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Physics 218
Lecture 21
Dr. David Toback
Physics 218, Lecture XXI
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Checklist for Today
• Things due Yesterday
– Chapters 12 & 13 in WebCT
• Things that are due for today
– Read Chapters 14-16
• Things that are due tomorrow for
Recitation
– Chapter 14 problems
– Read Lab hand out on webpage
• Things due next Monday
– Chapter 14 in WebCT
Physics 218, Lecture XXI
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The Schedule
This week (4/7)
• Mon: Chapter 12 & 13 material due in WebCT
• Today: Reading: Chap 14-16
• Wed: Recitation on Chap 14, Lab
• Thurs Lecture: 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, 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
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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
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Rotational Motion
Chapters 12 through 16 in six
combined lectures
• This is the 4th of the 6 lectures
• Concentrate on the relationship
between linear and angular variables
• Already did kinematics… Move to
dynamics just like earlier this
semester
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Angular Quantities
Last time:
• Position  Angle q
• Velocity  Angular Velocity w
• Acceleration  Angular Acceleration a
This time we’ll start by discussing the
vector nature of the variables and
then move forward on the others:
– Force
– Mass
– Momentum
– Energy
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Physics 218, Lecture XXI
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Angular Quantities
• Position  Angle q
• Velocity  Angular Velocity w
• Acceleration  Angular Acceleration a
Moving forward:
– Force
– Mass
– Momentum
– Energy
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Torque
• Torque is the analogue of Force
• Take into account the perpendicular
distance from axis
– Same force further from the axis leads
to more Torque
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Slamming a door
We know this from experience:
– If we want to slam a door
really hard, we grab it at the
end
– If we try to push in the
middle, we aren’t able to
make it slam nearly as hard
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Torque Continued
• What if we
change the
angle at which
the Force is
applied?
• What is the
“Effective
Radius?”
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Slamming a door
We also know this from
experience:
– If we want to slam a door
really hard, we grab it at the
end and “throw” perpendicular
to the hinges
– If we try to pushing towards
the hinges, the door won’t
even close
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Torque
• Torque is our “slamming” ability
• Need some new math to do Torque
Write Torque as
t
| t | | r || F | sin q



t  r F
• To find the direction of the torque, wrap
your fingers in the direction the torque
makes the object twist
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Vector Cross Product



C  A B
C  A B Sin 
This is the last way of
multiplying vectors we
will see
• Direction from the
“right-hand rule”
• Swing from A into B!
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Vector Cross Product Cont…
Multiply out, but use the
Sinq to give the
magnitude, and RHR to
give the direction
ˆ
i ˆ
i  0 (sin q  0 )
ˆ
i  ĵ  k̂
ˆ
i  k̂   ĵ
(sin q  1 )
(sin q  1 )
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Cross Product Example

A  AX

B  BX
ˆ
i AY ĵ
ˆ
i  BY ĵ


What is A  B using
Unit Vector notation?
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Torque and Force
Torque problems are like Force
problems
1. Draw a force diagram
2. Then, sum up all the torques
to find the total torque
Is torque a vector?
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Example: Composite Wheel
Two forces, F1 and
F2, act on
different radii of a
wheel, R1 and R2,
at different angles
1 and 2. 1 is a
right angle.
If the axis is fixed,
what is the net
torque on the
wheel?
2
F2
1
F1
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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
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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
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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
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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 XXI
r
dm

2
22
Torque and Moment of Inertia
• Force vs. Torque
F=ma

t = Ia
• Mass vs. Moment of Inertia
m  I 
2
mr

I 
or
2
r
 dm
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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
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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?
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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?
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Next Time
More on angular “Stuff”
–Angular Momentum
–Energy
• Get caught up on your
homework!!!
• Mini-practice exam 3 is
now available
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Example of Cross Product
The location of a body
is length r from the
origin and at an
angle q from the xaxis. A force F acts
on the body purely in
the y direction.
What is the Torque on
the body?
z
y
q
x
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Calculate Moment of Inertia
1.Calculate the
moment of inertia
for a ball of mass
m relative to the
center of the
circle R
2.What about lots
of points? For
example a wheel
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Rotating Rod
A uniform rod of mass m, length l, and moment of
inertia I = ml2/3 rotates around a pivot. It is
held horizontally and released.
Find the angular acceleration a and the linear
acceleration a at the end. Where, along the rod,
is a = g?
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Two weights on a bar
Find the
moment
of
inertia
for the
two
different
Axes
middle
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Schedule Changes
Please see the handout for schedule
changes
New Exam 3 Date:
Exam 3
Tuesday Nov. 26th
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Moments of Inertia
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