11. Rotational Vectors & Angular Momentum
1.
2.
3.
4.
5.
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 with
period ~26,000 yr.
Importance of rotation:
• Earth’s rotational axis  seasons.
• Angular momentum of protons in living tissues  MRI
• Rotating air  tornadoes.
• Rotating wheel  stabilizes bicycle.
11.1. Angular Velocity & Acceleration Vectors
Right-hand rule
Angular acceleration vector:
α  lim
t  0
 //   
ω
dω

t
dt
 //   
 change
direction
11.2. Torque & the Vector Cross Product
τr
τF
  r F sin 
τˆ  r
τˆ  F
Right hand rule
τ  r F
cross product
Cross Product
Cross product C of vectors A & B:
C  A B
C  A B sin 
Ĉ
Given by right-hand rule,
Dot product C of vectors A & B:
= area of A-B
parallelogram .
is a vector  A-B plane.
C  A  B  A B cos 
Properties of cross product :
1. Distributive
A   B  C  A  B  A  C
2. Anti-commutative
A  B  B  A

3. NOT associative
A   A  B
 A  A  B  0
is a vector in the A-B plane and 
A.
A A  0
xˆ
A  B  Ax
Bx
yˆ
Ay
By
zˆ
Az
Bz
  Ay Bz  Az B y  xˆ   Az Bx  Ax Bz  yˆ   Ax B y  Ay Bx  zˆ
xˆ
A  B  Ax
Bx
yˆ
Ay
By
zˆ
0
0
  Ax B y  Ay Bx  zˆ
GOT IT? 11.1.
Which numbered torque vector goes with each pair of force-radius vectors?
Neglect magnitudes.
3
2
1
Note: Wolfson gave 6 as the answer to (b).
4
11.3. Angular Momentum
Linear momentum:
Angular momentum:
pmv
L  rp  m r v
Iω
 I ω
particle
rigid body with axis of rotation
along principal axis
general case, I a tensor.
 L &  can have different directions.
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.
L  m rv
 m r v kˆ
 m r 2  kˆ
 m r2 ω
Iω
I  m r2
Torque & Angular Momentum
L   Li
System of particles:
  ri  pi
i

d Li
dL
 dr
dp 

   i  pi  ri  i 
dt
dt
dt 
i
i  dt
  ri 
i
d pi
dt
  ri  Fi
i

i
dL
τ
dt
d ri
 pi  v i  m v i  0
dt
  τi
i
rotational analog of 2nd law.
11.4. Conservation of Angular Momentum
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.
Before collapse:
After collapse:
L  L0
2
m r02 0
5
2
L  m r2 
5
L0 


20 10 km   1
r

0  
rev
/
day


2
r

 6 km   45
2
0
2
3
2
 2.5 105 rev / day  3 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?
Boy
Girl
Lb = 0   L = 0
I = Im + Ib
 
 L = Lg
I = Im + Ig
?
Making the Connection
A merry-go-round of radius R = 1.3 m has rotational inertia I = 240 kg m2
& is rotating freely at 1 = 11 rpm.
A boy of mass mb = 28 kg runs straight toward the center at vb = 2.5 m/s & leaps on.
At the same time, a girl of mass mg = 32 kg, running tangentially at speed vg = 3.7 m/s
in the same direction as the merry-go-round also leaps on.
Find the new angular speed 2 once both children are seated on the rim.
L0  I 1  mg R vg
Before :
After :
L  I 2  mb R2 2  mg R2 2
L  L0
 2 
I 1  mg R vg
I   mb  mg  R 2
 1

2
240
kg
m
11
rpm

32
kg
1.3
m
3.7
m
/
s
rev / rad   60 s / min 









 2

 12 rpm
2 
2
2
 240 kg m    28 kg  32 kg 1.3 m 
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, or
remain the same?
What about your direction of rotation?
(a) CW to keep L = 0.
(b) Same, CCW.
11.5. Gyroscopes & Precession
Gyroscope: spinning object whose rotational axis is fixed in space.
External torque required to change axis of rotation
 Higher spin rate  larger L  harder to change orientation
Usage:
• Navigation
• Missile & submarine guidance.
• Cruise ships stabilization.
• Space-based telescope like Hubble.
Precession
Precession: Continuous change of direction of rotation axis,
which traces out a circle.
dL
τ
dt
r L
 r  Fg

L  L
Rate of Precession
Precession occurs if   L.
dL

 τ  r    mg zˆ  
zˆ  Lˆ
dt
sin 
  rmg sin 
r̂ Lˆ
z
 L precesses CCW around z.
For L constant:

ˆ
dL

ˆ

zˆ  L
d t L sin 

x
ˆ   sin  cos  , sin  sin  , cos  
L
ˆ    sin  sin  , sin  cos  , 0
zˆ  L
   const

    sin  sin  ,   sin  cos  , 0     sin  , cos  , 0 
L
Rate of precession :
 

L sin 
y
Erath’s Precession
Earth’s precession (period ~ 26,000 y )
The equatorial bulge is highly exaggerated.
Perfect sphere
=0
=0
Oblate spheroid
 <

The equatorial bulge is highly exaggerated.
GOT IT? 11.3.
You push horizontally at right angles to the shaft of a spinning gyroscope.
Does the shaft move
(a)upward,
(b)downward,
(c)in the direction you push,
(d)opposite the direction you push?
Looking down at bike.
Bicycling
Direction of
bike’s motion
L+  t
wheel
L

τ  r  Fg
Wheel
turns
Biker leans
points into paper
L //   wheel turns to biker’s left