Physics 218
Lecture 14
Dr. David Toback
Physics 218, Lecture XIV
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Checklist for Today
•Things due awhile ago:
– Read Chapters 7, 8 & 9
•Things that were due Yesterday:
– Problems from Chap 7 on WebCT
•Things that are due Tomorrow in
Recitation
– Chapter 8
– Reading for Lab
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The Schedule
This week: (3/3)
• Chapter 7 due in WebCT
• 5th and 6th lectures (of six) on Chapters 7, 8 & 9
• Chapter 8 in recitation
Next week: (3/10) Spring Break!!!
Following Week: (3/17)
• Chapter 8 due in WebCT
• Reading for Chapters 10 & 11
• Lecture on Chapters 10 & 11
• Chapter 9 and Exam 2 Review in recitation
Following Week: (3/24)
• Chapter 9 due in WebCT
• Exam 2 on Tuesday
• Recitation on Chapters 10 & 11
• Reading for Chapters 12 & 13 for Thursday
• Lecture 12 & 13 on Thursday
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Chapters 7, 8 & 9 Cont
Before:
– Work and Energy
– The Work-Energy relationship
– Potential Energy
– Conservation of Mechanical Energy
This time and next time:
– Conservation of Energy
– Lots of problems
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Different Style Than the Textbook
I like teaching this material
using a different style than
the textbook
1. Teach you the concepts
2. Give you the important
equations
3. Then we’ll do lots of problems
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Mechanical Energy
• We define the total
mechanical energy in a
system to be the kinetic
energy plus the potential
energy
• Define E≡K+U
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Conservation of Mechanical Energy
• For some types of problems, Mechanical
Energy is conserved (more on this next
week)
• E.g. Mechanical energy before you drop a
brick is equal to the mechanical energy
after you drop the brick
K2+U2 = K1+U1
Conservation of Mechanical Energy
E2=E1
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Problem Solving
• What are the types of examples we’ll
encounter?
– Gravity
– Things falling
– Springs
• Converting their potential energy into
kinetic energy and back again
E = K + U =
2
½mv
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+ mgy
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Falling onto a Spring
We want to measure the
spring constant of a
certain spring. We drop a
ball of known mass m
from a known height Z
above the uncompressed
spring. Observe it
compresses a distance C.
Before After
Z
Z
C
What is the spring
constant?
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Quick Problem
A refrigerator with mass
M and speed V0 is sliding
on a dirty floor with
coefficient of friction m.
Is mechanical energy
conserved?
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Non-Conservative Forces
• We’ve talked about three different types
of forces:
1.Gravity: Conserves mechanical
energy
2.Normal Force: Conserves
mechanical energy (doesn’t do
work)
3.Friction: Doesn’t conserve
mechanical energy
• Since Friction causes us to lose mechanical
energy (doesn’t conserve mechanical
Physics 218, Lecture XIV
energy) it is a Non-Conservative
force! 12
Law of Conservation of Energy
• Mechanical Energy NOT always
conserved
• If you’ve ever watched a roller
coaster, you see that the friction
turns the energy into heating the
rails, sparks, noise, wind etc.
• Energy = Kinetic Energy + Potential
Energy + Heat + Others…
– Total Energy is what is
conserved!
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Conservative Forces
If there are only conservative forces in the
problem, then there is conservation of mechanical
energy
• Conservative: Can go back and forth along any
path and the potential energy and kinetic energy
keep turning into one another
– Good examples: Gravity and Springs
• Non-Conservative: As you move along a path, the
potential energy or kinetic energy is turned into
heat, light, sound etc… Mechanical energy is lost.
– Good example: Friction (like on Roller
Coasters)
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Law of Conservation of Energy
• Even if there is friction, Energy is
conserved
• Friction does work
– Can turn the energy into heat
– Changes the kinetic energy
• Total Energy = Kinetic Energy + Potential
Energy + Heat + Others…
– This is what is conserved
• Can use “lost” mechanical energy to
estimate things about friction
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Roller Coaster with Friction
A roller coaster of mass m starts at rest
at height y1 and falls down the path with
friction, then back up until it hits height
y2 (y1 > y2).
Assuming we don’t know anything about the
friction or the path, how much work is
done by friction on this path?
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Energy Summary
If there is net work done on an object, it
changes the kinetic energy of the object
(Gravity forces a ball falling from height h
to speed up  Work done.)
Wnet = DK
If there is a change in the potential energy,
some one had to do some work: (Ball falling
from height h speeds up→ work done → loss
of potential energy. I raise a ball up, I do
work which turns into potential energy for
the ball)
DUTotal = WPerson =-WGravity
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Energy Summary
If work is done by a non-conservative force
it does negative work (slows something
down), and we get heat, light, sound etc.
EHeat+Light+Sound.. = -WNC
If work is done by a non-conservative
force, take this into account in the total
energy. (Friction causes mechanical
energy to be lost)
K1+U1 = K2+U2+EHeat…
K1+U1 = K2+U2-WNC
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Friction and Springs
A block of mass
m is traveling on
a rough surface.
It reaches a
spring (spring
constant k) with
speed Vo and
compresses it a
total distance D.
Determine m
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Bungee Jump
You are standing on a
platform high in the air
with a bungee cord
(spring constant k)
strapped to your leg.
You have mass m and
jump off the platform.
1.How far does the cord
stretch, l in the picture?
2.What is the equilibrium
point around which you
will bounce?
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l
l
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Coming up…
• Lectures:
– Last lectures on Chaps 7, 8 and 9
• Chapter 7 was due in WebCT on
Monday
• For Recitation
– Chap 8 problems due
– Lab reading
• Reading for Lecture next week
– Chaps 10 & 11: Momentum
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Roller Coaster
You are in a roller coaster car of mass M
that starts at the top, height Z, with an
initial speed V0=0. Assume no friction.
a) What is the speed at the bottom?
b) How high will it go again?
c) Would it go as high if there were friction?
Z
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Energy
• Potential Energy & Conservation
of Energy problems
• The relationship between
potential energy and Force
• Energy diagrams and
Equilibrium
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Energy Review
If there is net work on an object, it changes the
kinetic energy of the object (Gravity forces a ball
falling from height h to speed up  Work done.)
Wnet = DK
If there is a change in the potential energy, some
one had to do some work: (Ball falling from height
h speeds up→ work done → loss of potential
energy. I raise a ball up, I do work which turns
into potential energy for the ball)
DUTotal = WPerson =-WGravity
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Energy Review
If work is done by a non-conservative force it is
negative work (slows something down), and we get
heat, light, sound etc.
EHeat+Light+Sound.. = -WNC
If work is done by a non-conservative force, take
this into account in the total energy. (Friction
causes mechanical energy to be lost)
K1+U1 = K2+U2+EHeat…
K1+U1 = K2+U2-WNC
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Potential Energy Diagrams
• For Conservative forces
can draw energy
diagrams
• Equilibrium points
– Motion will move
“around” the
equilibrium
– If placed there with
no energy, will just
stay (no force)
Fx  
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dU
dx
0
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Stable vs. Unstable Equilibrium Points
The force is zero at both maxima and minima but…
– If I put a ball with no velocity there would
it stay?
– What if it had a little bit of velocity?
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Roller Coaster with Friction
A roller coaster car of mass m starts at rest at
height y1 and falls down the path with friction,
then back up until it hits height y2 (y1 > y2).
Assuming we don’t know anything about the friction
or the path, how much work is done by friction
on this path?
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Roller Coaster with Friction
A roller coaster car of mass m starts at rest at
height y1 and falls down the path with friction,
then back up until it hits height y2 (y1 > y2). An
odometer tells us that the total scalar distance
traveled is d.
Assuming we don’t know anything about the friction
or the path, how much work is done by friction
on this path?
Assuming that the magnitude and angle of the
force of friction, F, between the car and the
track is constant, find |F|.
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Bungee Jump
A jumper of mass m
sits on a platform
attached to a bungee
cord with spring
constant k. The cord
has length l (it
doesn’t stretch until
it has reached this
length).
How far does the cord
stretch Dy?
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l
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A football is thrown
A 145g football starts at rest and is thrown
with a speed of 25m/s.
1. What is the final kinetic energy?
2. How much work was done to reach this
velocity?
We don’t know the forces exerted by the
arm as a function of time, but this allows
us to sum them all up to calculate the
work
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Robot Arm
A robot arm has a funny
Force equation in 1dimension
2

3x
F(x)  F0  1  2
x

where F0 and X0 are 0
constants.
What is the work done to
move a block from position
X1 to position X2?




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