Energy Conservation: Of Special Interest to Physics C Students

Practice problems for these concepts can be found at: Energy Conservation Practice Problems for AP Physics B & C

Potential Energy vs. Displacement Graphs

A different potential energy function can actually be derived for ANY conservative force. (A conservative force means that energy is conserved when this force acts … examples are gravity, spring, and electromagnetic forces; friction and air resistance are the most common nonconservative forces.) The potential energy U for a force is given by the following integral:

Note that this equation works for the gravitational force F = –mg (where – is the down direction) and the spring force F = –kx; the potential energy attributable to gravity integrates to mgh, and the spring potential energy becomes ½kx².

Chris on a Skateboard

Once a potential energy of an object is found as a function of position, making a U vs. x graph tells a lot about the long-term motion of the object. Consider the potential energy of a spring, ½kx². A graph of this function looks like a parabola, as shown in Figure 14.2.

You can get a general feel for how the mass on a spring moves by imagining that Chris is riding a skateboard on a ramp shaped like the graph. A ramp shaped like this looks like a half-pipe. If he starts from some height above the bottom, Chris will oscillate back and forth, going fastest in the middle, and turning around when he runs out of energy at the right or left end. Although this is not precisely how a mass on a spring moves—the mass only moves back and forth, for example—the long-term properties of Chris's motion and the motion of the mass on a spring are the same. The mass oscillates back and forth, with its fastest speed in the middle, just like Chris does.

Thinking about Chris on a skateboard works for all U vs. x graphs. Consider a model of the energy between two atoms that looks like the graph in Figure 14.3.

If Chris on his skateboard released himself from rest near position x₁, he'd just oscillate back and forth, much like in the mass on a spring problem. But if he were to let go near the position labeled x₂, he'd have enough energy to keep going to the right as far as he wants; in fact, he'd make it off the page, never coming back. This is what happens to the atoms in molecules, too. If a second atom is placed pretty close to a distance x₁ from the first atom, it will just oscillate back and forth about that position. However, if the second atom is placed very close to the first atom, it will gain enough energy to escape to a faraway place.

Practice problems for these concepts can be found at: Energy Conservation Practice Problems for AP Physics B & C