Schedule

1
Schedule
Feb. 9 review Feb. 11 review Feb. 13 Exam 1
Feb. 16 4.8-4.9 Feb. 18 5.1-5.2 Feb. 20 boardwork Quiz 4
Feb. 23 5.3,6.1-6.3 Feb. 25 6.4-6.7 Feb. 27 boardwork Quiz 5
March 1 6.8-6.10 March 3 review March 5 Exam 2
A circular motion problem.
Exam 2 Friday, March 5.
2
Example of Work A block of mass M slides a
distance D along a frictionless inclined plane,
which makes an angle ? with the horizontal. (a)
What force(s) do work? (b) How much work is done?
Litany for Work-Kinetic Energy Problems, step 1
draw a basic representative sketch of the process.
M
D
?
3
Step 2 free-body diagram.
N
M
y
M
D
a
?
x
Mg
D
?
I wanted to use W for weight, but didnt, to
avoid confusion with Work.
Forces think gravity and contact.
Show displacement vector. (Better to not touch
the mass.)
Show acceleration. Choose axes.
Show components of forces not parallel to axes.
Better label any angles!
4
Step 3 do not assume N mg!
?
The Litany also says to label all
non-perpendicular angles between forces and
displacements.
We have one such angle. We cant call it ?. Lets
call it ?. (Obnoxious special effect to help draw
your attention to the angle!)
5
Step 4 write down the appropriate OSE.
OSE WFi?f FD cos ?.
Danger! The ? in the OSE is the angle between the
force and displacement. An angle given as ? in
the problem may not be the correct angle to use
in the OSE!
Thats why I made you use different angle names
in step 3. The litany is designed to help you
avoid painful mistakes.
6
After step 4, the Litany is no longer applicable.
The Litany is intended for use with problems
involving work and kinetic energy (which I have
not yet introduced).
The Litany is still valid for setting up this
problem.
7
Remind me again, what am I trying to calculate?
Oh, yeswhich forces do work!
In this case N is perpendicular to the
displacement. The angle is 90 and cos(90)0.
Special dispensation any time a force F is
perpendicular to the displacement D, you may
immediately write WF0.
8
0
Thus, without further mathematics WN.
The force due to gravity does work
OSE WFi?f FD cos ?.
WMgi?f Mg D cos ?.
In the process of determining which forces did
work, I also answered part (b) of the
problemexcept that
9
The angle ? was not given in the original problem.
So I must write
WMgi?f Mg D cos ?.
WMgi?f Mg D cos (90-?).
Now I am done! (And sin ? would also be
acceptable.)
10
Dang, theres a lot of new stuff to learn here!
No, theres not. Just
OSE WFi?f FD cos ?
which I give you anyway. The hard part is
figuring out the correct angle.
11
6.2 Work Done by a Varying Force
Sorry, I have to skip this section.
If you promise to read this section, I wont test
you on it.
In problems with varying forces, you really need
to use calculus. The ideas are the same as those
you learn in this class the mathematics a bit
more complex.
Naturally, I wont give you a varying-force
problem.
12
6.3 Kinetic Energy and the Work-Energy Principle
What is kinetic energy?
Your text shows that the work to change the speed
of a mass m from v1 to v2 is W ½mv22 ½mv12
so we define kinetic energy as OSE K ½mv2
.
13
With the above definition of K, we have OSE
Wneti?f ?K .
This is often called the work-energy theorem. It
is one of the BIG DEALS of physics.
Note that ?(stuff) always means (stuff)final -
(stuff)initial.
So ?K means Kfinal - Kinitial.
Calling this a BIG DEAL will cause you to want
to apply it to every problem in this chapter.
There are even BIGGER DEALS later that are
usually better starting points for problems.
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Example 6-4. What is the KE of a baseball of mass
m thrown with a velocity of magnitude V? If the
ball started from rest, how much energy did it
take to make it reach this speed?
Giancoli does the second part the physicist
way. Lets do it the official way here (on the
blackboard).
15
Lets think for a minute about what I have done
so far in Chapter 6. First I introduced a
definition of work.
Using this definition, we found
OSE Wneti?f ?K .
This is often called the Work-Energy Theorem.
Now I will begin to introduce a REALLY BIG DEAL
equation, even bigger than the Work-Energy
theorem. Im going to change the order of
presentation a bit compared to Giancolis
development of the idea...
Giancoli calls a slightly different form of the
equation the work-energy principle.
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6.5 Conservative and Nonconservative Forces
Conservative force work done by the force does
not depend on path taken only depends on initial
and final positions (or states).
Gravitational force is the prime example of a
conservative force.
Let me pick our text off the floor and put it on
a table.
17
Calculate the work done by gravity in moving the
book from the floor to the table (not precisely
following the Litany)
1. Choose y-axis up, yi at floor, yf at table.
2. Indicate force due to gravity, displacement,
and angle between the two vectors (its the
smaller of the angles).
OSE WFi?f FD cos ?
Wgravi?f Fgrav D cos ?
18
3. But check out this trig
yf-yi
180-?
Wgravi?f Fgrav D cos ?
Wgravi?f Fgrav D (-cos (180-?))
Wgravi?f mg D ( - (yf-yi) / D )
Wgravi?f -mg(yf-yi)
Yes, gravity did negative work!
19
Let me digress on this negative work business
Wgravi?f -mg(yf-yi)
In moving the book, I exert a force on the book,
and in this case, do positive work.
The force of gravity opposes me, and does
negative work.
Your problems will often be set up so that
friction also does negative work.
Had I made the y-axis positive down, then yfltyi
and gravity does positive work (and I would have
done negative work).
As we have seen before in this class, the sign of
an answer depends on the choice of axes.
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Wgravi?f -mg(yf-yi)
The work done by gravity is independent of path
it depends only on the starting and stopping
locations.
But wait a minute? What about the work you did
in lifting the book?
Demonstrate professor doing work on the book.
Clearly, the work I do on the book depends on the
path I take. However, the work gravity does on
the book depends only on the initial and final
coordinates.
Gravity is a conservative force. The force
exerted by my hand is generally not a
conservative force.
21
Friction is a prime example of a non-conservative
force. Lets consider moving a book along a
table. Looking down at the tabletop.
A
Wfriction, ABC - µND1 µND2 ? µND3
Wfriction, AC
Work by friction depends on path, so friction is
a non-conservative force.
22
So why all this fuss about conservative and
nonconservative forces?
6.4 Potential Energy
KE is energy that an object possesses because it
is in motion.
The work done on an object by a conservative
force depends only on the objects initial and
final position (or configuration).
It is thus possible to define a function (in the
mathematical sense) whose value is the work done
on an object by a conservative force.
It is thus possible to define a function (in the
mathematical sense) whose value is the work done
on an object by a conservative force. We call
this function the potential energy.
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Potential energy is put into or taken out of
a system or object by the action of a
conservative force.
Nonconservative push force sets swing in motion.
Conservative gravitational force changes energy
between kinetic and potential
high
mid
Potential energy is an objects energy due to its
configuration or surroundings.
low
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In the case of gravitational force,
configuration refers to initial and final
positions.
The two conservative forces that we will work
with in this course are the force due to a
massless, frictionless spring, and the force due
to gravity.
These forces give rise to gravitational potential
energy and spring potential energy.
25
Potential Energy
We are going to use the letter U for potential
energy.
For reasons which might become clear to you
later, we define
OSE Uf Ui -WCi?f
26
Gravitational Potential Energy
Gravitational potential energy depends only on
the vertical distance between the two points of
interest.
Recall that you can choose your y-axis and origin
any where you want. Gravitational potential
energy differences are independent of your choice
of origin!
Recall that a bit earlier we found Wgravi?f
-mg(yf-yi).
If we take yi0 and yfy, we have
OSE Ugrav(y) mgy (near surface of earth,
y-axis up)
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Repeating the OSE, to make a point
OSE Ugrav(y) mgy (near surface of earth,
y-axis up)
You always get to choose where y0, and where
Ugrav0. Choose it wisely to simplify your
problem!
Spring Potential Energy
An unstretched spring wants to stay
unstretched. A stretched spring wants to go
back to its equilibrium length.
28
According to Hookes law, the force that restores
the spring to its equilibrium length is
proportional to the amount of stretch
29
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 158.
Lets take sgt0 to mean the spring is stretched,
and slt0 to mean the spring is compressed
(although because s is squared, the sign doesnt
matter for our results).
30
Remember, the symbol V may be used to indicate
potential energy. Dont get confused if I do
that.
Well do examples with potential energies soon
enough. For now, lets go back to section 6.5,
which I didnt quite finish.
31
6.5 Conservative and Nonconservative
Forces (continued)
Remember how our definition of potential energy,
Uf Ui -WCi?f
had that seemingly unnecessary minus sign in it?
Using our definition of U, we can do the
following Wneti?f ?K Wnet Wconservative
Wnonconservative Wconservative
Wnonconservative ?K -?U Wnonconservative
?K Wnonconservative ?K ?U.
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Wnonconservative ?K ?U.
Giancoli calls the above equation the
work-energy principle. The next section (6.6)
in Giancoli is a continuation of the ideas
developed in sections 6.4 and 6.5.
6.6 Mechanical Energy and its Conservation
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 Wotheri?f .
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If only conservative forces act on a system, the
total mechanical energy is conserved
OSE if Wotheri?f 0, then Ef Ei.
This is the Law of Conservation of Mechanical
Energy. It is a REALLY BIG IDEA.
Its a Hammer Equation!
34
If you have a great big nail to pound in, are you
going to pound it with a dinky little
screwdriver?
Or a hammer?
35
Ef Ei Wotheri?f
What goes into Ef and Ei?
Ks and Us. 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!
36
Ef Ei Wotheri?f
OK, springs and gravity have potential energies.
So what Us 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.
37
Ef Ei Wotheri?f
If springs and gravity go into Ef and Ei, what
goes into Wother?
Work done by any force that doesnt come from a
spring or gravity!
Work done by any force that doesnt 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. Thats a REALLY REALLY BIG IDEA.
Youll have to wait a bit for it.
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6.7 Problem Solving Using Conservation of
Mechanical Energy
Ill 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 rebounded to a height of H/2?
Neglect friction and air resistance.
Anybody try to care to solve this using
kinematics?
No, you dont want to use kinematics. In fact,
you cant. You dont know how to handle the
non-constant spring force.
But using energy methods makes this problem
easy...