Physics I
Class 11
Potential Energy and
Conservation of Energy
Rev. 14-Feb-03 GB
11-1
Work (Review)
 Work is a measure of the energy that a force puts into (+) or takes
away from (–) an object as it moves.
 Work is a scalar,
 calculated
  with a dot product if force is constant:
W  F  d  F d cos( )  Fx d x  Fy d y  Fz d z 
 For varying forces in one dimension:
xf
W   F dx
xi
11-2
Work Integral in
Multiple Dimensions
In multiple dimensions,
the work integral looks like this:

xf
 
W   F  dx

xi
What does it mean to “dot” the force with the variable of integration?

F

dx

xi
path of integration

xf
11-3
Does work depend on the path?
Conservative Forces
For general forces, the work does depend on the path that we take.
However, there are some forces for which work does not depend on
the path taken between the beginning and ending points.
These are called conservative forces.
A mathematically equivalent way to put this is that the work done by
a conservative force along any closed path is exactly zero.


F

d
x
 cons  0
(The funny integral symbol means a path that closes back on itself.)
11-4
Conservative Forces
Non-Conservative Forces
Examples of Conservative Forces:
 Gravity
 Ideal Spring (Hooke’s Law)
 Electrostatic Force (later in Physics 1)
Examples of Non-Conservative Forces:
 Human Pushes and Pulls
 Friction
11-5
How Conservative Forces
Help Us Calculate Work
If a force is conservative, then the work it does
on a particle that moves between two points is
the same for all paths connecting those points.
This is handy to know because it means that we can
indirectly calculate the work done along a
complicated path by calculating the work done along
a simple (for example, linear) path.
Also, if we have a closed path (return to start), then
W = 0.
11-6
Conservative Forces
and Potential Energy
If we are dealing with a conservative force, we can simplify the process
of calculating work by introducing potential energy.
1. Define a point where the potential energy is zero (our choice).
2. Find the work done from that point to any other point in space.
(This is not too hard for most conservative forces.)
3. Define the potential energy at each point as negative the work done
from the reference point to there. Call this function U.
4 The work done by the conservative force from any point A to any
point B is then simply W = U(A)–U(B).
11-7
Conservative Forces
and Potential Energy
W(A  B)  U(A)  U(B)   U
Point B:
 
U(B)    F  dx
B
 
Point A: U ( A )    F  dx
0
A
0
Point 0: U = 0 (defined)
11-8
Two Common Potential Energy
Functions in Physics 1
Gravitational Potential Energy
U g  m g (y  y0 )  m g h
(y0 is our choice to make the problem easier)
Spring Potential Energy
U s  12 k ( x  x 0 ) 2
(x0 is the equilibrium position and k is the spring constant)
11-9
Potential Energy, Kinetic Energy,
and Conservation of Energy
Recall the Work-Kinetic Energy Theorem:
 K  Wnet
And for conservative forces we have
Wcons   U
If the non-conservative forces are zero or negligible, then
Wnet  Wcons
Putting it together,
 K   U
or
K U  0
Another way to say this is the total energy, K+U, is conserved.
11-10
Example Problem
Skateboarder Going Up a Ramp
K0
U  mgh
K  12 m v 2
U0
d
v
h

m v2  0  0  m g h
v2
h
v2
h
d

2g
sin( ) 2 g sin( )
1
2
11-11
Class #11
Take-Away Concepts
1.
Multi-dimensional
form of work integral:

xf
 
W   F  dx

xi
2.
3.
Conservative force = work doesn’t depend on path.
Potential Energy defined for a conservative force:
 
U ( A )    F  dx
A
0
Ug  m g ( y  y0 )  m g h
U s  12 k ( x  x 0 ) 2
4.
Gravity:
5.
6.
Spring:
Conservation of energy if only conservative forces operate:
 K    U or  K   U  0
11-12
Class #11
Problems of the Day
___1. The sum of potential and kinetic energy for a system
of moving objects is conserved
A) only when no net external force acts on the objects
B) only when the objects move along closed paths
C) only when the work done by the resultant external
force is zero
D) always
E) none of the above
11-13
Answer to Problem 1 for Class #11
The answer is E.
The criterion for conservation of mechanical energy (kinetic plus
potential) is that only conservative forces do work on the system.
11-14
Class #11
Problems of the Day
2. A marble with mass 5 g is placed on a vertical spring that has
been compressed 5 cm from its equilibrium position. The spring
constant is 196 N/m. Neglecting friction, air resistance, and the
mass of the spring, how high does the marble go when the spring is
released, counting the marble’s position on the compressed spring
as zero height? Use g = 9.8 N/kg.
Hint: Make sure you use consistent units.
11-15
Answer to Problem 2 for Class #11
This would be a difficult problem to solve using F = m a and
kinematics with the known forces of gravity and the spring,
because the spring force is a function of position. However, with
conservation of mechanical energy it is straightforward. Both
gravity and the ideal spring force are conservative. The potential
energy of the spring is converted to the potential energy of gravity.
The potential energy of the marble on the compressed spring is
given by 0.5 * k * x2 = 0.5 * 196 * (0.05)2 = 0.245 J.
At the maximum height, the potential energy is given by m g h.
Solving for h, we have h = 0.245/(0.005 * 9.8) = 5 m.
11-16
Activity #11
Conservation of Energy
Objective of the Activity:
1.
2.
Use LoggerPro to study mechanical energy in a
simple system.
Consider how kinetic energy, potential energy,
and total mechanical energy vary with position.
11-17
Class #11 Optional Material
Work and Potential Energy
 
W (A  B)   F  dx 
B
B
A
A
 
 
 F  dx   F  dx 
0
B
A
0
  B 
  F  dx   F  dx 
A
0
0
0
U(A)  ( U(B))  U(A)  U(B)
If we change the defined zero point, we are simply adding the same
constant to U everywhere, so U(A)–U(B) does not change.
11-18