Potential Energy Curves
SO FAR WE HAVE DEALT WITH TWO
KINDS OF POTENTIAL ENERGY:
•G R A V I T A T I O N A L ( U = M G H )
•E L A S T I C ( U = 1 / 2 K X 2 )
POTENTIAL ENERGY GRAPHS CAN
PROVIDE INFORMATION ABOUT KINETIC
ENERGY, FORCE AND VELOCITY.
Force and Potential Energy
 We established that the relationship between work
and potential energy was:
W  U
which leads to…
F x  U
U
F
x
Force and Potential Energy
which leads to….
dU
F 
dx
d
Fg    mgh   mg
dx
d 1 2
Fs    kx   kx
dx  2

A conservative force always acts to push the system
toward a lower potential energy.
For example:
Example
A FORCE PARALLEL TO THE X -AXIS ACTS
ON A PARTICLE MOVING ALONG THE X AXIS. THE FORCE PRODUCES A
POTENTIAL ENERGY:
U ( X ) = 1 . 2 X 4.
WHAT IS THE FORCE WHEN THE
PARTICLE IS AT X=-0.8M?
Force and Potential Energy
This analysis can be extended to apply to three
dimensions:
 U ˆ U ˆ U
F  
i
j
y
x
 x

kˆ 

Potential Energy Curves for a Spring
Note:
 When the spring is either in a
state of maximum extension or
compression its potential energy
is also a maximum
 When the spring's displacement
is DOWN the restoring force is
UP
 When the potential energy
function has a negative slope, the
restoring force is positive and
vice-versa
 When the restoring force is zero,
the potential energy is zero
 At any point in the cycle, the total
energy is constant:
U + K = Umax = Kmax
Points of Equilibrium
When the force acting on the object is zero, the object
is said to be in a state of EQUILIBRIUM!
 STABLE EQUILIBRIUM – located at minimums, if
the object is displaced slightly it will tend back to
this location.
 UNSTABLE EQUILIBRIUM – located at maximums,
if the object is displaced slightly it will tend away
from this location.
 STATIC EQUILIBRIUM – located at plateaus, where
the net force equals zero.
Points of Equilibrium
 Stable Equilibrium
at x3 and x5.
 Unstable
Equilibrium at x4.
 Static Equilibrium
at x1 and x6.
Turning Points
 Define the boundaries
.
of the particle’s
motion.
 We know that E=K+U,
so where U=E, K=0 J
and the particle
changes direction.
 For instance, if E=4J,
there would be turning
point at x2.
Turning Points
If E = 1J, why is
the grey area
referred to as an
“energy well”?
Example
A particle of mass 0.5 kg obeys the
potential energy function:
U(x) = 2(x - 1) - (x - 2)3
 What is the value of U(0)?
 What are the values of x1 and x2?
Example
A particle of mass 0.5 kg obeys the
potential energy function:
U(x) = 2(x - 1) - (x - 2)3
 How much potential energy does the
particle have at position x1?
 If the object was initially released
from rest, how fast is it moving as it
passes through position x1?
Example
A particle of mass 0.5 kg obeys the
potential energy function:
U(x) = 2(x - 1) - (x - 2)3
 How much potential energy does
the mass have at x2?
 How fast is it moving through
position x2?
Example
A particle of mass 0.5 kg obeys the
potential energy function:
U(x) = 2(x - 1) - (x - 2)3
 Which position, x1 or x2, is a
position of stable equilibrium?
Example
A particle of mass 0.5 kg obeys the
potential energy function:
U(x) = 2(x - 1) - (x - 2)3
 How fast is the particle moving
when its potential energy, U(x) =
0J?
 If x3 = ½x1, then how fast is the
particle moving as it passes through
position x3?
Example
A particle of mass 0.5 kg obeys the
potential energy function:
U(x) = 2(x - 1) - (x - 2)3
 Sketch the graph of the particle's
acceleration as a function of x.
Indicate positions x1 and x2 on
your graph.