Equilibrium Points

Figure 8-10c shows three different values for superposed on the plot of the same potential energy function . Let us see how they would change the situation. If = 4.0 J (purple line), the turning point shifts from to a point between and x₂. Also, at any point to the right of , the system's mechanical energy is equal to its potential energy; thus, the particle has no kinetic energy and (by Eq. 8-20) no force acts on it, and so it must be stationary. A particle at such a position is said to be in neutral equilibrium.(A marble placed on a horizontal tabletop is in that state.)

If = 3.0 J (pink line), there are two turning points: One is between and , and the other is between and . In addition, is a point at which . If the particle is located exactly there, the force on it is also zero, and the particle remains stationary. However, if it is displaced even slightly in either direction, a nonzero force pushes it farther in the same direction, and the particle continues to move. A particle at such a position is said to be in unstable equilibrium.(A marble balanced on top of a bowling ball is an example.)

Next consider the particle's behavior if = 1.0 J (green line). If we place it at , it is stuck there. It cannot move left or right on its own because to do so would require a negative kinetic energy. If we push it slightly left or right, a restoring force appears that moves it back to : A particle at such a position is said to be in stable equilibrium.(A marble placed at the bottom of a hemispherical bowl is an example.) If we place the particle in the cuplike potential well centered at , it is between two turning points. It can still move somewhat, but only partway to or .