Potential Energy:

You have seen that the energy required to compress a spring is given by

When the spring is compressed (or stretched) by an amount x, we say that the spring has the potential to do work - if it is released, it will spring back! Thus, the work required to compress the spring is the same as the elastic potential energy of the spring in its compressed (or stretched) state.

Work is required to lift a body through a distance h:

If the body is released, it will gain speed (kinetic energy) as it falls and can do work on an object placed beneath it. Thus, the body has the potential to do work in its "elevated" state - we call this gravitational potential energy:

There are other kinds of potential energy, such as chemical (the bond energy), but these are not usually a factor in mechanical problems.

Conservative and Nonconservative Forces:

For conservative forces, such as weight, the work done does not depend on the path taken. For example, I might lift a 2.3 kg book 1.5 m straight up. The work done would be (2.3 kg)(9.8)(1.5 m) = 33.8 Joules. Or I might lift it vertically 0.8 m, then move it sideways 3 m, and then move it up the final 0.7 m, for a total vertical lift of 1.5 m. The work would be the same, since it takes no force to move it sideways. None, that is, if I don't include very short acceleration & deceleration times. There's no force to keep it moving at constant speed sideways. And there's really no work for the acceleration & deceleration part, either, because the deceleration force is opposite the displacement during deceleration - that work cancels out the work done during the acceleration phase. You'll do a problem based on this principle in the next module.

One very common nonconservative force is friction. Frictional work certainly depends on the path taken. Friction is a dissipative force; that is, the energy being used is dissipated as heat, most of which is unrecoverable without doing more work.

Conservation of Mechanical Energy:

Serway, like most physicists, separates out nonconservative forces from consideration in the principle of conservation of energy. He is, of course, correct to do so. It makes it a little harder to handle some kinds of problems. You will understand this better by studying the very consistent way in which he presents the material. But there is another way that engineers, especially chemical engineers, approach such problems - that of an energy balance. The advantage to this method is that you never have any negative terms, so you don't have to be careful with signs. Like Serway's presentation, however, the method makes some problems a little harder. Basically, it consists of summing the initial energy terms and equating them to the energy at the end of the process. Work done by nonconservative forces can be included in this method. The examples are the best way to illustrate this.

Example 1: The fox falls!

Example 2: A roller coaster

Example 3: Two Blocks on Two Inclines (an example from an earlier module done using energy methods)

Example 4: Springs and Inclines and Projectiles, Oh, my!

Example 5: Conservative or Liberal (oops, nonconservative)?

Example 6: A simpler incline-spring combination