Physics 201: Lecture 20, Pg 1 Lecture 20 Goals: Revisit vector cross product Introduce angular...

Physics 201: Lecture 20, Pg 1

Lecture 20Goals:Goals:

• Revisit vector cross product

• Introduce angular momentum

• Discuss conservation of angular momentum


Linear Momentum and Angular Momentum comparison

Newton’s 2nd Law (inertia times acceleration):

Units of angular momentum are kg m2/s

Momentum: (inertia time velocity)

vp

m

? IL

FrI a

mF

And the relationship to linear momentum?

IL

prI

L


Question: How does the angular momentum vary? A 10 kg cart is moving with a speed of 5 m/s

following the path shown How do the magnitudes of the angular momenta at

positions shown compare ( is changing)?

dtdIIpr

L m/s j0.5 m i0.1 11 vr

Axis

r1

r2

r3

/sm kg j50i0.1 211 pr

/sm kg k50

/sm kg 50)ji(

2

211

pr


Question: How does the angular momentum vary?

A 10 kg cart is moving with a speed of 5 m/s. How do the magnitudes of the angular momenta at

positions shown compare?

Ipr L

m/s j3.4i5.2 m j8.0i8.0 22 vr

Axis

r1

r2

r3

) j43i25() j8.0i8.0(22 pr

/sm kg k54

/sm kg )ij( 20)ji(34

2

2

k54)4530sin(|||| 2222 prpr


Question: How does the angular momentum vary?

A 10 kg cart is moving with a speed of 5 m/s How do the magnitudes of the angular momenta at

positions shown compare?

Ipr L

m/s i0.5 m j5.1 33 vr

Axis

r1

r2

r3

/sm kg k7590sin|||| 23333 prpr


Angular momentum question

Does a hockey puck sliding straight along the ice have angular momentum?

A. Yes

B. No

C. Not Sure

D. It depends on the mass

E. It depends on whether the puck is moving in a straight line or around a curved path

“axis of rotation”

v


Angular momentum comparison

Three pucks of equal mass are sliding smoothly along on a horizontal surface. There trajectories are given by the respective dashed lines.

How do the magnitudes of their angular momenta compare?

L = I = I v / r = m v r I = m r2

L= m r v

A. LA < LB < LC

B. LA = LB = 0 < LC

C. 0 < LA= LB < LC

D. LA = LB = LC

rv/2

Axis of rotation

rvv

A

B

C


Putting it all together

prL

)()( vmrprL dtd

dtd

dtd

vmrvmrL dtd

dtd

dtd

)(

FrLdtd

0

Fr

dtLd


Conservation of momentum

Fr

dtLd

0ext

dtLd

constantL


Example: A Two Disk Collision

A disk of mass M and radius R rotates around the z axis with angular velocity 0. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F.

0

z

F

z


Example: Two Disks A disk of mass M and radius R rotates around the z axis

with initial angular velocity 0. A second identical disk, at rest, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F.

0

z

F

z

No External Torque so Lz is constant

Li = Lf I i = I f ½ mR2 0 + 0 = ½ 2mR2 f


Another Demonstration:Conservation of Angular Momentum

Figure Skating :

A

z

B

z

Arm Arm

IA IB

A B

LA = LB

No External Torque so Lz is constant even if internal work done.


Demonstration:Conservation of Angular Momentum

Figure Skating :

A

z

B

z

Arm Arm

IA < IB

A > B

½ IAA2

> ½ IB B2 (work needs to be done)

IAA= LA = LB = IBB

No External Torque so Lz is constant even if internal work done.


Example: Bullet hitting stick

A uniform stick of mass M and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. The initial speed of the bullet is v1, and the final speed is v2.

What is the angular speed F of the stick after the collision? (Ignore gravity)

v1 v2

MF

before after

mD

D/4


Example: Bullet hitting stick What is the angular speed F of the stick after the

collision? (Ignore gravity). Process: (1) Define system (2) Identify Conditions

(1) System: bullet and stick (No Ext. torque, L is constant)

(2) Momentum is conserved (Istick = I = MD2/12 )

Linit = Lbullet + Lstick = m v1 D/4 + 0 = Lfinal = m v2 D/4 + I f

v1 v2

MF

before after

mD

D/4


For Tuesday

Statics: Read Chapter 12.1 to 12.3

Physics 201: Lecture 20, Pg 1 Lecture 20 Goals: Revisit vector cross product Introduce angular...

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