Download Not a Special Dispensation in Energy Problems!

Work Done by Friction
Review
Conservation of Energy
Power
Friction is a prime example of a non-conservative force. Let’s
consider moving a book along a table. Looking down at the
tabletop.
B
D1
μN
A
D2
μN
μN
D3
C
Wfriction, ABC = - μND1 – μND2  μND3 = Wfriction, AC
Work by friction depends on path, so friction is a nonconservative force.
Reminder: Gravitational Potential Energy
OSE
Ugrav(y) = mgy
(near surface of earth, y-axis up)
You always get to choose where y=0, and where Ugrav=0.
Choose it wisely to simplify your problem!
Reminder: Spring Potential Energy
An unstretched spring “wants” to stay unstretched. A
stretched spring “wants” to go back to its equilibrium length.
According to Hooke’s law, the force that restores the spring to
its equilibrium length is proportional to the amount of stretch:
If s (x in the picture below) is defined as the difference
between the equilibrium (unstretched) length of the spring
and the stretched (or compressed length of the spring, then
the potential energy of the spring is
OSE
Uspring(s) = ½ks2
Your text derives this
equation on page 147.
Let’s take s>0 to mean the spring is stretched, and s<0 to
mean the spring is compressed (although because s is
squared, the sign doesn’t “matter” for our results).
Reminder: Conservation of Mechanical Energy
We define the mechanical energy of a system to be
OSE
E=K+U.
With this definition, we can write the work-energy principle like
this:
OSE
Ef – Ei = [Wother]if .
If only conservative forces act on a system, the total
mechanical energy is conserved:
OSE
if [Wother]if = 0, then Ef – Ei.
This is the Law of Conservation of Mechanical Energy. It is a
REALLY BIG IDEA.
It’s a Hammer Equation!
If you have a great big nail to pound in, are you
going to pound it with a dinky little screwdriver?
Or a hammer?
Ef – Ei = [Wother]if
“What goes into Ef and Ei?”
K’s and U’s. Kinetic energies of any objects in our system.
Also potential energies.
“What kind of “things” do you know about that have potential
energies?”
Springs and Gravity!
Ef – Ei = [Wother]if
“OK, springs and gravity have potential energies. So what U’s
go into the E of the above OSE?”
Uspring and Ugrav! For every spring and mass in the system.
“Anything else?”
No, not until next semester.
Ef – Ei = [Wother]if
“If springs and gravity go into Ef and Ei, what goes into Wother?”
Work done by any force that doesn’t come from a spring or
gravity! DUH!
A look ahead: if we extend our definition of energy to include
forms other than mechanical, we find that the total energy of
a closed system is conserved. That’s a REALLY REALLY
BIG IDEA. You’ll have to wait a bit for it.
6.7 Problem Solving Using
Conservation of Mechanical Energy
I’ll work a simple example. A bowling ball of mass m is
dropped on a spring of force constant K from a height of H
above the spring. What is its speed after it has compressed
the spring a distance d?
Anybody try to care to solve this using kinematics?
No, you don’t want to use kinematics. In fact, you can’t. You
don’t know how to handle the non-constant spring force.
But using energy methods makes this problem easy...
Special Dispensations in Energy Problems
If two masses are connected by a
massless, taut rope (so that the tension
is the same everywhere along the rope),
it is not necessary to include the work
done by the tensions in Wother.
T
N
M
T
D
D m
W
w
If mass M undergoes a displacement D, mass m
undergoes a displacement of the same magnitude.
The two T’s are in “opposite” directions, relative to D, so
the net work is zero.
You do not need to justify this choice in your work, but
always THINK before you make this choice!
Special Dispensations in Energy Problems
If your diagram specifically shows that
the displacement of an object is
perpendicular to the normal force on
that object, you do not need to include
the work done by N in Wother.
T
N
M
T
D m
W
w
Why can you do this?
If your diagram shows it is valid, you do not need to
justify this choice in your work, but always THINK
before you make this choice!
Special Dispensations in Energy Problems
D
F
If your diagram specifically shows that the displacement of an
object is perpendicular to a force acting on the object, you may
immediately write in your solution, without further justification:
0
WF
Special Dispensations in Energy Problems
D
F
If your diagram specifically shows that the displacement of
an object is parallel to a force acting on the object, you may
immediately write in your solution, without further
justification:
WF = FD
Special Dispensations in Energy Problems
D
F
If your diagram specifically shows that the displacement of
an object is antiparallel to a force acting on the object, you
may immediately write in your solution, without further
justification:
WF = -FD
Not a Special Dispensation in Energy Problems!
T
θ
M
D
If there is a tension force on a single object, or if that
tension force is not perpendicular to the displacement, you
may not “zero out” the work due to the tension force.
6.8 Other Forms of Energy:
Energy Transformations and
the Law of Conservation of Energy
In a previous lecture, you learned that if only conservative
forces act, the total mechanical energy of a system is
conserved.
If you include in your calculations the energy that exits or
enters a system as a result of nonconservative forces, you find
that the total energy of the system is conserved.
“Total energy is neither increased nor decreased in
any process.”
Brief philosophical digression: conservation
of total energy is “demanded” by precision
experiments.
Most books are careful to point out that conservation of energy
has been verified by every experiment done so far.
This leaves you with a nagging feeling that maybe the
experiments aren’t precise enough; maybe someday we’ll find
that conservation of energy is not required.
It would be foolish to say that what we regard as truth 100
years from now will be the same as what we regard as truth
today.
Just as Newtonian mechanics has been shown to be an
approximation to quantum mechanics, valid only for “large”
objects, theories of today are likely to be shown to be a subset
of more encompassing theories.
However, the great conservation principles of physics, two of
which we will study this semester, follow mathematically from
symmetries in nature.
Conservation of energy is a mathematical consequence of the
symmetry of the universe under a translation in time.*
Conservation of energy is “bigger” than Newton’s laws, and I
would expect to see conservation of energy still with us 100
years from now.
*You get the same answer no matter when you set t=0.
The simple statement
“Total energy is neither increased nor decreased in
any process...”
has many important consequences. See
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html
We will see the impact of this Law of Conservation of Energy
a number of times this semester and next.
6.9 Energy Conservation
with Dissipative Forces
Forces which reduce the total mechanical energy of a system
are nonconservative. They are also called “dissipative” forces.
If you include the energy generated in dissipative processes,
you find that total energy in a closed system is conserved.
(The Law of Conservation of Energy discussed in the previous
section.)
In a sense, this information is already contained in our
OSE Ef – Ei = [Wother]if.
However, to be consistent with material shared with Physics
23, and to emphasize the importance of this idea, I’ll introduce
two more OSE’s:
OSE:
*OSE:
Etotal,i = Etotal,f
Etotal = E + 
(for a closed system)
*This is the BIG ONE.
Etotal = E + 
E is mechanical energy. What kinds of energies are included
in ?
“Thermal energy.” (Heat.)
Acoustic energy (sound).
Friction. (Heat.)
Internal energy.
Chemical energy.*
All of these energies have something to do with the motion or
configuration of atoms or molecules.
Because chemical energy is not associated with a dissipative force, we could (if we
wanted to) define a potential associated with chemical energy.
Next semester we will learn about electromagnetic energy. Our
law of conservation of energy will still work.
If you study special relativity (Einstein), you will find that the
Law has to be restated as the Law of Conservation of MassEnergy, but it still works.
Demonstration: trust in physics.
Examples: cannonball, lect10s.pdf
Here is our textbook’s companion website. If you are going to
take the MCAT, try some practice problems from chapters we
have covered.
6.10 Power
Power is defined as the rate at which work is done, or the rate
at which energy is transformed (from one form to another):
OSE :
WF
PF =
t
The unit of power is 1 Joule/s = 1 Watt.
PF =
WF F D cosθ
D
=
= F cosθ
t
t
t
OSE:
PF = F v cosθ
If F v then PF =F v .
Example: a 600 N marine climbs up a rope at
a constant speed of 1.5 m/s. How much
power does the force of gravity deliver to the
marine’s body?
PF = F v cosθ
P grav = Fgrav v cos 180

P grav = - mg v
Pgrav = -  600 N1.5m/s 
Fpull
Pgrav = - 900 W
v
180
The climber does positive work.
F
y
=Fpull,y +Fg,y =may
0
+Fpull - Fg = 0
+Fpull = Fg
But the angle is 0˚.
Fg=mg