Physics 151: Lecture 23 Todays Agenda

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Physics 151 Lecture 23Todays Agenda

• Topics
• More on Rolling Motion
• Ch. 11.1 Angular Momentum Ch. 11.3-5

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Example Rolling Motion

• A cylinder is about to roll down an inclined
  plane. What is its speed at the bottom of the
  plane ?

Cylinder has radius R
M
M
M
M
M
h
M
q
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Lecture 22, ACT 4aRolling Motion

• A race !!
• Two cylinders are rolled down a ramp. They have
  the same radius but different masses, M1 gt M2.
  Which wins the race to the bottom ?
• A) Cylinder 1
• B) Cylinder 2
• C) It will be a tie

M1
M2
h
M?
q
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Lecture 22, ACT 4bRolling Motion

• A race !!
• Two cylinders are rolled down a ramp. They have
  the same moment of inertia but different radius,
  R1 gt R2. Which wins the race to the bottom ?
• A) Cylinder 1
• B) Cylinder 2
• C) It will be a tie

R1
R2
animation
h
M?
q
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Lecture 22, ACT 4cRolling Motion

• A race !!
• A cylinder and a hoop are rolled down a ramp.
  They have the same mass and the same radius.
  Which wins the race to the bottom ?
• A) Cylinder
• B) Hoop
• C) It will be a tie

M1
M2
animation
h
M?
q
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Remember our roller coaster. Perhaps now we can
get the ball to go around the circle without
anyone dying.
Note Radius of loop R Radius of ball r
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How high do we have to start the ball ?
h
h 2.7 R (2R 1/2R) 2/10 R
-gt The rolling motion added an extra 2/10 R to
the height)
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Angular MomentumDefinitions Derivations
See text 11.3
pmv

• We have shown that for a system of particles
• Momentum is conserved if
• What is the rotational version of this ??

Animation
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Definitions Derivations...
See text 11.3

• First consider the rate of change of L

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Definitions Derivations...
See text 11.3

• Recall that

???
?EXT

• Which finally gives us

• Analogue of !!

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What does it mean?
See text 11.5

• where
  and

Total angular momentum is conserved
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Angular momentum of a rigid bodyabout a fixed
axis
See text 11.4

• Consider a rigid distribution of point particles
  rotating in the x-y plane around the z axis, as
  shown below. The total angular momentum around
  the origin is the sum of the angular momenta of
  each particle

(since ri , vi , are perpendicular)
v1
We see that L is in the z direction.
m2
j
Using vi ? ri , we get
r2
m1
r1
i
v2
?
r3
m3
v3
I
?
Analogue of p mv !!
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Lecture 23, ACT 2Angular momentum
In the figure, a 1.6-kg weight swings in a
vertical circle at the end of a string having
negligible weight. The string is 2 m long. If the
weight is released with zero initial velocity
from a horizontal position, its angular momentum
(in kg m2/s) at the lowest point of its path
relative to the center of the circle is
approximately a. 40 b. 10 c. 30 d. 20 e. 50
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Angular momentum of a rigid bodyabout a fixed
axis
See text 11.4

• In general, for an object rotating about a fixed
  (z) axis we can write LZ I ?
• The direction of LZ is given by theright hand
  rule (same as ?).
• We will omit the Z subscript for
  simplicity,and write L I ?

z
?
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Lecture 23, ACT 2Angular momentum

• Two different spinning disks have the same
  angular momentum, but disk 1 has more kinetic
  energy than disk 2.
• Which one has the biggest moment of inertia ?

(a) disk 1 (b) disk 2 (c) not
enough info
I1 lt I2
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Example Two Disks

• A disk of mass M and radius R rotates around the
  z axis with angular velocity ?0. 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. What is ?F ?

?0
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Example Two Disks

• First realize that there are no external torques
  acting on the two-disk system.
• Angular momentum will be conserved !
• Initially, the total angular momentum is due
  only to the disk on the bottom

2
1
?0
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Example Two Disks

• First realize that there are no external torques
  acting on the two-disk system.
• Angular momentum will be conserved !
• Finally, the total angular momentum is dueto
  both disks spinning

z
2
1
?F
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Example Two Disks

• Since LINI LFIN

An inelastic collision, since E is not
conserved (friction) !
LINI
LFIN
?0
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Example Two Disks

• Lets use conservation of energy principle

EINI EFIN
1/2 I w02 1/2 (I I) wF2 wF2 1/2 w02 wF
w0 / 21/2
EINI
EFIN
?0
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Example Two Disks

• Using conservation of angular momentum
• LINI LFIN we got a different
  answer !

wF w0 / 21/2
Conservation of energy !
Conservation of momentum !
wF w0 / 2
wF gt wF
Which one is correct ?
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Example Two Disks

• Is the system conservative ?
• Are there any non-conservative forces involved ?
• In order for top disc to turn when in contact
  with the bottom one there has to be friction !
  (non-conservative force !)
• So, we can not use the conservation of energy
  here.
• correct answer wF w0/2
• We can calculate work being done due to this
  friction !

W DE 1/2 Iw02 - 1/2 (II) (w0/2)2 1/2
Iw02 (1 - 2/4) 1/4 I w02 1/8 MR2
w02
This is 1/2 of initial Energy !
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Lecture 23, ACT 2Angular momentum
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Recap of todays lecture

• Chapter 11.1-5,
• Rolling Motion
• Angular Momentum
• For next time Read Ch. 11.1-11.