Potential energy diagrams for a particle are useful inpredicting the motion of that particle. These diagrams allow one todetermine the direction of the force acting on the particle at anypoint, the points of stable and unstable equilibrium, theparticle's kinetic energy, etc.

Consider the potential energy diagram shown. The curve represents the value ofpotential energy as a function of the particle's coordinate . Thehorizontal line above the curve represents the constant value ofthe total energy of the particle . The total energy is the sum ofkinetic ( ) and potential ( ) energies of the particle.

The key idea in interpreting the graph can be expressed in theequation

where is the x component of the net force as functionof the particle's coordinate . Note the negative sign: It meansthat the x component of the net force is negative when thederivative is positive and vice versa. For instance, if theparticle is moving to the right, and its potential energy isincreasing, the net force would be pulling the particle to theleft.

If you are still having trouble visualizing this, consider thefollowing: If a massive particle is increasing its gravitationalpotential energy (that is, moving upward), the force of gravity ispulling in the opposite direction (that is, downward).

If the x component of the net force is zero, theparticle is said to be in equilibrium. There are two kindsof equilibrium:

Stable equilibrium means that small deviations fromthe equilibrium point create a net force that accelerates theparticle back toward the equilibrium point (think of a ball rollingbetween two hills).
Unstable equilibrium means that small deviations fromthe equilibrium point create a net force that accelerates theparticle further away from the equilibrium point (think of a ballon top of a hill).
In answering the following questions, we will assume that thereis a single varying force acting on the particle along thex axis. Therefore, we will use the term forceinstead of the cumbersome x component of the netforce.

part A

The force acting on the particle at point A is .

part B

The force acting on the particle at point C is .

par C

The force acting on the particle at point B is .

part D

The acceleration of the particle at point B is .

part E

If the particle is located slightly to the left of point B, itsacceleration is .

part F

If the particle is located slightly to the right of point B, itsacceleration is .

part G

Name all labeled points on the graph corresponding tounstable equilibrium.

List your choices alphabetically, with no commas or spaces; forinstance, if you choose points B, D, and E, type your answer asBDE.

part H

Name all labeled points on the graph corresponding tostable equilibrium.

List your choices alphabetically, with no commas or spaces; forinstance, if you choose points B, D, and E, type your answer asBDE

part I

Name all labeled points on the graph where the acceleration ofthe particle is zero.

List your choices alphabetically, with no commas or spaces; forinstance, if you choose points B, D, and E, type your answer asBDE

part J

Name all labeled points such that when a particle is releasedfrom rest there, it would accelerate to the left.

List your choices alphabetically, with no commas or spaces; forinstance, if you choose points B, D, and E, type your answer asBDE.

part K

Consider points A, E, and G. Of these three points, which onecorresponds to the greatest magnitude of acceleration of theparticle?

part L

What point on the graph corresponds to the maximum kineticenergy of the moving particle?

part M

At what point on the graph does the particle have the lowestspeed?