. The block slides from position x0, with initial velocity v0 = 0, to x1. Our task is to find the velocity v1 as a function of x1.

Model: Figure 11.15 shows that the block is affected by the normal force N , friction, f , and by gravity, G. Because the block is affected by a non-conservative force, f , we cannot use energy conservation to find the velocity as a function of position, but we can still use the energy principle: The change in energy is equal to the work done by the non-conservative forces.