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.
The curve represents the value of potential energy as
a function of the particle's coordinate . The horizontal line above the curve represents the constant value
of the total energy of the particle
energies of the particle.
. The total energy
is the sum of kinetic (
) and potential (
)
The key idea in interpreting the graph can be expressed in the equation
where
is the x component of the net force as function of the particle's coordinate . 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 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.
1. The force acting on the particle at point A is __________.
directed to the right
directed to the left
equal to zero
F=-dU/dx < 0 at A
2. The force acting on the particle at point C is __________.
directed to the right
directed to the left
equal to zero
F=-dU/dx > 0 at C
3. The force acting on the particle at point B is __________.
directed to the right
directed to the left
equal to zero
F=-dU/dx = 0 at B
4. If the particle is located slightly to the left of point B, its acceleration is __________.
directed to the right
directed to the left
equal to zero
F = -dU/dx =m*a<0 at the left of B
5. Name all labeled points on the graph corresponding to unstable equilibrium.
List your choices alphabetically, with no commas or spaces; for instance, if you choose
points B, D, and E, type your answer as BDE.
BF
Because dU/dx is zero at these two points but if the particle is moved slightly away these
points, they can not go back to these equilibrium points.
6. Name all labeled points such that when a particle is released from rest there, it would
accelerate to the left.
List your choices alphabetically, with no commas or spaces; for instance, if you choose
points B, D, and E, type your answer as BDE.
AE
F = -dU/dx =m*a<0 at these two points
7. Consider points A, E, and G. Of these three points, which one corresponds to the
greatest magnitude of acceleration of the particle?
A
E
G
F = -dU/dx =m*a
8. If the total energy of the particle is known, one can also use the graph of
conclusions about the kinetic energy of the particle since
to draw
.
As a reminder, on this graph, the total energy
is shown by the horizontal line.
What point on the graph corresponds to the maximum kinetic energy of the moving
particle?
D
The lowest potential energy point will corresponds to the maximum kinetic energy point.
9. At what point on the graph does the particle have the lowest speed?
B