Physics 2211: Lecture 39 Todays Agenda

1
Physics 2211 Lecture 39Todays Agenda

• ANGULAR MOMENTUM
• EXAMPLES

2
STATIC EQUILIBRIUM
1.
AND
2.
3
A uniform beam (length 8 m, mass 50 kg) is
supported by a pin at its left end and a cable
attached to its right end. If a crate of mass
M80 kg is at a distance d2m from the wall,
what is the tension in the cable? (1) 491 N
(2) 576 N (3) 1274 N (4) 241 N (5) 982 N
(6) 1663 N
M
d
50 deg
19141
4
Torque and the Right Hand Rule

• The right hand rule can tell you the direction of
  torque
• Point your hand along the direction from the axis
  to the point where the force is applied.
• Curl your fingers in the direction of the force.
• Your thumb will point in the directionof the
  torque.

F
y
r
x
?
z
5
Torque the Cross Product

• So we can define torque as
• ? r X F
• rF sin ?
• ?X rY FZ - FY rZ y FZ - FY z
• ?Y rZ FX - FZ rX z FX - FZ x
• ?Z rX FY - FX rY x FY - FX y

F
?
r
6
Angular MomentumDefinitions Derivations
p mv

• We have shown that for a system of particles
• Momentum is conserved if
• What is the rotational version of this??

7
Angular 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

(1) increase (2) decrease (3) stay
the same
w2
w1
8
Conceptual 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.

w2
w
9
Definitions Derivations...

• First consider the rate of change of L

10
Definitions Derivations...

• Recall that

???
?EXT

• Which finally gives us

• Analogue of !!

11
What does it mean?

• where
  and

Total angular momentum is conserved
12
Angular momentum of a rigid bodyabout a fixed
axis

• 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

(since ri and vi are perpendicular)
v1
We see that L is in the z direction.
m2
j
Using vi ? ri , we get
r2
m1
r1
i
v2
?
r3
m3
v3
I
?
Analogue of p mv!!
13
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
14
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)?

y
x
d
v
15
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
x
r
d
pmv
?
?
16
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
d
p
17
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)

M
?F
D
m
D/4
v1
v2
initial
final
18
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).

M
D
D/4
m
v1
initial
19
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.

?F
D/4
v2
final
20
Example Bullet hitting stick...

• Set Li Lf using

M
?F
D
m
D/4
v1
v2
initial
final
21
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 ?

z
?
22
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
23
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

2
1
?0
24
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

z
2
1
?f
25
Example Two Disks

• Since Li Lf

An inelastic collision, since E is not
conserved (friction)!
Li
Lf
?f
26
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?

M
v
d
?F
I
I
top view initial final
27
Example Throwing ball from stool...

• Conserve angular momentum (since there are no
  external torques acting on the student-stool
  system)
• Li 0
• Lf 0 I?F - Mvd

M
v
d
?F
I
I
top view initial final
28
Recap of todays 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

29
Rotations

• A girl is riding on the outside edge of a
  merry-go-round turning with constant w. 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?

(2)
(1)
(3)
(4)
w
30
Solution

• Just before release, the velocity of the ball is
  tangent to the circle it is moving in.

w
31
Solution

• After release it keeps going in the same
  direction since there are no forces acting on it
  to change this direction.

32
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?

?f
?i
If
Ii
33
Example Rotating Table...

• Again, there are no external torques acting on
  the student-stool system, so angular momentum
  will be conserved.
• Initially Li Ii?i
• Finally Lf If ?f

?f
?i
If
Ii
Lf
Li
34
Angular Momentum

• A student sits on a freely turning stool and
  rotates with constant angular velocity w1. She
  pulls her arms in, and due to angular momentum
  conservation her angular velocity increases to
  w2. In doing this her kinetic energy

(a) increases (b) decreases (c) stays
the same
w2
w1
I2
I1
L
L
35
Solution

(using L I?)
36
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!

w2
w1
I2
I1
L
L
37
Solution

• The angular momentum is due to the girl, the
  merry-go-round and the ball. LNET LMGR LGIRL
  LBALL

38
Solution

• Since LBALL is the same before after, w must
  stay the same to keep the rest of LNET
  unchanged.

w
w