Transcript 幻灯片 1

Chapter 8 More on Angular Momentum
and Torque
Main Points of Chapter 8
• Any moving object can have angular
momentum around a fixed point
• Angular momentum and torque can be
expressed as vector products
• Angular momentum is conserved
• Precession
8-1 Generalization of Angular Momentum
We can give the definition of angular
momentum to a particle
(with respect to the point O)
S

O
Discussion
• Angular momentum of a particle depends on both its
momentum and position vector (on the point O)
• We find that the angular
momentum is constant for an
object moving at constant
velocity.
8-2 Generalization of Torque
We can write an analogous
expression for the torque with
respect to the point O
Ｏ
h
  Fr sin  Fh
P
The torque depends on both the force and
the reference point O
The relation between the torque and the angular
momentum of a particle with respect to the same
point O
This is the rotational version of Newton’s second law
8-3 The Dynamics of Rotation
For system of particles

 dL
 
dt

 dL
 
dt
• Torque and angular momentum can be measured
from (the same) any reference point
• The angular momentum of a system of particles
can be changed only by the external torques.
• The angular momentum of a particle in the
system can be changed by the internal torques.


dLcm
• In center-of-mass reference frame,  cm 
dt
holds even center-of-mass is accelerating.
For a rigid body (rotating about a fixed axis)
z

vi
dLz
z 
dt
 I z
Rigid
O
body
fixed axis

ri
P
Angular impulse
1. for system of particles
angular impulse
The change in angular momentum of system of
particles during a time interval equals the angular
impulse that acts on the particles during that interval
2. for a rigid body (rotating about a fixed axis)
The angular impulse acts on the a rigid body
rotating about a fixed axis equals to the change
of angular momentum about the same axis.
 Example A slender uniform rod of length l and mass m
is initially lying on a frictionless horizontal surface. A
horizontal impulse FDt exerts on it at a distance d from
C , describe the subsequent motion of the rod.
Solution
Consider CM motion
d
c
FDt
Consider rotation motion about CM
CAI
8-4 Conservation of Angular Momentum
If the net external torque acting on a system is zero, the
angular momentum
of the system remains constant,
no matter what changes take place within the system.
1. For a particle

  0  L  Constant Vector

a central force
For central forces, angular momentum around
origin is conserved and motion is in a plane

L

This leads to Kepler’s second law:
The radius vector of a planetary
orbit sweeps out equal areas in
equal times.

dr 
 m
r
Example A space vehicle is launched to a planet with mass
M, radius R, when the space vehicle reaches the planet at a
distance of 4R, it launches an apparatus with mass m at
velocity v0. If this apparatus can even sweep the surface of
the planet, find the speed of landing and angularθ
m
v0
Solution:

conservation of
mechanical energy and
angular momentum
v
R
r0
OM
1
GMm 1
GMm
2
2
mv 0 
 mv 
2
r0
2
R
mv 0 r0 sin   mvR
1/ 2
1
3GM 

sin   1 
2
4  2 Rv 0 
1/ 2

3GM 

v  v 0 1 
2
 2 Rv 0 
Act A puck with mass m slides in a circular path on
a horizontal frictionless table. It is held by a string
thread through a frictionless hole at the center of the
table. If you pull on the string such that the radius
decreases from r0 to r1, How much work have you
done?
Solution:
r0 r1

Conservation of Angular Momentum
mv 0 r0  mvr
r0
v  v0 
r1
v0
v0
F
1
1
2
mv 1  mv 02
2
2
2
2






r
r
1
1
1
W  mv 02  0   mv 02  mv 02  0   1
2
2
 r1 

 r1  2
Work-Energy theorem W 

Example a particle of mass m is shooting horizontally
in a hemispherical bowl which is frictionless. The
particle can just reach the rim of bowl. Find v 0
Solution:
conservation of mechanical energy
1
1
2
mv 0  mv 2  mgrCos  0
2
2
z
o

N
v0

mg
conservation of angular momentum in z direction (Z=0)
mv 0 rSin 0  mvr
2 gr
v0 
Cos 0
2. For a rigid body (rotating about a fixed axis)
Demo:
the gyroscope compass
the helicopter
For a nonrigid body (rotating about a fixed axis)
• Can change rotational inertia through internal
forces
Examples: rotating skater, collapsing interstellar
dust cloud
• Angular momentum must remain constant
• Decreased rotational inertia means increased
angular speed, and vice versa
video
 Act A student sits on a rotating stool with his arms
extended and a weight in each hand. He then pulls
his hands in toward his body. In doing this her
kinetic energy
(a) increases
(b) decreases
(c) stays the same
f
i
Ii
If
3. For system of particle and rigid body
If there is no net external torque on a system,
angular momentum is conserved
 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 the final speed is v2.
– What is the angular speed F of the stick after
the collision?
M
m
D D/4
F
v1
v2
initial
final
Solution
M
m
D D/4
F
v1
v2
initial
final
Conserve angular momentum around the pivot (z) axis!
 Act A uniform stick of mass m and length l is
moving on a frictionless table with velocity v. It
hits the nail at a distance l/4 from one end, so it
rotates about the nail. Find the angular velocity.
Solution
Conservation of angular
momentum
r
8-5 Precession
Consider a top that is supported at one point O. If the
top is spinning in a very high speed, one possible motion
is a steady circular motion of the axis combined with the
spin motion of the top about the symmetry axis
We call this phenomenon
precession

The angular momentum of top

The angular momentum of top changes
only its direction, but not its magnitude.

rc

mg
O
• A torque perpendicular to the axis of rotation can cause
the axis itself to rotate
• If there is no such torque, the axis will not rotate – this
leads to the stability of gyroscopes
Find the angular velocity of Precession
d
The Dynamics of Rotation

the angular frequency of precession
Discussion

O
1. the angular frequency of precession depends on
, is independent of 
2. the direction of angular velocity of Precession
depends on the direction of
3. There is another type of motion called nutation
when  is not large.
CAI
• Suppose you have a spinning top (gyroscope) in the
configuration shown below. If the left support is
removed, what will happen?
– The gyroscope does not fall down! Instead it
precesses around its pivot axis !
support
o
pivot
d
o
pivot
mg
d
top view
Demo: the gyroscope in the form of a wheel
pivot
Summary of Chapter 8
• Angular momentum can be defined for any moving
object; for an object moving at constant velocity, it is
constant
•Rotational quantities are analogous to linear
quantities
• Angular momentum is conserved in the absence of
external torques
Summary of Chapter 8, cont.
• Nonrigid objects can change rotational
inertia through internal forces
• Angular momentum, Iω, remains constant
• The axis of rotation can itself experience a
torque, which tends to change its direction