Transcript New Phenomena: Recent Results and Prospects from the Fermilab

Physics 218
Lecture 17
Dr. David Toback
Physics 218, Lecture XVII
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Before we begin
• Chapter 8HW was due Monday
• Ch. 10 reading assignment: due
today
• Already covered most of chapter
9 in lecture, but I’m changing the
Chapter 9 HW due date to next
Wednesday
• Ch. 10HW still due the following
Monday
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Rotational Motion
Chapters 9 and 10 in four combined
lectures
• This is the 2nd of the 4 lectures
• Concentrate on the relationship
between linear and angular variables
• Last time started 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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Angular Velocity and Acceleration
Are w and a vectors?
w and a clearly have
magnitude
Do they have direction?
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Right-Hand Rule
Yes!
Define the
direction to
point along
the axis of
rotation
Right-hand
Rule
This is true for
Q, w and a
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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
• 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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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 Q1 and Q2.
Q1 is a right angle.
If the axis is fixed,
what is the net
torque on the
wheel?
Q2
F2
Q1
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   r dm
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Torque and Moment of Inertia
• Force vs. Torque
F=ma t = Ia
• Mass vs. Moment of Inertia
m  I   mr
2
or
I   r dm
2
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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
• The rest of Chapter 10
– More on angular “Stuff”
– Angular Momentum
– Energy
• Get caught up on your homework!!!
• Chapter 8 was due Monday, Chapter 9
homework due Wednesday, Chapter 10
is due the Monday after that
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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 x-axis. 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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