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Physics 207, Lecture 16, Oct. 29

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Title: Physics 207, Lecture 16, Oct. 29


1
Physics 207, Lecture 16, Oct. 29
  • Agenda Chapter 13
  • Center of Mass
  • Torque
  • Moment of Inertia
  • Rotational Energy
  • Rotational Momentum
  • Assignment
  • Wednesday is an exam review session, Exam will be
    held in rooms B102 B130 in Van Vleck at 715 PM
  • MP Homework 7, Ch. 11, 5 problems,
  • NOTE Due Wednesday at 4 PM
  • MP Homework 7A, Ch. 13, 5 problems, available soon

2
Chap. 13 Rotational Dynamics
  • Up until now rotation has been only in terms of
    circular motion with ac v2 / R and aT d
    v / dt
  • Rotation is common in the world around us.
  • Many ideas developed for translational motion are
    transferable.

3
Conservation of angular momentum has consequences
How does one describe rotation (magnitude and
direction)?
4
Rotational Dynamics A childs toy, a physics
playground or a students nightmare
  • A merry-go-round is spinning and we run and jump
    on it. What does it do?
  • We are standing on the rim and our friends spin
    it faster. What happens to us?
  • We are standing on the rim a walk towards the
    center. Does anything change?

5
Rotational Variables
  • Rotation about a fixed axis
  • Consider a disk rotating aboutan axis through
    its center
  • How do we describe the motion
  • (Analogous to the linear case )

6
Rotational Variables...
  • Recall At a point a distance R away from the
    axis of rotation, the tangential motion
  • x ? R
  • v ? R
  • a ? R

7
Summary (with comparison to 1-D kinematics)
  • Angular Linear

And for a point at a distance R from the rotation
axis
x R ????????????v ? R ??????????a ? R
8
Lecture 15, Exercise 5Rotational Definitions
  • A friend at a party (perhaps a little tipsy) sees
    a disk spinning and says Ooh, look! Theres a
    wheel with a negative w and positive a!
  • Which of the following is a true statement about
    the wheel?
  1. The wheel is spinning counter-clockwise and
    slowing down.
  2. The wheel is spinning counter-clockwise and
    speeding up.
  3. The wheel is spinning clockwise and slowing down.
  4. The wheel is spinning clockwise and speeding up

9
Lecture 15, Exercise 5Rotational Definitions
  • A goofy friend sees a disk spinning and says
    Ooh, look! Theres a wheel with a negative w and
    with antiparallel w and a!
  • Which of the following is a true statement about
    the wheel?

(A) The wheel is spinning counter-clockwise and
slowing down. (B) The wheel is spinning
counter-clockwise and speeding up. (C) The wheel
is spinning clockwise and slowing down. (D) The
wheel is spinning clockwise and speeding up
10
Example Wheel And Rope
  • A wheel with radius r 0.4 m rotates freely
    about a fixed axle. There is a rope wound around
    the wheel.
  • Starting from rest at t 0, the rope is
    pulled such that it has a constant acceleration a
    4m/s2.
  • How many revolutions has the wheel made after
    10 seconds?
  • (One revolution 2? radians)

11
Example Wheel And Rope
  • A wheel with radius r 0.4 m rotates freely
    about a fixed axle. There is a rope wound around
    the wheel. Starting from rest at t 0, the rope
    is pulled such that it has a constant
    acceleration a 4 m/s2. How many revolutions
    has the wheel made after 10 seconds?
    (One revolution 2? radians)
  • Revolutions R (q - q0) / 2p and a a r
  • q q0 w0 t ½ a t2 ?
  • R (q - q0) / 2p 0 ½ (a/r) t2 / 2p
  • R (0.5 x 10 x 100) / 6.28

12
System of Particles (Distributed Mass)
  • Until now, we have considered the behavior of
    very simple systems (one or two masses).
  • But real objects have distributed mass !
  • For example, consider a simple rotating disk and
    2 equal mass m plugs at distances r and 2r.
  • Compare the velocities and kinetic energies at
    these two points.

w
1
2
13
System of Particles (Distributed Mass)
  • An extended solid object (like a disk) can be
    thought of as a collection of parts.
  • The motion of each little part depends on where
    it is in the object!
  • The rotation axis matters too!

1 K ½ m v2 ½ m (w r)2
w
2 K ½ m (2v)2 ½ m (w 2r)2
14
System of Particles Center of Mass
  • If an object is not held then it rotates about
    the center of mass.
  • Center of mass Where the system is balanced !
  • Building a mobile is an exercise in finding
  • centers of mass.

mobile
15
System of Particles Center of Mass
  • How do we describe the position of a system
    made up of many parts ?
  • Define the Center of Mass (average position)
  • For a collection of N individual pointlike
    particles whose masses and positions we know

RCM
m2
m1
r2
r1
y
x
(In this case, N 2)
16
Sample calculation
  • Consider the following mass distribution

XCM (m x 0 2m x 12 m x 24 )/4m meters YCM
(m x 0 2m x 12 m x 0 )/4m meters XCM 12
meters YCM 6 meters
17
System of Particles Center of Mass
  • For a continuous solid, convert sums to an
    integral.

dm
r
y
where dm is an infinitesimal mass element.
x
18
Rotational Dynamics What makes it spin?
A force applied at a distance from the rotation
axis
  • ?TOT r FTang r F sin f
  • Torque is the rotational equivalent of force
  • Torque has units of kg m2/s2 (kg m/s2) m N m
  • A constant torque gives constant angular
    acceleration iff the mass distribution and the
    axis of rotation remain constant.

19
Lecture 16, Exercise 1Torque
  • In which of the cases shown below is the torque
    provided by the applied force about the rotation
    axis biggest? In both cases the magnitude and
    direction of the applied force is the same.
  • Remember torque requires F, r and sin q
  • or the tangential force component times
    perpendicular distance
  1. Case 1
  2. Case 2
  3. Same

20
Lecture 16, Exercise 1Torque
  • In which of the cases shown below is the torque
    provided by the applied force about the rotation
    axis biggest? In both cases the magnitude and
    direction of the applied force is the same.
  • Remember torque requires F, r and sin f
  • or the tangential force component times
    perpendicular distance

L
F
F
(A) case 1 (B) case 2 (C) same
L
axis
case 1
case 2
21
Rotational Dynamics What makes it spin?
A force applied at a distance from the rotation
axis
  • ?TOT r FTang r F sin f
  • Torque is the rotational equivalent of force
  • Torque has units of kg m2/s2 (kg m/s2) m N m

?TOT r FTang r m a
r m r a m r2 a
For every little part of the wheel
22
?TOT m r2 a and inertia
  • The further a mass is away from this axis the
    greater the inertia (resistance) to rotation
  • This is the rotational version of FTOT ma
  • Moment of inertia, I m r2 , (here I is just a
    point on the wheel) is the rotational equivalent
    of mass.
  • If I is big, more torque is required to achieve
    a given angular acceleration.

23
Calculating Moment of Inertia
  • where r is the distance from the mass
    to the axis of rotation.

Example Calculate the moment of inertia of four
point masses (m) on the corners of a square whose
sides have length L, about a perpendicular axis
through the center of the square
m
m
L
m
m
24
Calculating Moment of Inertia...
  • For a single object, I depends on the rotation
    axis!
  • Example I1 4 m R2 4 m (21/2 L / 2)2

I1 2mL2
I 2mL2
I2 mL2
m
m
L
m
m
25
Lecture 16, Exercise 2Moment of Inertia
  • A triangular shape is made from identical balls
    and identical rigid, massless rods as shown. The
    moment of inertia about the a, b, and c axes is
    Ia, Ib, and Ic respectively.
  • Which of the following is correct

(A) Ia gt Ib gt Ic (B) Ia gt Ic gt Ib (C)
Ib gt Ia gt Ic
26
Lecture 16, Home Exercise Moment of Inertia
  • Ia 2 m (2L)2 Ib 3 m L2 Ic m (2L)2
  • Which of the following is correct

a
(A) Ia gt Ib gt Ic (B) Ia gt Ic gt Ib (C)
Ib gt Ia gt Ic
L
b
L
c
27
Calculating Moment of Inertia...
  • For a discrete collection of point masses we
    found
  • For a continuous solid object we have to add up
    the mr2 contribution for every infinitesimal mass
    element dm.
  • An integral is required to find I

dm
r
28
Moments of Inertia
  • Some examples of I for solid objects
  • Solid disk or cylinder of mass M and radius
    R, about perpendicular axis through its center.
  • I ½ M R2

29
Moments of Inertia...
  • Some examples of I for solid objects

Solid sphere of mass M and radius R, about an
axis through its center. I 2/5 M R2
R
Thin spherical shell of mass M and radius R,
about an axis through its center. Use the table
R
See Table 13.3, Moments of Inertia
30
Moments of Inertia
  • Some examples of I for solid objects

Thin hoop (or cylinder) of mass M and radius R,
about an axis through it center, perpendicular
to the plane of the hoop is just MR2
R
R
Thin hoop of mass M and radius R, about an axis
through a diameter.
31
Rotation Kinetic Energy
  • Consider the simple rotating system shown below.
    (Assume the masses are attached to the rotation
    axis by massless rigid rods).
  • The kinetic energy of this system will be the sum
    of the kinetic energy of each piece
  • K ½ m1v12 ½ m2v22 ½ m3v32 ½ m4v42

m4
m1
r1
?
r4
r2
m3
r3
m2
32
Rotation Kinetic Energy
  • Notice that v1 w r1 , v2 w r2 , v3 w r3 ,
    v4 w r4
  • So we can rewrite the summation
  • We recognize the quantity, moment of inertia or
    I, and write

33
Lecture 16, Exercise 2Rotational Kinetic Energy
  • We have two balls of the same mass. Ball 1 is
    attached to a 0.1 m long rope. It spins around at
    2 revolutions per second. Ball 2 is on a 0.2 m
    long rope. It spins around at 2 revolutions per
    second.
  • What is the ratio of the kinetic energy
  • of Ball 2 to that of Ball 1 ?
  1. ¼
  2. ½
  3. 1
  4. 2
  5. 4

34
Lecture 16, Exercise 2Rotational Kinetic Energy
  • K2/K1 ½ m wr22 / ½ m wr12 0.22 / 0.12 4
  • What is the ratio of the kinetic energy of Ball 2
    to that of Ball 1 ?
  • (A) 1/4 (B) 1/2 (C) 1 (D) 2 (E)
    4

35
Rotation Kinetic Energy...
  • The kinetic energy of a rotating system looks
    similar to that of a point particle
  • Point Particle Rotating System

v is linear velocity m is the mass.
? is angular velocity I is the moment of
inertia about the rotation axis.
36
Moment of Inertia and Rotational Energy
  • So where
  • Notice that the moment of inertia I depends on
    the distribution of mass in the system.
  • The further the mass is from the rotation axis,
    the bigger the moment of inertia.
  • For a given object, the moment of inertia depends
    on where we choose the rotation axis (unlike the
    center of mass).
  • In rotational dynamics, the moment of inertia I
    appears in the same way that mass m does in
    linear dynamics !

37
Work (in rotational motion)
  • Consider the work done by a force F acting on an
    object constrained to move around a fixed axis.
    For an infinitesimal angular displacement d?
    where dr R d?
  • ?dW FTangential dr
  • dW (FTangential R) d?
  • ?dW ? d? (and with a constant torque)
  • We can integrate this to find W ? ?
    (?f-ti) ?
  • Analogue of W F ?r
  • W will be negative if ? and ? have opposite sign
    !

axis of rotation
38
Work Kinetic Energy
  • Recall the Work Kinetic-Energy Theorem ?K
    WNET
  • This is true in general, and hence applies to
    rotational motion as well as linear motion.
  • So for an object that rotates about a fixed axis

39
Lecture 16, Home exercise Work Energy
  • Strings are wrapped around the circumference of
    two solid disks and pulled with identical forces
    for the same linear distance. Disk 1 has a
    bigger radius, but both are identical material
    (i.e. their density r M/V is the same). Both
    disks rotate freely around axes though their
    centers, and start at rest.
  • Which disk has the biggest angular velocity
    after the pull?

W ? ? F d ½ I w2 (A) Disk 1 (B) Disk
2 (C) Same
w2
w1
F
F
start
d
finish
40
Lecture 16, Home exercise Work Energy
  • Strings are wrapped around the circumference of
    two solid disks and pulled with identical forces
    for the same linear distance. Disk 1 has a
    bigger radius, but both are identical material
    (i.e. their density r M/V is the same). Both
    disks rotate freely around axes though their
    centers, and start at rest.
  • Which disk has the biggest angular velocity
    after the pull?

W F d ½ I1 w12 ½ I2 w22 w1 (I2 / I1)½ w2
and I2 lt I1 (A) Disk 1 (B) Disk 2 (C) Same

w2
w1
F
F
start
d
finish
41
Example Rotating Rod
  • A uniform rod of length L0.5 m and mass m1 kg
    is free to rotate on a frictionless pin passing
    through one end as in the Figure. The rod is
    released from rest in the horizontal position.
    What is
  • (A) its angular speed when it reaches the lowest
    point ?
  • (B) its initial angular acceleration ?
  • (C) initial linear acceleration of its free end
    ?

42
Example Rotating Rod
  • A uniform rod of length L0.5 m and mass m1 kg
    is free to rotate on a frictionless hinge passing
    through one end as shown. The rod is released
    from rest in the horizontal position. What is
  • (B) its initial angular acceleration ?
  • 1. For forces you need to locate the Center of
    Mass
  • CM is at L/2 ( halfway ) and put in the Force on
    a FBD
  • 2. The hinge changes everything!

S F 0 occurs only at the hinge
but tz I az r F sin 90 at the center of
mass and (ICM m(L/2)2) az (L/2) mg and solve
for az
mg
43
Example Rotating Rod
  • A uniform rod of length L0.5 m and mass m1 kg
    is free to rotate on a frictionless hinge passing
    through one end as shown. The rod is released
    from rest in the horizontal position. What is
  • (C) initial linear acceleration of its free end
    ?
  • 1. For forces you need to locate the Center of
    Mass
  • CM is at L/2 ( halfway ) and put in the Force on
    a FBD
  • 2. The hinge changes everything!

a a L
mg
44
Example Rotating Rod
  • A uniform rod of length L0.5 m and mass m1 kg
    is free to rotate on a frictionless hinge passing
    through one end as shown. The rod is released
    from rest in the horizontal position. What is
  • (A) its angular speed when it reaches the lowest
    point ?
  • 1. For forces you need to locate the Center of
    Mass
  • CM is at L/2 ( halfway ) and use the Work-Energy
    Theorem
  • 2. The hinge changes everything!

L
W mgh ½ I w2
m
W mgL/2 ½ (ICM m (L/2)2) w2 and solve for
w
mg
L/2
mg
45
Connection with CM motion
  • If an object of mass M is moving linearly at
    velocity VCM without rotating then its kinetic
    energy is
  • If an object of moment of inertia ICM is rotating
    in place about its center of mass at angular
    velocity w then its kinetic energy is
  • What if the object is both moving linearly and
    rotating?

46
Connection with CM motion...
  • So for a solid object which rotates about its
    center of mass and whose CM is moving

VCM
?
47
Rolling Motion
  • Now consider a cylinder rolling at a constant
    speed.

VCM
CM
The cylinder is rotating about CM and its CM is
moving at constant speed (VCM). Thus its total
kinetic energy is given by
48
Lecture 16, Example The YoYo
  • A solid uniform disk yoyo of radium R and mass M
    starts from rest, unrolls, and falls a distance
    h.
  • (1) What is the angular acceleration?
  • (2) What will be the linear velocity of the
    center of mass after it falls h meters?
  • (3) What is the tension on the cord ?

T
w
M
h
49
Lecture 16, Example The YoYo
  • A solid uniform disk yoyo of radium R and mass M
    starts from rest, unrolls, and falls a distance
    h.
  • Conceptual Exercise
  • Which of the following pictures correctly
    represents the yoyo after it falls a height h?
  • (A) (B) (C)

h
50
Lecture 16, Example The YoYo
  • A solid uniform disk yoyo of radium R and mass M
    starts from rest, unrolls, and falls a distance
    h.
  • Conceptual Exercise
  • Which of the following pictures correctly
    represents the yoyo after it falls a height h?
  • (A) (B) No Fx, no ax (C)

T
w0
M
h
h
Mg
51
Lecture 16, Example The YoYo
  • A solid uniform disk yoyo of radium R and mass M
    starts from rest, unrolls, and falls a distance
    h.
  • (1) What is the angular acceleration?
  • (2) What will be the linear velocity of the
    center of mass after it falls h meters?
  • (3) What is the tension on the cord ?

T
Choose a point and calculate the torque
St I az Mg R T0 ( ½ MR2 MR2 ) az
Mg R az Mg /(3/2 MR) 2 g / (3R)
w
M
X
h
52
Lecture 16, Example The YoYo
  • A solid uniform disk yoyo of radium R and mass M
    starts from rest, unrolls, and falls a distance
    h.
  • (1) What is the angular acceleration?
  • (2) What will be the linear velocity of the
    center of mass after it falls h meters?
  • (3) What is the tension on the cord ?
  • az Mg /(3/2 MR) 2 g / (3R)
  • and aCM az R 2g/3 (down)
  • Can use kinetics or work energy
  • Mgh ½ mv2 ½ ICM w2
  • Mgh ½ Mv2 ½ ( ½ MR2) (v/R)2
  • gh ¾ v2 ? v 2 (gh/3)½

T
w
M
X
h
53
Lecture 16, Example The YoYo
  • A solid uniform disk yoyo of radium R and mass M
    starts from rest, unrolls, and falls a distance
    h.
  • (1) What is the angular acceleration?
  • (2) What will be the linear velocity of the
    center of mass after it falls h meters?
  • (3) What is the tension on the cord ?
  • aCM az R -2g/3
  • MaCM - 2Mg/3 T Mg
  • T Mg/3
  • or from torques
  • I az TR ½ MR2 (2g/3R)
  • T Mg/3

T
w
M
X
h
54
Rolling Motion
  • Again consider a cylinder rolling at a constant
    speed.

2VCM
CM
VCM
55
Example Rolling Motion
  • A cylinder is about to roll down an inclined
    plane. What is its speed at the bottom of the
    plane ?

56
Example Rolling Motion
  • A cylinder is about to roll down an inclined
    plane. What is its speed at the bottom of the
    plane ?
  • Use Work-Energy theorem

Mgh ½ Mv2 ½ ICM w2 Mgh ½ Mv2 ½ (½ M R2
)(v/R)2 ¾ Mv2 v 2(gh/3)½
57
Rolling Motion
  • Now consider a cylinder rolling at a constant
    speed.

VCM
CM
The cylinder is rotating about CM and its CM is
moving at constant speed (VCM). Thus its total
kinetic energy is given by
58
Motion
  • Again consider a cylinder rolling at a constant
    speed.

Both with VTang VCM
Rotation only VTang wR
Sliding only
2VCM
VCM
CM
CM
CM
VCM
59
Angular Momentum
  • We have shown that for a system of particles,
  • momentum
  • is conserved if
  • What is the rotational equivalent of this?
  • angular momentum
  • is conserved if

60
Example Two Disks
  • 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
61
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.

No External Torque so Lz is constant Li Lf ?
I wi i I wf ? ½ mR2 w0 ½ 2mR2 wf
62
Lecture 16, Oct. 29
  • Assignment
  • Wednesday is an exam review session, Exam will be
    held in rooms B102 B130 in Van Vleck at 715 PM
  • MP Homework 7, Ch. 11, 5 problems,
  • NOTE Due Wednesday at 4 PM
  • MP Homework 7A, Ch. 13, 5 problems, available
    soon

63
Angular MomentumDefinitions Derivations

p mv
  • We have shown that for a system of particles
  • Momentum is conserved if
  • What is the rotational equivalent of this?
  • The rotational analog of force F is torque ?
  • Define the rotational analog of momentum p to be
    angular momentum, L or

64
Recall from Chapter 9 Linear Momentum
  • Definition For a single particle, the momentum
    p is defined as

p mv
(p is a vector since v is a vector)
So px mvx etc.
  • Newtons 2nd Law

F ma
  • Units of linear momentum are kg m/s.

65
Linear Momentum and Angular Momentum
  • So from
  • Newtons 2nd Law

F ma
  • Units of angular momentum are kg m2/s.

66
Putting it all together
  • In the absence of external torques

Total angular momentum is conserved
67
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 momentum of
    each particle

(since ri , vi , are perpendicular)
v1
We see that L is in the z direction.
m2
Using vi ? ri , we get
r2
r1
i
v2
r3
v3
m3
68
Example Two Disks
  • 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
69
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.

No External Torque so Lz is constant Li Lf ?
I wi i I wf ? ½ mR2 w0 ½ 2mR2 wf
70
DemonstrationConservation of Angular Momentum
  • Figure Skating

z
Arm
Arm
?B
LA LB
IA IB
wA wB
No External Torque so Lz is constant even if
internal work done.
71
DemonstrationConservation of Angular Momentum
  • Figure Skating

z
Arm
Arm
?B
?A
IAwA LA LB IBwB
IA lt IB wA gt wB ½ IAwA2 gt ½
IB wB2 (work needs to be done)
No External Torque so Lz is constant even if
internal work done.
72
Angular Momentum Conservation
  • A freely moving particle has a well defined
    angular momentum about any given axis.
  • If no torques are acting on the particle, its
    angular momentum remains constant (i.e., will be
    conserved).
  • In the example below, the direction of L is along
    the z axis, and its magnitude is given by LZ pd
    mvd.

y
x
d
v
m
73
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
D
?F
m
D/4
v1
v2
before
after
74
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 wf

M
D
?F
m
D/4
v1
v2
before
after
75
Example Throwing ball from stool
  • A student sits on a stool, initially at rest, but
    which is free to rotate. The moment of inertia
    of the student plus the stool is I. They throw a
    heavy ball of mass M with speed v such that its
    velocity vector moves a distance d from the axis
    of rotation.
  • What is the angular speed ?F of the
    student-stool system after they throw the ball ?

M
v
d
?F
I
I
Top view before after
76
Example Throwing ball from stool
  • What is the angular speed ?F of the student-stool
    system after they throw the ball ?
  • Process (1) Define system (2) Identify
    Conditions
  • (1) System student, stool and ball (No Ext.
    torque, L is constant)
  • (2) Momentum is conserved
  • Linit 0 Lfinal m v d I wf

M
v
d
?F
I
I
Top view before after
77
An example Neutron Star rotation
Neutron star with a mass of 1.5 solar masses has
a diameter of 11 km. Our sun rotates about once
every 37 days wf / wi Ii / If ri2 / rf2
(7x105 km)2/(11 km)2 4 x 109 gives millisecond
periods!
period of pulsar is 1.187911164 s
78
Angular Momentum as a Fundamental Quantity
  • The concept of angular momentum is also valid on
    a submicroscopic scale
  • Angular momentum has been used in the development
    of modern theories of atomic, molecular and
    nuclear physics
  • In these systems, the angular momentum has been
    found to be a fundamental quantity
  • Fundamental here means that it is an intrinsic
    property of these objects

79
Fundamental Angular Momentum
  • Angular momentum has discrete values
  • These discrete values are multiples of a
    fundamental unit of angular momentum
  • The fundamental unit of angular momentum is h-bar
  • Where h is called Plancks constant

80
Intrinsic Angular Momentum
photon
81
Angular Momentum of a Molecule
  • Consider the molecule as a rigid rotor, with the
    two atoms separated by a fixed distance
  • The rotation occurs about the center of mass in
    the plane of the page with a speed of

82
Angular Momentum of a Molecule (It heats the
water in a microwave over)
E h2/(8p2I) J (J1) J 0, 1, 2, .
83
Center of Mass Example Astronauts Rope
  • Two astronauts are initially at rest in outer
    space and 20 meters apart. The one on the right
    has 1.5 times the mass of the other (as shown).
    The 1.5 m astronaut wants to get back to the ship
    but his jet pack is broken. There happens to be
    a rope connected between the two. The heavier
    astronaut starts pulling in the rope.
  • (1) Does he/she get back to the ship ?
  • (2) Does he/she meet the other astronaut ?

M 1.5m
m
84
Example Astronauts Rope
  1. There is no external force so if the larger
    astronaut pulls on the rope he will create an
    impulse that accelerates him/her to the left and
    the small astronaut to the right. The larger
    ones velocity will be less than the smaller
    ones so he/she doesnt let go of the rope they
    will either collide (elastically or
    inelastically) and thus never make it.

85
Example Astronauts Rope
  • (2) However if the larger astronaut lets go of
    the rope he will get to the ship. (Too bad for
    the smaller astronaut!)
  • In all cases the center of mass will remain fixed
    because they are initially at rest and there is
    no external force.
  • To find the position where they meet all we need
    do is find the Center of Mass
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