## Illustration 7.3: Potential Energy Diagrams

Please wait for the animation to completely load.

A large 2-N/m spring is shown attached to a 1-kg red ball that is initially displaced 5 m **(position is given in meters, time is given in seconds, and energy on the bar graph is given in joules)**. The total energy and the potential energy are shown in the graph. Two bar graphs that depict the kinetic and potential energy are also shown. Finally, the values of the energy are shown in the table. Restart.

The potential energy diagram is an important diagram because it depicts the potential energy function, often just called the potential. This terminology is unfortunate since it can lead to confusion with the electric potential. The potential energy function is plotted vs. position, and therefore it tells you the potential energy of an object if you know its position. The potential energy function for a mass on a spring is just PE(x) = 0.5*k*x^{2}. Here PE(x) = x^{2}. Note that, depending on your text, you may have seen the potential energy function represented as either V(x) or U(x). We use the book-independent version PE(x). In addition to the potential energy function, a horizontal teal line represents the total energy of the system.

Because of the form of the above potential energy function, it is easy to get confused as to what it is actually showing and what it represents. If you have not done so already, run the animation. The red dot on the potential energy curve does NOT represent the actual motion of a particle on a bowl or roller coaster. In other words, it does NOT represent the two-dimensional motion of an object. It represents the one-dimensional motion of an object, here the one-dimensional motion of a mass attached to a spring. The motion of the red mass is limited to between the turning points represented by where the total energy is equal to the potential energy.

Now also show the kinetic energy on the graph. Watch the kinetic energy and potential change as the mass moves and the spring ceases to be stretched and then gets compressed. Notice that the potential energy added to the kinetic energy always adds up to the total energy. Therefore, if you know the total energy and the potential energy function, you know the kinetic energy of the object at any position in its motion.

Clearly, the force depicted is a spring force. How can we be sure? Well, there is a relationship between the force and the potential energy function. This relationship is expressed as F_{x} = - d (PE)/dx. Therefore, since PE(x) = x^{2}, F_{x} = - 2 x, which tells us that k = 2 N/m (as stated in the first line of the Illustration).