Angular Momentum

Definition: prL

(for a particle)

iii prLL

(for a system of particles)

Units: sJsmNsmkgL /][ 2

dt

Ld

Frvmv

dt

pdrp

dt

rdpr

dt

d

dt

Ld

compare with:dt

pdF

Angular momentum and second Newton's law

sinrpL depends on the choice of origin, since it involves the position of vector of the particle relative to the origin

L

Conservation of angular momentum

ABF

BAF

BAAB FF

ABr

BAr

sinrBABAABAB rr sinsin

BAAB Fr

sinrF

0 0 If fiext LLdt

Ld

0 BAABBA

dt

Ld

dt

Ld

compare with conservation of linear momentum

When the external torque acting on a system is zero, the total angular momentum is conserved

Angular momentum for a rigid body rotation around a symmetry axis

Conservation of angular momentum for a rigid body rotation around a symmetry axis

IrmLLL

mrRmrLLL

LL

RmrRmvRpRpL

iiizz

zz

xx

2

221

21

sinsin

0

sin

IL ILIL zz

0 If extffii II

I

LIK rot 2

22

21 Kinetic energy:

I

dt

dI

dt

Ldext

1R

2R

1r

2r

1L

2L

z

Noether’s Theorem

1882-1935

According to Noether’s Theorem, for each conserved there exists a symmetry of the laws of physicswhich “generates” it. Emmy Amalie Noether

So far we have learned about threeconserved quantities

EnergyMomentum Angular Momentum

Conserved quantities are precious in theoretical physics because…

Example: A person of mass 75 kg stands at the center of a rotating marry-go-round platform of radius 2.0 m and moment of inertia 900 kgm2. The platform rotates without friction with angular velocity 2.0 rad/s.The person walks radially to the edge of the platform. Calculate the angular velocity when the person reaches the edge.

m = 75 kgr = 2.0 mIp = 900 kgm2

ω1 = 2.0 rad/sω2 - ?

L1 = L2

IL I1 = Ip

I2 = Ip + mr2

Ip ω1 = (Ip + mr2 )ω2

122 mrI

I

p

p

sradsradsradmkgmkg

mkg/5.1/0.2

1200

900/0.2

0.275900

90022

2

2

Dv1 v2

ω

D/4

Example: 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 its final speed is v2. What is the angular speed ω of the stick after the collision?

1 bef ore 4z

DL mv

2 af ter 4 zz

DL mv I

1 24 4 z

D Dmv mv I

2 1

4z

mD v v

I

2 13 v vmM D

2121 MDI

External forces: weight of the stick and force on the stick by the pivoting axle produce no torque. Weight of the bullet is negligible. No external torque → Angular momentum conserved

1 bef ore 4z

DL mv

af ter 4 zz

DL mv I

11

2 2

1241 1 3 416 12

z

Dmv vm

m M DmD MD

2 2

4 4z z z

D Dm I m I

4z

Dv

Dv1

ω’

D/4

Total linear momentum totalp

is not conserved, because

net,ext 0F

Example: What if instead of a stick we have a thicker block so the bullet embeds itself in it?

Example: A student sits on a rotating stool and holds a rotating horizontal bicycle wheel by a rod through its axis. The stool is initially at rest. The student flips the axis of rotation of the wheel by 180°. What happens to the stool?

A. It rotates in the same direction as the wheel after the flip.

B. It rotates in the same direction as the wheel before the flip

C. Nothing! Why would it rotate at all?

LwLw

Ls+

s

Ltotal

Example: A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater center-of-mass speed?

1) case (a)1) case (a)2) case (b)2) case (b)3) no difference3) no difference4) It depends on the rotational 4) It depends on the rotational inertia of the dumbbell.inertia of the dumbbell.

In which case does the dumbbell acquire the greater energy?

Example: A spherical shell rotates about an axis through its center of mass. It has an initial radius Ri and angular speed ωi. By applying a radial force, we can cause the sphere to collapse to Rf = Ri/3. What is the ratio of the final and the initial angular speed, ωf/ωi ?

i i iL I

ff fL I i i ffI I 2 2

f i i i2 2

i ff f

9 I MR R

I MR R