Angular Momentum
Module 1, Lecture 4
• Objects rotating about a fixed axis
ƒ Angular Momentum
ƒ Vector Cross Product
L = Iω
• Can relate to Newton’s second law
∑τ =
Quick Review:
• How did we calculate momentum?
• Conservation of momentum
• Was momentum a vector or a scalar?
• Total angular momentum remains __________ if
net external torque acting on system is ______.
• When did Conservation of Momentum apply?
EF 152 Fall, 2008 Lecture 1-4
dL
dt
1
EF 152 Fall, 2008 Lecture 1-4
Example: Clutch
Example: Circular platform, part I
Given: In a clutch, a disc with I = 1.2 kg-m2 is brought
into contact with a spinning flywheel having I = 1.8 kg-m2
Hurricane Katrina
and ω = 72 rad/s.
Required: Angular velocity after the disc is moving with
the flywheel.
Given: A 60 kg person stands at the edge of a 6.0 m diameter
circular platform with I = 1800 kg-m2 that is initially at rest.
Hurricane Katrina
Required: Angular velocity of the platform if person runs at:
a. 4.2 m/s with respect to the earth
b. 4.2 m/s with respect to the platform
A.
B.
C.
D.
2
28.8 rad/s
43.2 rad/s
72.0 rad/s
108 rad/s
EF 152 Fall, 2008 Lecture 1-4
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EF 152 Fall, 2008 Lecture 1-4
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Example: Circular platform, part II
R = P ×Q
Vector Cross-Product
You are standing at the edge of a rotating circular platform.
You walk towards the center. What happens? Hurricane Katrina
a. platform slows down
b. platform speeds up
c. rotation speed is unchanged
Magnitude
R
Direction: ___________
to plane containing P and
Q; determined using
______________ rule
Hurricane Katrina
Q
P
Not commutative:
Q× P ≠ P ×Q
EF 152 Fall, 2008 Lecture 1-4
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Vector Cross Product: Applications
EF 152 Fall, 2008 Lecture 1-4
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Vector Cross Product: Basic Cases
• General 3-dimensional motion
a = α × r + ω × (ω × r )
v=
P
θ =0
o
• Torque
Q
τ =
θ = 90o
• Angular momentum
P
P ×Q =
P ×Q =
Q
• General definition of angular momentum of a
particle about a point O is:
L=
iˆ × iˆ =
ˆj × kˆ =
iˆ × ˆj =
ˆj × iˆ =
^
k
^
i
EF 152 Fall, 2008 Lecture 1-4
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EF 152 Fall, 2008 Lecture 1-4
^
j
8
Vector Cross Product: General Solution
iˆ
px
ˆj
py
kˆ
pz
qx
qy
qz
EF 152 Fall, 2008 Lecture 1-4
P ×Q
Torque (Moment): Definition, units, direction
Handle:
Force:
(3iˆ − 1 ˆj + 1kˆ) ft
(− 12iˆ + 7 ˆj + 18kˆ)lb
Determine torque about origin
9
z
EF 152 Fall, 2008 Lecture 1-4
F
x
y
10