Download Potential Energy Curves

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 )
SOMETIMES IT IS MORE HELPFUL WHEN
YOU HAVE POTENTIAL ENERGY AS A
FUNCTION OF POSITION TO FIND
POTENTIAL ENERGY FIRST AND DERIVE
THE FORCE FROM THERE.
Force and Potential Energy
 We established last class 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
A conservative force always acts to push the system
toward a lower potential energy.
d
Fg    mgh   mg
dx
d 1 2
Fs    kx   kx
dx  2

Example
A FORCE PARALLEL TO THE X -AXIS
ACTS ON A PARTICLE MOVING ALONG
THE X-AXIS. THE FORCE PRODUCES
A P O T E N T I A L E N E R G Y : 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 

Check Point
A PARTICLE MOVING ALONG THE X-AXIS IS
ACTED ON BY A CONSERVATIVE FORCE. AT A
CERTAIN POINT, THE FORCE IS ZERO.
WHAT DOES THIS TELL YOU ABOUT THE
VALUE OF THE POTENTIAL ENERGY
FUNCTION AT THIS POINT?
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
Potential Energy Curve for a function
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.
Example
A particle of mass 0.5 kg obeys the potential energy
function:
U(x) = 2(x - 1) - (x - 2)3
 At what value of x does the particle experience it
greatest negative acceleration?
 What is the value of its potential energy at this
position?
 How much kinetic energy does it have at this
position?
 What force is being exerted upon it at this
position?
 What is the value of its acceleration at this
position?