Download Rotation, angular motion & angular momentom

-Conservation of angular momentum
-Relation between conservation laws &
symmetries
Lect 4
Rotation
Rotation
d1
d2
The ants moved different
distances: d1 is less than d2
Rotation
q
q1
q2
Both ants moved the
Same angle: q1 = q2 (=q)
Angle is a simpler quantity than distance
for describing rotational motion
Angular vs “linear” quantities
Linear quantity
distance
velocity
=
change in d
elapsed time
symb.
d
v
Angular quantity
symb.
angle
angular vel.
change in q
= elapsed
time
q
w
Angular vs “linear” quantities
Linear quantity
distance
velocity
acceleration
=
change in v
elapsed time
symb.
d
v
a
Angular quantity
symb.
angle
q
angular vel. w
angular accel. a
change in w
= elapsed
time
Angular vs “linear” quantities
Linear quantity
symb.
distance
velocity
acceleration
mass
d
v
a
m
resistance to change in the
state of (linear) motion
moment
arm
x
Angular quantity
symb.
angle
q
angular vel. w
angular accel. a
Moment of Inertia
I (= mr2)
resistance to change in the
state of angular motion
M
Moment of inertia
= mass x (moment-arm)2
Moment of inertial
M
M
x
I  Mr2
r
I=small
r
r = dist from axis of rotation
I=large
(same M)
easy to turn
harder to turn
Moment of inertia
Angular vs “linear” quantities
Linear quantity
distance
velocity
acceleration
mass
Force
symb.
Angular quantity
symb.
d
angle
q
v
angular vel. w
a
angular accel. a
m
moment of inertia I
F (=ma)
torque
t (=I a)
Sameforce;
force;
Same
bigger
torque
even
bigger
torque
torque = force x moment-arm
Teeter-Totter
His weight
produces a
larger torque
F
Forces are
the same..
but Boy’s moment-arm is larger..
F
Torque = force x moment-arm
t=Fxd
F
“Moment Arm” = d
“Line of action”
Opening a door
d
small
d
large
F
difficult
F
easy
Angular vs “linear” quantities
Linear quantity
symb.
distance
velocity
acceleration
mass
Force
momentum
d
v
a
m
F (=ma)
p (=mv)
p
x
L= p x moment-arm = Iw
Angular momentum
is conserved: L=const
Angular quantity
angle
symb.
q
angular vel. w
angular accel. a
moment of inertia I
torque
t (=I a)
angular mom. L (=I w)
Iw = Iw
Conservation of angular
momentum
w
I
Iw
Iw
High Diver
Iw
w
I
Iw
Conservation of angular
momentum
Iw
w
I
Conservation of angular
momentum
Angular momentum is a vector
Right-hand
rule
Torque is also a vector
example:
pivot
point
another
right-hand rule
F
t is out of
the screen
Thumb in
t direction
F
wrist by
pivot point
Fingers in
F direction
Conservation of angular
momentum
L has no vertical
component
No torques possible
Around vertical axis
vertical component of L= const
Girl spins:
net vertical
component of L
still = 0
Turning bicycle
L
These compensate
Spinning wheel
t
F
wheel precesses
away from viewer
Angular vs “linear” quantities
Linear quantity
symb.
distance
velocity
acceleration
mass
d
v
a
m
Force
momentum
F (=ma)
p (=mv)
kinetic energy
½ mv2
I
w
V
Angular quantity
angle
symb.
q
angular vel. w
angular accel. a
moment of inertia I
torque
t (=I a)
angular mom. L (=I w)
rotational k.e. ½ I w2
KEtot = ½ mV2 + ½ Iw2
Hoop disk sphere race
Hoop disk sphere race
hoop
I
I
disk
I
sphere
Hoop disk sphere race
hoop
KE = ½ mv2 + ½
v
KE = ½ m
disk
I
2
+ ½
v
KE = ½ m
sphere
w2
Iw
2
2+½ Iw
2
Hoop disk sphere race
Every sphere beats every disk
& every disk beats every hoop
Kepler’s 3 laws of planetary
motion
Johannes Kepler
1571-1630
• Orbits are elipses with
Sun at a focus
• Equal areas in equal time
• Period2  r3
2
2
(
2

r
)
(
2

r
)
1
2
v 


2
3
P
r
r
1

r
Basis of Kepler’s laws
Laws 1 & 3 are consequences of the nature of the gravitational force
F G
M sunM planet
r2
The 2nd law is a consequence of conservation of angular momentum
v2
r2
A1=r1v1T
L1=Mr1v1
r1
v1
A2=r2v2T
L2=Mr2v2
L1=L2 v1r1 =v2r2
Symmetry and Conservation laws
Lect 4a
Hiroshige 1797-1858
36 views of Fuji
View 4
View 14
Hokusai 1760-1849
24 views of Fuji
View 18
View 20
Temple of heaven (Beijing)
Snowflakes
600
Kaleidoscope
Start with a random pattern
Include a reflection
Use mirrors
to repeat it
over & over
The attraction
is all in the
symmetry
Rotational symmetry
qq2 1
No matter which way I turn a perfect sphere
It looks identical
Space translation symmetry
Mid-west corn field
Timetranslation
symmetry
in music
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
Kepler: planetary orbits are
ellipses; not perfect circles
Galileo:There are mountains
on the Moon; it is not a
perfect sphere!
Critique of Newton’s 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.
Newton’s 2nd Law
F = ma
?????
But what is F?
whatever gives you the
correct value for
ma
Is this a law of nature?
or a definition of force?
But Newton’s laws led us to
discover Conservation Laws!
• Conservation of
Momentum
• Conservation of
Energy
• Conservation of
Angular Momentum
Newton’s 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
Newton’s laws have time-translation symmetry
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
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.
Newton’s laws have space-translation symmetry
rotational symmetry
F=ma
F
Same rule for
all directions
a
(no “preferred” directions
in space.)
a
F
Newton’s laws have
rotation symmetry
Symmetry recovered
Symmetry resides in the laws of nature,
not necessarily in the solutions to these laws.
Emmy Noether
Conserved
Symmetry:
Conservation
quantities:
something
laws are
stay
the
same
that
stays
consequences
throughout
the
same a
of symmetries
process a
throughout
process
1882 - 1935
Symmetries Conservation laws
Conservation law
Symmetry

Angular momentum
Space translation

Momentum
Time translation

Energy
Rotation
Noether’s
discovery:
Conservation laws are a consequence
of the simple and elegant
properties of space and time!
Content of Newton’s laws is in their
symmetry properties