1. No category

Conservative Forces
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
Is it possible for the
gravitational potential
energy of an object to
be negative?
A) yes
B) no
Is it possible for the
gravitational potential
energy of an object to
be negative?
A) yes
B) no
Gravitational PE is mgh, where height h is
measured relative to some arbitrary
reference level where PE = 0. For example, a
book on a table has positive PE if the zero
reference level is chosen to be the floor.
However, if the ceiling is the zero level, then
the book has negative PE on the table. It is
only differences (or changes) in PE that have
any physical meaning.
Last Time
Discussed kinetic energy
Work-energy theorem
Today
Conservative and nonconservative forces
Gravitational potential energy
Other kinds of potential energy
Conservation of mechanical energy
Conservation of Energy
• A conservative force does zero total work on
any closed path.
•
• The work done by a conservative force in
going from an arbitrary point A to an
arbitrary point B is independent of the path
from A to B.
•B
A•
Doing Work Against Gravity
Energy is reclaimed in this case.
Doing Work Against Gravity
Work done by gravity = -mgh
d
d
mg
Energy is reclaimed in this case.
Work Done by Gravity
on a Closed Path is Zero.
Work Done by Friction on
a Closed Path is Not Zero.
Floor (top view)
f  k mg
Conservative Forces
Gravity
Springs
Nonconservative Forces
Friction
Tension
Potential Energy
When we do work, say to lift a
box off the floor, then we give the
box energy. We call that energy
potential energy. Potential
energy, in a sense, has potential to
do work. It is like stored energy.
However, it only works for
conservative forces.
Do potential energy demo. Burn
string and let large mass drop.
Notes on potential energy
Potential energy is part of the workenergy theorem. Potential energy
can be changed into kinetic energy.
Think about gravity for a good
example to use.
There is no single “equation” to use
for potential energy.
Remember that it is only useful for
conservative forces.
Potential Energy
In raising a mass m to a height
h, the work done by the
external force is
Wext = Fext ×d = mgh
= mg ( y2 - y1 )
We therefore define the
gravitational potential
energy at a height y above
some reference point:
.
U grav = mgy
Copyright © 2009 Pearson Education, Inc.
.
Definition of potential energy
We will (sometimes) use a subscript
on Wc to remind us about conservative
forces. This doesn’t work for friction.
Wc U U   (U  U )   U
i
f
f
i
SI unit is the joule (still energy).
Remember gravity
The work done by a conservative force
is equal to the negative of the change in
potential energy.
Hold a box up. It has potential energy.
Drop the box. Gravity does positive
work on the box. The change in the
gravitational potential energy is
negative. The box has less potential
energy when it is on the floor.
More potential energy (PE) notes
 Gravitational potential energy = mgh
 Only change in potential energy U is
important.
 There is no absolute value of PE.
 We choose the zero of PE to be at the most
convenient position to solve problem.
Gravitational potential energy
Wc  mgy
U  U i  U f  Wc  mgy
Ui
y
U i  mgy  U f
Ui  U f
Uf
Because we can choose the “zero” of
potential energy anywhere we want, it
might be convenient to place it at y = 0
(but not always!).
Where might we choose the zero of potential
energy to be here?
Do demos
Loop the loop
Bowling ball (wrecking ball video)
Hopper popper
http://www.youtube.com/watch?v=R
x28g0aqfIk
is B.
An object can have potential energy by virtue
of its surroundings.
Familiar examples of potential energy:
• A compressed (or wound-up) spring
• A stretched elastic band
• An object at some height above the ground
Potential Energy
General definition of gravitational
potential energy:
2
D U = - WG = -
F
×
d
G
ò
1
For any conservative force:
2
D U = U 2 - U1 = -
ò F ×d = - W
1
In one dimension,
U ( x) = -
ò F ( x) dx + C
We can invert this equation to find F(x)
if we know U(x):
dU ( x)
F ( x) = dx
In three dimensions:
¶U
¶U
¶U
F ( x, y , z ) = - i
- j
- k
¶x
¶y
¶z
Example: U  xy 2  5 xyz
U
 2 xy  5 xz (other variables remain constant
y
in partial derivatives)
Gravitational Potential Energy
Boy does +mgy work
W  F d  mgy
to climb up to y.
(Gravity does
negative work, -mgy).
He has potential
energy mgy. Gravity
mg d
does work on boy to
Wg  mgd bring him down. The
potential energy is
converted into kinetic
energy.
Conservation of mechanical energy
Mechanical energy E is defined to be
the sum of K + U.
E  K U
Mechanical energy is conserved.
Only happens for conservative forces.
Conservation of Mechanical Energy
In the image on the left,
the total mechanical
energy at any point is:
E= K+U
1 2
= mv + mgy
2
Solving a Kinematics Problem
Using Conservation of Energy
E = mgh
E=0
High Jump. In the high jump, the
kinetic energy of an athlete is
transformed into gravitational
potential energy without the aid of a
pole. With what minimum speed
must the athlete leave the ground in
order to lift his center of mass 2.10 m
and cross the bar with a speed of 0.70
m/s?
Conceptual Quiz
You see a leaf falling to the ground
with constant speed. When you
first notice it, the leaf has initial
total energy PEi + KEi. You watch
the leaf float down until just before
it hits the ground, at which point it
has final total energy PEf + KEf.
How do these total energies
compare?
A) PEi + KEi > PEf + KEf
B) PEi + KEi = PEf + KEf
C) PEi + KEi < PEf + KEf
D) impossible to tell from
the information provided
Conceptual Quiz
You see a leaf falling to the ground
with constant speed. When you
first notice it, the leaf has initial
total energy PEi + KEi. You watch
the leaf float down until just before
it hits the ground, at which point it
has final total energy PEf + KEf.
How do these total energies
compare?
A) PEi + KEi > PEf + KEf
B) PEi + KEi = PEf + KEf
C) PEi + KEi < PEf + KEf
D) impossible to tell from
the information provided
As the leaf falls, air resistance exerts a force on it opposite to
its direction of motion. This force does negative work, which
prevents the leaf from accelerating. This frictional force is a
nonconservative force, so the leaf loses energy as it falls, and
its final total energy is less than its initial total energy.
Follow-up: What happens to leaf’s KE as it falls? What net work is done?