11. Rotational Vectors & Angular Momentum
1. Angular Velocity & Acceleration Vectors
2. Torque & the Vector Cross Product
3. Angular Momentum
4. Conservation of Angular Momentum
5. Gyroscopes & Precession
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Angular Momentum and Rotation: Torque, Vector Cross Product, and Conservation, Slides of Physics
Various concepts related to angular momentum and rotation, including angular velocity and acceleration vectors, torque and the vector cross product, and conservation of angular momentum. It also touches upon the importance of rotation in various contexts, such as earth's seasons and mri scans. Examples and problem-solving exercises.
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2012/2013
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11. Rotational Vectors & Angular Momentum
Angular Velocity & Acceleration Vectors
Torque & the Vector Cross Product
Angular Momentum
Conservation of Angular Momentum
Gyroscopes & Precession
Earth isn’t quite round.How does this affect its rotation axis,and what’s this got to do with ice ages?(The deviation from roundness is exaggerated.)
Axis precesses withperiod ~26,000 yr.
11.1. Angular Velocity & Acceleration Vectors
Right-hand rule
Angular acceleration vector:
0
lim^ t^
t
^
ω
α
d d t
ω
change direction
//
//
11.2. Torque & the Vector Cross Product
τ
r
τ
F
τ
r
F
Right hand rule
cross product
sin
r F
τ
r
τ
F
ˆ^
ˆ^
x^
y^
z
x^
y^
z
A
A
A
B
B
B
x
y
z
A
B
^
^
^
^
^
ˆ^
ˆ^
y^
z^
z^
y^
z^
x^
x^
z^
x^
y^
y^
x
A B
A B
A B
A B
A B
A B
x
y
z
ˆ^
ˆ^
x^
y
x^
y
A
A
B
B
x
y
z
A
B
^
^
x^
y^
y^
x
A B
A B
z
GOT IT? 11.1.
Which numbered torque vector goes with each pair of force-radius vectors?Neglect magnitudes.
^1
^2
^3
^4
Note: Wolfson gave
6
as the answer to (b).
Example 11.1. Single Particle
A particle of mass
m
moves CCW at speed
v
around a circle of radius
r
in the
x-y
plane.
Find its angular momentum about the center of the circle,express the answer in terms of its angular velocity.
m
L
r
v^ ˆ
m r v
k 2
m r
k
2
m r
ω
I
ω
2
I^
m r
Torque & Angular Momentum
i
i
L
L
System of particles:
i^
i
i
r
p
i
i
d
d dt
dt
L
L
i^
i
i^
i
i
d
d
dt
dt
^
^
r
p
p
r
i
i i
d dt
p
r
i
i^
i^
i
d
m
dt
r
p
v
v
i^
i
i
r
F
i
i
τ
d dt
L
τ
rotational analog of 2
nd
law.
Example 11.2. Pulsars
A star rotates once every 45 days.It then undergoes supernova explosion, hurling most of its mass into space.The inner core of the star, whose radius is initially 20 Mm,
collapses into a neutron star only 6 km in radius. The rotating neutron star emits regular pulses of radio waves, making it a pulsar.Calculate the pulse rate ( = rotation rate ).Assume core to be a uniform sphere & no external torque.
2
0
0
0
L
m r
(^20)
0
2 r r
Before collapse:
2
L
m r
After collapse:
0
L
L
^
^
2
3
2
km
rev
day
km
^
^
^
5
rev
day
rev
s
Conceptual Example 11.1. Playground
A merry-go-round is rotating freely when a boy runs straight toward the center & leaps on.Later, a girl runs tangentially in the same direction as the merry-go-round also leaps on.Does the merry-go-round’s speed increase, decrease, or stays the same in each case?
L
b^
= 0
L
= 0
I = I
m
+ I
b
Boy
Girl
L
=
L
g
I = I
m
+ I
g
?
Demonstration of Conservation of Angular Momentum
GOT IT? 11.2.
If you step on a non-rotating table holding a non-rotating wheel.(a)if you spin the wheel CCW as viewed from above, which way do you rotate?(b)If you then turn the wheel upside down, will your rotation rate increase, decrease, orremain the same?
What about your direction of rotation?
(a) CW to keep L = 0.(b)
Same, CCW.
Precession
Precession: Continuous change of direction of rotation axis,
which traces out a circle.
d dt
L
τ
g
r
F
r
L
L
L
Rate of Precession
Precession occurs if
L
. ˆ
ˆ
d d t
L
τ
z
L
L
precesses CCW around
z
.
ˆ^
d d t
L
L
z
L
For
L
constant:^
ˆ^
sin
cos
, sin
sin
, cos
L
^
ˆ^
sin
sin
, sin
cos
z
L
const
^
^
^
sin
sin
,^
sin
cos
sin
sin
, sin
cos
^ L
^
^ L
Rate of precession :
x
^
y
z
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