Physics 2211 Spring 2005

1
Physics 2211 Lecture 34

• Potential Energy Force
• Energy Diagrams

2
Simple Pendulum
3
The del operator acts on a scalar function
and returns a vector. It is a vector that points
in the physical direction of the greatest rate of
increase in the scalar function. The vector
produced is called the gradient of f.
4

• Example Suppose the air temperature varies as
• In what direction do you move one meter to see
  the greatest rise in temperature?
• The x-direction
• The y-direction
• The z-direction
• Some other direction

5
(No Transcript)
6
The gradient is the inverse of the line
integral. Are completely equivalent statements.
Either completely defines the relationship
between a conservative force and its related
potential function.
7
Potential Energy Force

• For a conservative force we define the potential
  energy function (1D)

• Consider some potential energy functions we know,
  and
• find the forces

Spring
Gravity (near earth)
Gravity (general)
8
Rather than a graph of Energy vs. Time, it is
often more convenient to make a graph of Energy
vs. Position. This is called an energy diagram.
Example If gravity is the only force present,
the potential energy is
9
The particle is projected upward. The energy is
entirely kinetic
The energy is entirely potential at the turning
point.
The particle has gained potential energy lost
kinetic energy
The particle gains kinetic energy and loses
potential energy
The energy is entirely potential at the turning
point.
10
Potential Energy Diagrams

• Consider a block sliding on a frictionless
  surface, attached to an ideal spring.

m
x
U
x
0
11
Turning Points
Total E1
Total E2
12
Here?
Or Here?
Or Here?
13
What kind of motion happens with this potential
energy curve?
Total E
KE
Point of Maximum Speed
Turning Point
PE
14

• The particle never reaches point A
• The particle bounces at point C
• The particle accelerates between D and C
• The particle decelerates between D and C
• The particle moves with constant speed

15
(No Transcript)
16
Equilibrium

• F -dU/dx -slope
• So F 0 if slope 0.
• This is the case at the minimum or maximum of
  U(x).
• This is called an equilibrium position.
• If we place the block at rest at x 0, it wont
  move.

m
x
y
U(x)
x
0
17
Equilibrium

• If small displacements from the equilibrium
  position result in a force that tends to move the
  system back to its equilibrium position, the
  equilibrium is said to be stable.
• This is the case if U is a minimum at the
  equilibrium position.
• In calculus language, the equilibrium is stable
  if the curvature (second derivative) is positive.

F
m
x
y
U(x)
F
x
0
18
Equilibrium
U

• Suppose U(x) looked like this
• This has two equilibrium positions, one is
  stable ( curvature) and one is unstable (-
  curvature).
• Think of a small object sliding on the U(x)
  surface
• If it wants to keep sliding when you give it a
  little push, the equilibrium is unstable.
• If it returns to the equilibrium position when
  you give it a little push, the equilibrium is
  stable.
• If the curvature is zero (flat line) the
  equilibrium is neutral.

unstable
neutral
stable
x
0
19
A Look Ahead
force
force
20
Conservation of Mechanical Energy

• If only conservative forces are present, the
  total kinetic plus potential energy of an
  isolated system (no external work done on system)
  is conserved.
• 
  E K U is constant! or DK
  DU 0
• Both K and U can change, but E K U remains
  constant.

E K U ?E ?K ?U W ?U W
(-W) 0
using ?K W using ?U -W
21
Potential Energy Diagrams

• Consider a block sliding on a frictionless
  surface, attached to an ideal spring.
• F -dU/dx -slope

F
x
U
DU
Dx
x
x
0
22
Potential Energy Diagrams

• The potential energy of the block is the same as
  that of an object sliding in a frictionless
  parabolic bowl
• Ug mgy 1/2 kx2 Us

m
y
U(x)
y is the height of an object in the bowl at
position x .
x
0
23
Energy Diagrams
24
Energy Diagrams