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