Fri. 11.1 Angular Momentum Quiz 10 RE 11.a; HW10: 13*, 21, 30, “39”
Mon.
Tues.
11.2-.3, (.12) Rotational + Translational
RE 11.b
EP10
Mon.
Tues.
Wed.
Lab
Fri.
11.4-.6, (.13) Angular Momentum & Torque
11.7 - .9, (.11) Torque
L11 Rotation Course Evals
11.10 Quantization, Quiz 11
RE 11.c
EP11
RE 11.d
RE 11.e
Mon.
Review for Final (1-11) HW11: Ch 11 Pr’s 39, 57, 64, 74, 78 &
Practice Exam
Introducing Angular Momentum The measure of motion about a point
sun
Earthsunr
towardp
aroundp
p
sin90cos ppparound
sinEarthsunEarthsunaround rprpL
sun
Earthsunr
p
Only ‘around’ component of momentum counts
sinEarthsunrr
rprprpL EarthsunEarthsunaround sin
Magnitude
Using Angular Momentum The measure of motion about a point
What is the magnitude of the angular momentum about location K, for the object shown
below? The magnitude of the object's momentum |
|= 7 kg·m/s, the distance |
|= 0.6 m,
and the angle = 150 °
Magnitude
sinrprprpL around
Using Angular Momentum The measure of motion about a point
Determine the magnitude of the translational angular momentum of the particle at
location O relative to each point: A, B, C, D, E, F, G, and H.
sinrprprpL around
AL
BL
CL
DL
b=
9m
h =
12m
w= 11m
EL
FL
GL
HL
|p| = 50 kg · m/s
Magnitude
Using Angular Momentum The measure of motion about a point
Direction
x
y
z
.
(tip of z-axis arrow pointing at you)
p
r
p
p
p
The one direction momentum and position vectors never point is
z – Axis of rotation
But that’s also true for p
r
p
p
p
Distinguish with Right Hand Rule
Orient Right hand so fingers
curl with motion, then thump
points in conventional
direction of angular
momentum
Using Angular Momentum The measure of motion about a point
Direction
x
y
z.
(tip of z-axis arrow pointing at you)
2p
1p
1r
Distinguish with Right Hand Rule
Orient Right hand so fingers
curl with motion, then thump
points in conventional
direction of angular
momentum
Example
1
A
2r
What are the directions of Angular Momentum for particle 1 about point A and particle 2
about point A
a)
=
=
b)
=
=
c)
=
=
d)
=
=
A comet orbits the Sun, in the xy plane. Its
momentum is shown by the red arrow.
What is the direction of the comet's
angular momentum about the Sun?
1) +x
2) –x
3) +y
4) –y
5) +z
6) –z
7) toward the sun
8) away from the sun y
x
z (out of the page)
Using Angular Momentum The measure of motion about a point
Determine the direction of the translational angular momentum of the particle at location
O relative to each point: A, B, C, D, E, F, G, and H.
AL
BL
CL
DL
b=
9m
h =
12m
w= 11m
EL
FL
GL
HL
|p| = 50 kg · m/s
Direction Distinguish with Right Hand Rule
A ball falls straight down in the xy plane. Its
momentum is shown by the red arrow.
What is the direction of the ball's angular
momentum about location A?
1) +x
2) –x
3) +y
4) –y
5) +z
6) –z
7) zero magnitude
y
x
z (out of the page)
A 4m
10 kg m/s
1) 10 kg m2/s
3) 40 kg m2/s
5) 0
Given these values, what is the magnitude
of the ball’s angular momentum about A?
Using Angular Momentum The measure of motion about a point
Magnitude and Direction
x
y
z.
=
=
=
Similarly for position and momentum in the y-z
xy
z
.
=
=
=
and for position and momentum in the x-z =
Most General Expression =
=
Cross Product
09_Cross-product.py
Using Angular Momentum The measure of motion about a point
Magnitude and Direction
Example: say you have a mass that, at some instant, has linear momentum
and is from some point A. What is its
angular momentum about this point?
=
=
smkgp /0,2,4 mrA 0,3,5
What is the direction of
< 0, 0, 3> x < 0, 4, 0>?
1) +x
2) –x
3) +y
4) –y
5) +z
6) –z
7) zero magnitude
What is the direction of
< 0, 4, 0> x < 0, 0, 3>?
What is the direction of
< 0, 0, 6> x < 0, 0, -3>?
If an object is traveling at a constant speed in a vertical circle, how does the object's
angular momentum change as the object goes from the top of the circle to the bottom
of the circle?
1.
increases
2.
decreases
3.
stays the same but the direction of
changes
4. The direction and magnitude of
remain the same
Using Angular Momentum The measure of motion about a point
Effect of a radial force (like gravity or electric) sun
sunEarthr
p
sunEarthF
dt
pdrp
dt
rdpr
dt
dL
dt
d ESEE
SEESESE
sunEarthSEEESE FrpvLdt
d
Parallel Parallel
= 0
constantSEL
Orbit noncircular.py
Kepler and Planetary Orbits:
Sweeping our equal area in equal time: If t1= t2, then A1 = A2
sin221
2 mvrm
tA
t1 A1
A2
A=A=b*h=
r
A2= ½
2tv
r
htv 2 sin2rtv sin2rtv2tvb=
h ri h
2tv 2tv
Some mathematical manipulation…
Some Geometric manipulation
sin221
prm
tL
m
trp
m
t 221
221
Since L is constant (and m is constant), A is the same for the same time interval
Relating Energy, Radius and Angular Momentum in Circular Orbit
rmvrpLorbitAngular Momentum: (r & p perpendicular)
Kinetic energy:
Kinetic and Gravitational Potential Energy: UKE
2
2
2
2
1
2mr
LmvK orbit
Potential energy: r
MmGU
Gravitational Force and Circular motion: r
vmFnet
2
Kr
22r
MmG
r
U
U K2
r
MmG 2
2
22
mr
Lorbit
mGMm
Lr
2
KUK
2
2
4mr
LUK orbit
Later in chapter will apply same reasoning to electric interaction
Fri. 11.1 Angular Momentum Quiz 10 RE 11.a; HW10: 13*, 21, 30, “39”
Mon.
Tues.
11.2-.3, (.12) Rotational + Translational
RE 11.b
EP10
Mon.
Tues.
Wed.
Lab
Fri.
11.4-.6, (.13) Angular Momentum & Torque
11.7 - .9, (.11) Torque
L11 Rotation Course Evals
11.10 Quantization, Quiz 11
RE 11.c
EP11
RE 11.d
RE 11.e
Mon.
Review for Final (1-11) HW11: Ch 11 Pr’s 39, 57, 64, 74, 78 &
Practice Exam