Class content > Energy: The Quantity of Motion > The conservation of mechanical energy
Prerequisites
When we have a moving object for which mechanical energy can be treated as conserved (resistive forces can be ignored), plotting a graph of the potential energy can be a very useful tool for understanding the motion. In this case, the work energy theorem simply becomes that as the system moves, there is no change in the kinetic plus potential energy. Or we can say that the energy at an initial time is equal to the energy at a final time:
A particularly useful way of writing this is to take the value of the total energy at the initial time  call it E_{0}  and use the "initial energy = final energy" equation to write that the energy at any time is equal to E_{0}.
The potential energy, U, is a function of position. This equation then tells us how the kinetic energy (and therefore the speed) changes as the object's position changes. The potential energy can be any combination of the potential energies we have looked at  or could be any potential energy that arises from a more complex (but conservative) force.
Example 1: Two positive charges
Here's an example. Consider a light, movable positive charge (q) approaching another positive charge (Q), that one heavy and fixed in space. In this case, the total energy will be the KE of the first charge plus the PE of the interaction. We know that the PE is k_{c}qQ/r. The graph of the PE as a function of the separation of the two charges is shown below (in unspecified units).
Suppose that charge q approaches charge Q starting from very far away with a KE = E_{0}. As it gets closer (r gets smaller), the PE grows. When charge q reaches r = 4, the total energy is still E_{0}, but some of it is now PE so the KE is reduced (to KE_{4}). When it reaches r = 3, the total energy is still E0, but more of it is PE so the KE is reduced further, to KE_{3}, as shown. As it goes in, towards the origin, the KE continues to drop as the PE rises, until, when the charge reaches r = 1/2, the energy is now all potential. The KE is 0, so the charge stops. It cannot get to a distance closer than 1/2 since it doesn't have enough energy.
The key in reading a PE graph is to see that the KE is the gap between the total energy and the PE curve. Since KE is ½mv^{2}, it can never be negative.
After the charge stops momentarily, it will now "roll back down the hill", going through the increasing values of KE as the PE decreases. Since KE is a scalar and doesn't care about direction, at any point where we determine the object's KE, it could be going in either direction. The graph only tells us about what the value of the KE is at any position.
Example 2: Mass on a spring
As a second example, let's consider a cart on a horizontal spring. If we choose our coordinate so the the cart is at position x = 0 when the spring is at its unstretched length, the PE of the spring is just ½kx^{2}. The graph looks like the one shown below.
For this case, lets start our cart connected to the spring at its rest length but moving to the right with a KE of E_{0}.
The point on the graph representing this instant is at x = 0. The green arrow shows that KE is the full energy. If the cart is now moving to positive values of x, the spring pulls on it, doing work and slowing it down, at the same time increasing the PE stored in the spring. As we follow the cart to larger and larger values of x, the KE keeps decreasing until, when x = 2.9, the KE is 0  the PE is equal to the total energy  so the cart stops. There is a force on it so it begins to move back and runs "down hill" through decreasing values of PE and increasing values of KE (green arrows). As it runs to the left  to negative values of x, where the spring is compressed  the KE reduces again until it is brought to 0 at x = 2.9 and the cart again comes to a stop. The pattern of exchanging PE and KE keeps repeating as the cart oscillates back and forth. Since we have assumed there are no resistive forces, it will keep oscillating forever (or, rather, for as long as that model continues to be reasonable).
Example 3: A complex PE
Suppose that there were a more complex potential energy, U, such as the one shown in the figure at the right. What would the motion of an object in this PE look like if it has an energy E_{0}? E_{1}?
This situation will become relevant when we look at the binding of atoms into molecules and chemical reactions.


Joe Redish 11/5/11
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