Physics 111: Lecture 22, Pg 1 Physics 111: Lecture 22 Today’s Agenda Angular Momentum: Definitions & Derivations What does it mean? Rotation about a fixed axis L = IExample: Two disks Student on rotating stool Angular momentum of a freely moving particle Bullet hitting stick Student throwing ball
Physics 111: Lecture 22 Today’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. - PowerPoint PPT Presentation
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Physics 111: Lecture 22, Pg 1
Physics 111: Lecture 22
Today’s Agenda Angular Momentum:
Definitions & DerivationsWhat does it mean?
Rotation about a fixed axisL = IExample: Two disksStudent on rotating stool
Angular momentum of a freely moving particleBullet hitting stickStudent throwing ball
Physics 111: Lecture 22, Pg 2
Lecture 22, Act 1Rotations
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 111: Lecture 22, Pg 3
Lecture 22, Act 1 Solution
Just before release, the velocity of the ball is tangent to the circle it is moving in.
Physics 111: Lecture 22, Pg 4
Lecture 22, Act 1 Solution
After release it keeps going in the same direction since there are no forces acting on it to change this direction.
Physics 111: Lecture 22, Pg 5
Angular Momentum:Definitions & Derivations
We have shown that for a system of particles
Momentum is conserved if
What is the rotational version of this??
F pEXT
ddt
FEXT 0
L r p
r F The rotational analogue of force F is torque
Define the rotational analogue of momentum p to be
angular momentum
p = mv
Physics 111: Lecture 22, Pg 6
Definitions & Derivations...
First consider the rate of change of L: ddt
ddt
L r p
ddt
ddt
ddt
r p r p r p
v vm0
So ddt
ddt
L r p (so what...?)
Physics 111: Lecture 22, Pg 7
Definitions & Derivations...
ddt
ddt
L r p
F pEXT
ddt
EXTFdtd rL Recall that
Which finally gives us: EXTddt
L
Analogue of !! F pEXT
ddt
EXT
Physics 111: Lecture 22, Pg 8
What does it mean?
where andEXTddt
L EXT EXT r FL r p
EXTddt
L 0 In the absence of external torques
Total angular momentum is conserved
Physics 111: Lecture 22, Pg 9
i
j
Angular momentum of a rigid bodyabout 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:
r1
r3
r2
m2
m1
m3
v2
v1
v3
We see that L is in the z direction.
Using vi = ri , we get
LI
(since ri and vi are perpendicular)
Analogue of p = mv!!
krmLi
2ii
ˆ
Rolling chain
Physics 111: Lecture 22, Pg 10
Angular momentum of a rigid bodyabout 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 111: Lecture 22, Pg 11
Example: 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 111: Lecture 22, Pg 12
Example: 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
f2
11i MR21L I
2
1
Physics 111: Lecture 22, Pg 13
Example: 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 111: Lecture 22, Pg 14
Example: Two Disks
Since Li = Lf
f
z
f
z
Li Lf
f2
i2 MRMR
21
if 21
An inelastic collision,since E is not
conserved (friction)!
Wheel rimdrop
Physics 111: Lecture 22, Pg 15
Example: 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 111: Lecture 22, Pg 16
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
fII
i
Ii
f
If
LiLf
Drop mass from stool
Student on stool
Physics 111: Lecture 22, Pg 17
Lecture 22, Act 2Angular 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 111: Lecture 22, Pg 18
Lecture 22, Act 2 Solution
I2
LI21K
22 (using L = I)
L is conserved:
I2 < I1 K2 > K1 K increases!
1 2
I2 I1
L L
Physics 111: Lecture 22, Pg 19
Lecture 22, Act 2 Solution
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 111: Lecture 22, Pg 20
Angular Momentum of aFreely 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
v
L r p
Physics 111: Lecture 22, Pg 21
Angular Momentum of aFreely 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
v
md
y
Physics 111: Lecture 22, Pg 22
Angular Momentum of aFreely Moving Particle...
We need to figure out The magnitude of the angular momentum is:
Since r and p are both in the x-y plane, L will be in the z direction (right hand rule):
y
xp=mvd
r
approachclosestofcetandisxp
pdsinrpsinrp prL
L pdZ
L r p
Physics 111: Lecture 22, Pg 23
Angular Momentum of aFreely Moving Particle...
So we see that the direction of L 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
p
d
Physics 111: Lecture 22, Pg 24
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
initial final
m D D/4
Physics 111: Lecture 22, Pg 25
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
4DmvapproachclosestofcedisxpL 1i )tan(
Physics 111: Lecture 22, Pg 26
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 4DmvL I
Physics 111: Lecture 22, Pg 27
Example: Bullet hitting stick...
Set Li = Lf using
v1 v2
MF
initial final
m D D/4
I 1
122MD
mv D mv D MD F1 22
4 41
12 F
mMD
v v 3
1 2
Physics 111: Lecture 22, Pg 28
Example: 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 111: Lecture 22, Pg 29
Example: Throwing ball from stool...
Conserve angular momentum (since there are no external torques acting on the student-stool system):Li = 0Lf = 0 = IF - Mvd
top view: initial final
d
vM
I I
F
FMvd
I
Physics 111: Lecture 22, Pg 30
Lecture 22, Act 3Angular Momentum
1
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
2
Physics 111: Lecture 22, Pg 31
Lecture 22, Act 3 Solution
The angular momentum is due to the girl, the merry-go-round and the ball. LNET = LMGR + LGIRL + LBALL
Initial: L mR vR
mvRBALL
I 2
v
Rm
Final: L mvRBALL
vm
same
Physics 111: Lecture 22, Pg 32
Lecture 22, Act 3 Solution
Since LBALL is the same before & after, must stay the same to keep “the rest” of LNET unchanged.
Physics 111: Lecture 22, Pg 33
Lecture 22, Act 3Conceptual 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.
2
Physics 111: Lecture 22, Pg 34
Recap of today’s lecture
Angular Momentum: (Text: 10-2, 10-4)Definitions & DerivationsWhat does it mean?
Rotation about a fixed axis (Text: 10-2, 10-4)L = IExample: Two disksStudent on rotating stool
Angular momentum of a freely moving particle(Text: 10-2, 10-4)