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Physics 218
Lecture 18
Dr. David Toback
Physics 218, Lecture XVIII
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Announcements
• Chapter 9 HW due Wed Nov
• Chapter 10 HW due Monday
th
Nov 13 , as usual
th
8
• First Announcement of Exam 3:
st
–November 21
–Tuesday before Thanksgiving!
Physics 218, Lecture XVIII
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Rotational Motion
Chapters 9 and 10 in four lectures
• Lecture three of the four lectures
• Concentrate on the relationship
between linear and angular variables
• Today: Finish up topics
• Thursday: Hard problems
Physics 218, Lecture XVIII
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Angular Quantities
• Position  Angle q
• Velocity  Angular Velocity w
• Acceleration  Angular Acceleration a
• Force  Torque t
• Mass  Moment of Inertia I
Today we’ll finish:
– Momentum
– Energy
Physics 218, Lecture XVIII
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Momentum
Momentum vs. Angular Momentum:




p  mv  L  Iw
Newton’s Laws:


 dp
 dL
F
t 
dt
dt
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Angular Momentum


L  Iw
First way to define the Angular Momentum L:




dw d ( Iw ) d ( L ) dL
t  Ia  I dt  dt  dt  dt

 dL

t  Ia  dt


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Angular Momentum Definition
Another
definition:
  
L rp
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Angular Motion of a Particle
Determine the
angular
momentum, L, of
a particle, with
mass m and speed
v, moving in
circular motion
with radius r
Physics 218, Lecture XVIII
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Conservation of Angular Momentum

 dL
t  dt
if t  0  L  Const
By Newton’s laws, the angular momentum of a
body can change, but the angular momentum for a
system cannot change
Conservation of Angular Momentum
Same as for linear momentum
Physics 218, Lecture XVIII
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Ice Skater
• This one you’ve
seen on TV
• Try this at home in
a chair that rotates
• Get yourself
spinning with your
arms and legs
stretched out, then
pull them in


L  Iw
Physics 218, Lecture XVIII
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Problem Solving
For Conservation of Angular Momentum
problems:
BEFORE and AFTER
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Conservation of Angular Momentum
Before
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Conservation of Angular Momentum
After
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Clutch Design
As a car engineer, you
model a car clutch as
two plates, each with
radius R, and masses MA
and MB (IPlate = ½MR2).
Plate A spins with speed
w1 and plate B is at rest.
you close them so they
spin together
Find the final angular
velocity of the system
Physics 218, Lecture XVIII
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Angular Quantities
• Position  Angle q
• Velocity  Angular Velocity w
• Acceleration  Angular Acceleration a
• Force  Torque t
• Mass  Moment of Inertia I
Today we’ll finish:
– Momentum  Angular Momentum L
– Energy
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Rotational Kinetic Energy
2
½mv
KEtrans =
2
 KErotate = ½Iw
Conservation of Energy must take
rotational kinetic energy into account
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Rotation and Translation
• Objects can both Rotate and
Translate
• Need to add the two
KEtotal = ½ mv2 + ½Iw2
• Rolling without slipping is a
special case where you can
relate the two
 V = wr
Physics 218, Lecture XVIII
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Rolling Down an Incline
You take a solid ball of mass m and radius R and
hold it at rest on a plane with height Z. You then
let go and the ball rolls without slipping.
What will be the speed of the ball at the bottom?
What would be the speed if the ball didn’t roll and
there were no friction?
Note:
Isphere =
2
2/5MR
Z
Physics 218, Lecture XVIII
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A bullet strikes a cylinder
A bullet of speed V and
mass m strikes a solid
cylinder of mass M
and inertia I=½MR2,
at radius R and
sticks. The cylinder is
anchored at point 0
and is initially at rest.
What is w of the
system after the
collision?
Is energy Conserved?
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Rotating Rod
A rod of mass uniform
density, mass m and
length l pivots at a
hinge. It has moment
of inertia I=ml/3 and
starts at rest at a
right angle. You let it
go:
What is w when it
reaches the bottom?
What is the velocity of
the tip at the bottom?
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Less Spherical Heavy Pulley
A heavy pulley, with radius R,
starts at rest. We pull on an
attached rope with constant
force FT. It accelerates to
final angular speed w in
time t.
A better estimate takes into
account that there is
friction in the system. This
gives a torque (due to the
axel) we’ll call this tfric.
What is this better estimate of
the moment of Inertia?
Physics 218, Lecture XVIII
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Person on a Disk
A person with mass m
stands on the edge of a
disk with radius R and
moment ½MR2.
Neither is moving.
The person then starts
moving on the disk
with speed V.
Find the angular velocity
of the disk
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Same Problem: Forces
Same problem but with Forces
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Next Time
• More problems on Chapters 9 & 10
–Please get caught up on
homework!!!
–Believe it or not exam 3 is just
st
around the bend, Nov 21
–Tuesday before Thanksgiving!
• Chap 9 HW due Wednesday Nov 8th
th
• Chap 10 HW due Nov 13
Physics 218, Lecture XVIII
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Exam II
• Mean = 75
– Please check to make sure they added your
points correctly AND entered them into WebCT
correctly!!!
• Average on first two exam = 76%
– Straight scale so far…
• Reading quizzes should be passed back in
recitation
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Next Time
• Chapter 11
– Reading Questions: Q11.X & Q11.X
– XXX FIXME!!!
– Math, Torque, Angular Momentum, Energy
again, but more sophisticated
– The material will not be on the 3rd exam, but
will help with the exam. It will all be on the
final
• HW 10 Due Monday
• Exam 3 is next Thursday, April 22nd
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Angular Quantities
• Position  Angle q
• Velocity  Angular Velocity w
• Acceleration  Angular Acceleration a
• Force  Torque t
Today we’ll finish:
– Mass
– Momentum
– Energy
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Calculating Moments of Inertia
I   mr
2
I   r dm
2
Here r is the distance from the axis of
each little piece of mass
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Calculate the Moment of Inertia
A pulley has mass M,
uniform density,
radius R, and
rotates around its
fixed axis
Calculate its moment
of inertia
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Calculate the Moment of Inertia
Better example
here…
R
Calculate its moment
of inertia
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Hollow Cylinder
Consider a hollow
cylinder with
uniform density,
inner radius R1,
outer radius R2
and total Mass M.
Find the moment
of Inertia
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Parallel-Axis Theorem
• Quick Trick for calculating Moments
• I = Icm + Mh2
• Example
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