Transcript Physics 106P: Lecture 1 Notes

Rotational KE, Angular
Momentum
Rotational Energy
 It
is moving so it is a type of Kinetic
Energy (go back and rename the first)
Translational KE
Rotaional KE
Example: cylinder rolling

Consider a cylinder with radius R and mass M, rolling
w/o slipping down a ramp. Determine the ratio of the
translational to rotational KE.
Friction causes object to roll, but if it rolls w/o
slipping friction does NO work!
W = F d cos q
contact
d is zero for point in
H
No dissipated work, energy is conserved
Need to include both translation and rotation
kinetic energy.
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Example: cylinder rolling

Consider a cylinder with radius R and mass M, rolling
w/o slipping down a ramp. Determine the ratio of the
translational to rotational KE.
Translational:
Rotational:
use
1
I  MR 2
2
and

V
R
Ratio:
H
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Example: cylinder rolling

What is the velocity of the cylinder at the bottom of the
ramp?
H
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Angular Momentum
Momentum
Angular Momentum
p = mV
L = I
conserved if Fext = 0
conserved if ext =0
Vector
Vector!
units: kg-m/s
units: kg-m2/s
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Direction of Angular Momentum
 Right
Hand Rule
 Wrap fingers of right hand around direction of
rotation, thumb gives direction of angular
momentum.
Example: Two Disks

A disk of mass M and radius R rotates around the z axis
with angular velocity i. A second identical disk, initially
not rotating, is dropped on top of the first. There is
friction between the disks, and eventually they rotate
together with angular velocity f. Find f.
z
z
i
f
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Example: Merry Go Round
Four students (mass 70kg) are riding on a merry-go-round (solid disk
of mass = 90kg rotating with angular velocity =3 rad/s. Initially all
four students are on the outer edge. Suddenly 3 of students pull
themselves to within 0.25m of the center. What is the final angular
velocity of the merry-go-round?
Before
1.2m
After
.25
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Example: Merry Go Round
Before
1.2m
After
.25
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Example: Merry Go Round
What is the centripetal acceleration felt by each of the students?
Before
1.2m
After
.25
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Demo
You are sitting on a freely rotating bar-stool with your arms
stretched out and a weight in each hand. Your professor gives you a
twist and you start rotating around a vertical axis though the center
of the stool. You can assume that the bearing the stool turns on is
frictionless, and that there is no net external torque present once
you have started spinning.
You now pull your arms and hands (and weights) close to your
body.
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Demo
What happens to the angular momentum as you pull
in your arms?
1. it increases
2. it decreases
3. it stays the same
L
L
1
2
What happens to your angular velocity as you pull in
your arms?
1. it increases
2. it decreases
3. it stays the same
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Demo
What happens to your kinetic energy as you pull in
your arms?
1. it increases
2. it decreases
3. it stays the same
K
1
1 2 2
I 2 
I 
2
2I

1 2
L
2I
(using L = I )
2
1
I2
I1
L
L
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Turning the bike wheel
A student sits on a barstool holding a bike wheel. The wheel
is initially spinning CCW in the horizontal plane (as viewed
from above) L= 25 kg m2/s She now turns the bike wheel
over. What happens?
A. She starts to spin CCW.
B. She starts to spin CW.
C. Nothing
Start w/ angular momentum L pointing up from wheel. When
wheel is flipped, no more angular momentum from it pointing
up, so need to spin person/stool to conserve L!
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Gyroscopic Motion:
 Suppose
you have a spinning gyroscope
in the configuration shown below:
 If the left support is removed, what will
happen??
support

pivot
g
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Gyroscopic Motion...
 Suppose
you have a spinning gyroscope
in the configuration shown below:
 If the left support is removed, what will
happen?
The gyroscope does not fall down!

pivot
g
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Gyroscopic Motion...
Bicycle wheel
 ...
instead it precesses around its pivot
axis !

pivot
Summary
 
L
=Ia
=I
Right Hand Rule gives direction
If  = 0, L is conserved