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Physics 218
Lecture 22
Dr. David Toback
Physics 218, Lecture XXII
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Checklist for Today
• Things due Monday
– Chapters 12 & 13 in WebCT
• Things that are due Tuesday
– Read Chapters 14-16
• Things that were due yesterday
– Chapter 14 problems
– Read Lab hand out on webpage
• Things due next Monday
– Chapter 14 in WebCT
• Next Tuesday
– Exam 3, Chapters 10-13
– Mini-practice exam and bonus points available
Physics 218, Lecture XXII
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The Schedule
This week (4/7)
• Mon: Chapter 12 & 13 material due in WebCT
• Tues: Reading: Chap 14-16
• Wed: Recitation on Chap 14, Lab
• Today: Chap 15, Part 1
Next Week (4/14)
• Monday: Chapter 14 due in WebCT
• Tues: Exam 3 (Chaps 10-13)
• Wed: Recitation on Chap 15, Lab
• Thurs: Lecture on Chap 15, Part 2
Week after that (4/21)
• Monday: Chapter 15 & 16 due in WebCT
• Tues: Reading for Chapter 18
• Tues: Lecture on Chapter 18
• Wed: Recitation on Chapter 18, Lab
• Thurs: Last lecture, Chapter 18
Week after that (4/28)
• No lectures or recitations
Week after that (5/5)
• Final: Monday May 5th, 1PM-3PM
in this
room
Physics 218,
Lecture
XXII
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Overview
• Chapters 12-16 are about Rotational
Motion
• While we’ll do Exam 3 on Chapters 1013, we’ll do the lectures on 12-16 in six
combined lectures
• Give extra time after the lectures to
Study for the exam
• The book does the math, I’ll focus on
the understanding and making the issues
more intuitive
Physics 218, Lecture XXII
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Rotational Motion
Chapters 12 through 16 in six
combined lectures
• This is the 5th of the 6 lectures
• Concentrate on the relationship
between linear and angular variables
Today: Finish up topics
Next time: Hard problems
Physics 218, Lecture XXII
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Physics 218, Lecture XXII
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Angular Quantities
• Position  Angle q
• Velocity  Angular Velocity w
• Acceleration  Angular Acceleration a
Moving forward:
– Force  Torque t
– Mass
– Momentum
– Energy
Physics 218, Lecture XXII
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Analogue of Mass
The analogue of
Mass is called
Moment of
Inertia
Example: A ball of
mass m moving in a
circle of radius R
around a point has a
moment of inertia
F=ma

t=Ia
Physics 218, Lecture XXII
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Calculate Moment of Inertia
Calculate the
moment of
inertia for a
ball of mass
m relative to
the center of
the circle R
Physics 218, Lecture XXII
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Moment of Inertia
• To find the mass of an
object, just add up all the
little pieces of mass
To find the moment of
inertia around a point, just
add up all the little moments
I 
 mr
2
or
I 
Physics 218, Lecture XXII
r
dm

2
10
Torque and Moment of Inertia
• Force vs. Torque
F=ma

t = Ia
• Mass vs. Moment of Inertia
m
 m r
m 
 I
i
i
i
i i
2
or m 
dm

or I 
r
dm

Physics 218, Lecture XXII
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Pulley and Bucket
A heavy pulley, with
radius R, and known
moment of inertia I
starts at rest. We
attach it to a bucket
with mass m. The
friction torque is tfric.
Find the angular
acceleration a
Physics 218, Lecture XXII
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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 XXII
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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
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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
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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
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Problem Solving
For Conservation of Angular Momentum
problems:
BEFORE and AFTER
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Conservation of Angular Momentum
Before
Physics 218, Lecture XXII
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Conservation of Angular Momentum
After
Physics 218, Lecture XXII
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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 XXII
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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
Physics 218, Lecture XXII
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Rotational Kinetic Energy
2
½mv

KEtrans =
2
KErotate = ½Iw
Conservation of Energy must take
rotational kinetic energy into account
Physics 218, Lecture XXII
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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 XXII
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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 = 2/5MR2
Z
Physics 218, Lecture XXII
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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?
Physics 218, Lecture XXII
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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?
Physics 218, Lecture XXII
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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 XXII
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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
Exam 3!!!
• Covers Chapters 10-13
• Get caught up on your
homework!!!
• Mini-practice exam 3 is now
available
Thursday:
- Finish up angular “Stuff”
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Physics 218, Lecture XXII
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Spherical Heavy Pulley
A heavy pulley, with
radius R, starts at
rest. We pull on an
attached rope with a
constant force FT. It
accelerates to an
angular speed of w in
time t.
What is the moment of
inertia of the pulley?
Physics 218, Lecture XXII
R
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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 XXII
R
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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
Physics 218, Lecture XXII
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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
Physics 218, Lecture XXII
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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
Physics 218, Lecture XXII
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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
Physics 218, Lecture XXII
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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
Physics 218, Lecture XXII
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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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• Old stuff
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Kepler’s
nd
2
Law
2nd Law: Each
planet moves
so that an
imaginary line
drawn from
the Sun to the
planet sweeps
out area in
equal periods
of time
Physics 218, Lecture XXII
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Atwood’s Machine
A pulley with a fixed
center (at point O),
radius R0 and moment of
inertia I, has a massless
rope wrapped around it
(no slipping). The rope
has two masses, m1 and
m2 attached to its ends.
Assume m2>m1
• What is the acceleration
of the system?
• Do some checks.
Physics 218, Lecture XXII
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Why does the Bicycle Wheel Turn to the
Right?
Physics 218, Lecture XXII
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Angular Momentum
Again we use the
Cross Product:
  
L rp
Derivation of
St = dL/dt
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L for a system of many bodies
• Have to be careful with Angular Momentum
– t = dl/dt for a single particle
– St = S(dl/dt) for a system of many
particles
– All internal torques cancel because of
Newton’s law (all internal forces are
equal and opposite)
• Reference Frame matters. Only true for:
– The origin is an inertial Reference Frame
– The center of mass
Physics 218, Lecture XXII
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L for a Rigid Body
Find the angular
momentum, L, for
this body given that
it is rotating around
the Z axis with
angular velocity w.
Physics 218, Lecture XXII
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