Learning Goal:

To be able to interpret potential energy diagrams and predict the corresponding motion of a particle.

Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle’s kinetic energy, etc.

Consider the potential energy diagram shown. (Figure 1) The curve represents the value of potential energy U as a function of the particle’s coordinate x. The horizontal line above the curve represents the constant value of the total energy of the particle E. The total energy E is the sum of kinetic (K) and potential (U) energies of the particle.

The key idea in interpreting the graph can be expressed in the equation                                                                       Fx(x)=−dU(x) ⁄ dx,

where Fx(x) is the x component of the net force as function of the particle’s coordinate x. Note the negative sign: It means that the x component of the net force is negative when the derivative is positive and vice versa. For instance, if the particle is moving to the right, and its potential energy is increasing, the net force would be pulling the particle to the left.

If you are still having trouble visualizing this, consider the following: If a massive particle is increasing its gravitational potential energy (that is, moving upward), the force of gravity is pulling in the opposite direction (that is, downward). If the x component of the net force is zero, the particle is said to be in equilibrium. There are two kinds of equilibrium:

Stable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point (think of a ball rolling between two hills). Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill). In answering the following questions, we will assume that there is a single varying force F acting on the particle along the x axis. Therefore, we will use the term force instead of the cumbersome x component of the net force.

Part A

The force acting on the particle at point A is

A. directed to the right

B. directed to the left

C. equal to zero

Part B

The force acting on the particle at point B is 

A. directed to the right

B. directed to the left

C. equal to zero Submit

Part C

The force acting on the particle at point C is

A. directed to the right

B. directed to the left

C. equal to zero Submit

Part D Complete previous part(s)

Part E If the particle is located slightly to the left of point its acceleration is