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Physics 211: Lecture 22, Pg 1
Physics 211: Lecture 22Physics 211: Lecture 22
Today’s AgendaToday’s Agenda Angular Momentum:
Definitions & Derivations What does it mean?
Rotation about a fixed axis L = I Example: Two disks Student on rotating stool
Angular momentum of a freely moving particle Bullet hitting stick Student throwing ball
Original: Mats SelenModified:W.Geist
Physics 211: Lecture 22, Pg 2
Lecture 22, Lecture 22, Act 1Act 1RotationsRotations
A girl is riding on the outside edge of a merry-go-round turning with constant . She holds a ball at rest in her hand and releases it. Viewed from above, which of the paths shown below will the ball follow after she lets it go?
(c)
(b)(a)
(d)
Physics 211: Lecture 22, Pg 3
Lecture 22, Lecture 22, Act 1Act 1 SolutionSolution
Just before release, the velocity of the ball is tangent to the circle it is moving in.
Physics 211: Lecture 22, Pg 4
Lecture 22, Lecture 22, Act 1Act 1 SolutionSolution
After release it keeps going in the same direction since there are no forces acting on it to change this direction.
Physics 211: Lecture 22, Pg 5
Angular Momentum:Angular Momentum:Definitions & DerivationsDefinitions & Derivations
We have shown that for a system of particles
Momentum is conserved if
What is the rotational version of this??
Fp
EXTddt
FEXT 0
L r p
r F The rotational analogue of force F F is torque
Define the rotational analogue of momentum pp to be
angular momentum
p = mv
Physics 211: Lecture 22, Pg 6
Definitions & Derivations...Definitions & Derivations...
First consider the rate of change of LL: ddt
ddt
Lr p
ddt
ddt
ddt
r pr
p rp
v vm
0
So ddt
ddt
Lr
p (so what...?)
Physics 211: Lecture 22, Pg 7
Definitions & Derivations...Definitions & Derivations...
ddt
ddt
Lr
p
Fp
EXTddt
EXTFdtd r
L Recall that
Which finally gives us: EXTddt
L
Analogue of !! Fp
EXTddt
EXT
Physics 211: Lecture 22, Pg 8
What does it mean?What does it mean?
where andEXTddt
L EXT EXT r FL r p
EXTddt
L0 In the absence of external torquesIn the absence of external torques
Total angular momentum is conservedTotal angular momentum is conserved
Physics 211: Lecture 22, Pg 9
i
j
Angular momentum of a rigid bodyAngular momentum of a rigid bodyabout a fixed axis:about a fixed axis:
k̂vrmmi
iiii
iiiii
i vrprL
Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin is the sum of the angular momenta of each particle:
rr1
rr3
rr2
m2
m1
m3
vv2
vv1
vv3
We see that LL is in the z direction.
Using vi = ri , we get
LI
(since ri and vi are
perpendicular)
Analogue of p = mv!!
krmLi
2
iiˆ
Rolling chain
Physics 211: Lecture 22, Pg 10
Angular momentum of a rigid bodyAngular momentum of a rigid bodyabout a fixed axis:about a fixed axis:
In general, for an object rotating about a fixed (z) axis we can write LZ = I
The direction of LZ is given by theright hand rule (same as ).
We will omit the Z subscript for simplicity,and write L = I
LZ I
z
Physics 211: Lecture 22, Pg 11
Example: Two DisksExample: Two Disks
A disk of mass M and radius R rotates around the z axis with angular velocity i. 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.
i
z
f
z
Physics 211: Lecture 22, Pg 12
Example: Two DisksExample: Two Disks
First realize that there are no external torques acting on the two-disk system. Angular momentum will be conserved!
Initially, the total angular momentum is due only to the disk on the bottom:
0
z
ii MRL 211 2
1I
2
1
Physics 211: Lecture 22, Pg 13
Example: Two DisksExample: Two Disks
First realize that there are no external torques acting on the two-disk system. Angular momentum will be conserved!
Finally, the total angular momentum is dueto both disks spinning:
f
z
f2
2211f MRL II21
Physics 211: Lecture 22, Pg 14
Example: Two DisksExample: Two Disks
Since Li = Lf
f
z
f
z
Li Lf
f2
i2 MRMR
2
1
if 2
1 An inelastic collision,
since E is not conserved (friction)!
Wheel rim
drop
Physics 211: Lecture 22, Pg 15
Example: Rotating TableExample: Rotating Table
A student sits on a rotating stool with his arms extended and a weight in each hand. The total moment of inertia is Ii, and he is rotating with angular speed i. He then pulls his hands in toward his body so that the moment of inertia reduces to If. What is his final angular speed f?
i
Ii
f
If
Physics 211: Lecture 22, Pg 16
Example: Rotating Table...Example: Rotating Table...
Again, there are no external torques acting on the student-stool system, so angular momentum will be conserved. Initially: Li = Iii
Finally: Lf = If f f
i
i
f
I
I
i
Ii
f
If
LiLf
Student on stool
Physics 211: Lecture 22, Pg 17
Lecture 22, Lecture 22, Act 2Act 2Angular MomentumAngular Momentum
A student sits on a freely turning stool and rotates with constant angular velocity 1. She pulls her arms in, and due to angular momentum conservation her angular velocity increases to 2. In doing this her kinetic energy:
(a) increases (b) decreases (c) stays the same
1 2
I2 I1
L L
Physics 211: Lecture 22, Pg 18
Lecture 22, Lecture 22, Act 2Act 2 SolutionSolution
I2
LI
21
K2
2 (using L = I)
L is conserved:
I2 < I1 K2 > K1 K increases!
1 2
I2 I1
L L
Physics 211: Lecture 22, Pg 19
Lecture 22, Lecture 22, Act 2Act 2 SolutionSolution
Since the student has to force her arms to move toward her body, she must be doing positive work! The work/kinetic energy theorem states that this will increase the kinetic energy of the system!
1
I1
2
I2
L L
Physics 211: Lecture 22, Pg 20
Angular Momentum of aAngular Momentum of aFreely Moving ParticleFreely Moving Particle
We have defined the angular momentum of a particle about the origin as
This does not demand that the particle is moving in a circle! We will show that this particle has a constant angular
momentum!
y
x
vv
L r p
Physics 211: Lecture 22, Pg 21
Angular Momentum of aAngular Momentum of aFreely Moving Particle...Freely Moving Particle...
Consider a particle of mass m moving with speed v along the line y = -d. What is its angular momentum as measured from the origin (0,0)?
x
vv
md
y
Physics 211: Lecture 22, Pg 22
Angular Momentum of aAngular Momentum of aFreely Moving Particle...Freely Moving Particle...
We need to figure out The magnitude of the angular momentum is:
Since rr and pp are both in the x-y plane, LL will be in the z direction (right hand rule):
y
x
pp=mvvd
r r
approachclosestofcetandisxp
pdsinrpsinrp prL
L pdZ
L r p
Physics 211: Lecture 22, Pg 23
Angular Momentum of aAngular Momentum of aFreely Moving Particle...Freely Moving Particle...
So we see that the direction of LL is along the z axis, and its magnitude is given by LZ = pd = mvd. L is clearly conserved since d is constant (the distance of closest approach of the particle to the origin) and p is
constant (momentum conservation).
y
x
pp
d
Physics 211: Lecture 22, Pg 24
Example: Bullet hitting stickExample: 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
M
F
initial final
mD
D/4
Physics 211: Lecture 22, Pg 25
ACTACT
What is the angular momentum around the pivot of the bullet stick system before the collision?
A) zero
B)(m+M) v1D/4
C) mv1D/4
D) none of the above
v1 v2
M
F
initial final
mD
D/4
approachclosestofcetandisxp
pdsinrpsinrp prL
Physics 211: Lecture 22, Pg 26
Example: Bullet hitting stick...Example: Bullet hitting stick...
Conserve angular momentum around the pivot (z) axis! The total angular momentum before the collision is due
only to the bullet (since the stick is not rotating yet).
v1
D
M
initial
D/4m
4
DmvapproachclosestofcedisxpL 1i )tan(
Physics 211: Lecture 22, Pg 27
ACTACT
What is the angular momentum around the pivot of the bullet - stick system after the collision?
A) zero
B)(m+M) v2D/4
C) mv2D/4 - I ωF
D) mv2D/4 + IωF
v1 v2
M
F
initial final
mD
D/4
Physics 211: Lecture 22, Pg 28
Example: Bullet hitting stick...Example: Bullet hitting stick...
Conserve angular momentum around the pivot (z) axis! The total angular momentum after the collision has
contributions from both the bullet and the stick.
where I is the moment of inertia of the stick about the
pivot.
v2
F
final
D/4
F2f 4
DmvL I
Physics 211: Lecture 22, Pg 29
Example: Bullet hitting stick...Example: Bullet hitting stick...
Set Li = Lf using
v1 v2
M
F
initial final
mD
D/4
I 1
122MD
mvD
mvD
MD F1 22
4 4
1
12 F
m
MDv v
31 2
Physics 211: Lecture 22, Pg 30
Example: Throwing ball from stoolExample: Throwing ball from stool
A student sits on a stool which is free to rotate. The moment of inertia of the student plus the stool is I. She throws a heavy ball of mass M with speed v such that its velocity vector passes a distance d from the axis of rotation. What is the angular speed F of the student-stool
system after she throws the ball?
top view: initial final
d
vM
I I
F
Physics 211: Lecture 22, Pg 31
Example: Throwing ball from stool...Example: Throwing ball from stool...
Conserve angular momentum (since there are no external torques acting on the student-stool system): Li = 0 Lf = 0 = IF - Mvd
top view: initial final
d
vM
I I
F
FMvd
I
Physics 211: Lecture 22, Pg 32
Lecture 22, Lecture 22, Act 3Act 3Angular MomentumAngular Momentum
A student is riding on the outside edge of a merry-go-round rotating about a frictionless pivot. She holds a heavy ball at rest in her hand. If she releases the ball, the angular velocity of the merry-go-round will:
(a) increase (b) decrease (c) stay the same
Physics 211: Lecture 22, Pg 33
Lecture 22, Lecture 22, Act 3Act 3 SolutionSolution
The angular momentum is due to the girl, the merry-go-round and the ball. LNET = LMGR + LGIRL + LBALL
Initial: L mRvR
mvRBALL
I 2
v
R
m
Final: L mvRBALL
vm
same
Physics 211: Lecture 22, Pg 34
Lecture 22, Lecture 22, Act 3Act 3 SolutionSolution
Since LBALL is the same before & after, must stay the same to keep “the rest” of LNET unchanged.
Physics 211: Lecture 22, Pg 35
Lecture 22, Lecture 22, Act 3Act 3Conceptual answerConceptual answer
Since dropping the ball does not cause any forces to act on the merry-go-round, there is no way that this can change the angular velocity. Just like dropping a weight from a level coasting car does not affect the speed of the car.
Physics 211: Lecture 22, Pg 36
Lecture 22, Lecture 22, Act 3Act 3Angular MomentumAngular Momentum
A student is riding on the outside edge of a merry-go-round rotating about a frictionless pivot. She holds a heavy ball at rest in her hand. If she throws the ball perpendicular to the
tangential direction, the angular velocity of the merry-go-round will:
(a) increase (b) decrease (c) stay the same
Physics 211: Lecture 22, Pg 37
Recap of today’s lectureRecap of today’s lecture
Angular Momentum: Definitions & Derivations What does it mean?
Rotation about a fixed axis L = I Example: Two disks Student on rotating stool
Angular momentum of a freely moving particle Bullet hitting stick Student throwing ball
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