-Conservation of angular momentum -Relation between conservation laws

1
-Conservation of angular momentum-Relation
between conservation laws symmetries

• Lect 4

2
Rotation
3
Rotation
d2
d1
The ants moved different distances d1 is less
than d2
4
Rotation
q
q2
q1
Both ants moved the Same angle q1 q2 (q)
Angle is a simpler quantity than distance for
describing rotational motion
5
Angular vs linear quantities
Linear quantity symb. Angular quantity
symb.
distance d angle q

velocity v
angular vel. w
change in d elapsed time

change in q elapsed time

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Angular vs linear quantities
Linear quantity symb. Angular quantity
symb.
distance d angle q

velocity v
angular vel. w
acceleration a
angular accel. a
change in w elapsed time
change in v elapsed time


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Angular vs linear quantities
Linear quantity symb. Angular quantity
symb.
distance d angle q

velocity v
angular vel. w
acceleration a
angular accel. a
mass m
Moment of Inertia I ( mr2)
resistance to change in the state of (linear)
motion
resistance to change in the state of angular
motion
moment arm
M
Moment of inertia mass x (moment-arm)2
x
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Moment of inertial
M
M
x
I ? Mr2
r
r
r dist from axis of rotation
Ismall
Ilarge (same M)
easy to turn
harder to turn
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Moment of inertia
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Angular vs linear quantities
Linear quantity symb. Angular quantity
symb.
distance d angle q

velocity v
angular vel. w
acceleration a
angular accel. a
mass m
moment of inertia I
Force F (ma)
torque t (I a)
Same force bigger torque
Same force even bigger torque
torque force x moment-arm
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Teeter-Totter
His weight produces a larger torque
F
Forces are the same..
but Boys moment-arm is larger..
F
12
Torque force x moment-arm
t F x d
F
Moment Arm d
Line of action
13
Opening a door
d small
d large
F
F
difficult
easy
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Angular vs linear quantities
Linear quantity symb. Angular quantity
symb.
distance d angle q

velocity v
angular vel. w
acceleration a
angular accel. a
mass m
moment of inertia I
torque t (I a)
Force F (ma)
angular mom. L (I w)
momentum p (mv)
Iw Iw
p
x
L p x moment-arm Iw
Angular momentum is conserved Lconst
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Conservation of angular momentum
Iw
Iw
Iw
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High Diver
Iw
Iw
Iw
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Conservation of angular momentum
Iw
Iw
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Conservation of angular momentum
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Angular momentum is a vector
Right-hand rule
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Torque is also a vector
example
pivot point
another right-hand rule
F
t is out of the screen
Thumb in t direction
wrist by pivot point
F
Fingers in F direction
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Conservation of angular momentum
Girl spins net vertical component of L still 0
L has no vertical component
No torques possible Around vertical
axis? vertical component of L const
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Turning bicycle
These compensate
L
L
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Spinning wheel
t
wheel precesses away from viewer
F
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Angular vs linear quantities
Linear quantity symb. Angular quantity
symb.
distance d angle q

velocity v
angular vel. w
acceleration a
angular accel. a
mass m
moment of inertia I
torque t (I a)
Force F (ma)
momentum p (mv)
angular mom. L (I w)
kinetic energy ½ mv2
rotational k.e. ½ I w2
w
I
V
KEtot ½ mV2 ½ Iw2
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Hoop disk sphere race
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Hoop disk sphere race
I
hoop
I
disk
I
sphere
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Hoop disk sphere race
KE ½ mv2 ½ Iw2
I
hoop
KE ½ mv2 ½ Iw2
I
disk
KE ½ mv2½ Iw2
I
sphere
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Hoop disk sphere race

• Every sphere beats every disk
• every disk beats every hoop

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Keplers 3 laws of planetary motion
Johannes Kepler 1571-1630

• Orbits are elipses with Sun at a focus
• Equal areas in equal time
• Period2 ? r3

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Basis of Keplers laws
Laws 1 3 are consequences of the nature of the
gravitational force The 2nd law is a
consequence of conservation of angular momentum

v2
r2
A2r2v2T
A1r1v1T
r1
L2Mr2v2
L1Mr1v1
L1L2 ?v1r1 v2r2
v1
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Symmetry and Conservation laws

• Lect 4a

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Hiroshige 1797-1858
36 views of Fuji
View 4
View 14
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Hokusai 1760-1849
24 views of Fuji
View 18
View 20
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Temple of heaven (Beijing)
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Snowflakes
600
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Kaleidoscope
Start with a random pattern
Include a reflection
rotate by 450
The attraction is all in the symmetry
Use mirrors to repeat it over over
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Rotational symmetry
q1
q2
No matter which way I turn a perfect sphere It
looks identical
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Space translation symmetry
Mid-west corn field
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Time-translation symmetry in music
repeat
repeat again
again
again
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Prior to Kepler, Galileo, etc

• God is perfect, therefore nature must be
  perfectly symmetric
• Planetary orbits must be perfect circles
• Celestial objects must be perfect spheres

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Kepler planetary orbits are ellipses not
perfect circles
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GalileoThere are mountains on the Moon it is
not a perfect sphere!
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Critique of Newtons Laws
Law of Inertia (1st Law) only works in
inertial reference frames.
Circular Logic!!
What is an inertial reference frame?
a frame where the law of inertia works.
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Newtons 2nd Law
F m a
?????
But what is F?
whatever gives you the correct value for m a
Is this a law of nature? or a definition of force?
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But Newtons laws led us to discover Conservation
Laws!

• Conservation of Momentum
• Conservation of Energy
• Conservation of Angular Momentum

These are fundamental (At least we think so.)
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Newtons laws implicitly assume that they are
valid for all times in the past, present future
Processes that we see occurring in these distant
Galaxies actually happened billions of years ago
Newtons laws have time-translation symmetry
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The Bible agrees that nature is time-translation
symmetric
Ecclesiates 1.9
The thing that hath been, it is that which shall
be and that which is done is that which shall
be done and there is no new thing under the
sun
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Newton believed that his laws apply equally well
everywhere in the Universe
Newton realized that the same laws that cause
apples to fall from trees here on Earth, apply to
planets billions of miles away from Earth.
Newtons laws have space-translation symmetry
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rotational symmetry
F m a
a
F
Same rule for all directions (no preferred
directions in space.)
a
Newtons laws have rotation symmetry
F
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Symmetry recovered

• Symmetry resides in the laws of nature, not
  necessarily in the solutions to these laws.

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Emmy Noether
Conserved quantities stay the same throughout
a process
Symmetry something that stays the same
throughout a process
Conservation laws are consequences of symmetries
1882 - 1935
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Symmetries ??Conservation laws
Conservation law
Symmetry
Angular momentum
Rotation
??
Space translation
??
Momentum
Time translation
??
Energy
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Noethers discovery

• Conservation laws are a consequence of the simple
  and elegant properties of space and time!
• Content of Newtons laws is in their symmetry
  properties