chap10 Angular momentum

Special case: when choosing the origin to lie in the plane of the circulating particle, l ω
2. Rigid body (made up of many particles) If the body is axial symmetry, L ω → L = I ω the upper bearing may be removed, and the shaft remains parallel to z axis. If the body is asymmetry L is not parallet to ω , the upper bearing is required to keep the shaft in the fixed direction. The issue of wobble(摇晃) is particular serious for objects that rotate at high speed, such as turbine(涡轮) rotors. Note: For any rigid body, L z = I ω z
For fixed axis (say, z axis)
∑τ
→
net ,z t2
d Lz d (Iω ) = = dt dt
net ,z
∫ ∑τ
t1
dt = Iω
2
− Iω1
angular impulse →Theorem of angular momentum impulse.
1）The theorem is also valid
dt
d ( mg ) R = ( I ω + mvR ) = I α + mRa dt 1 Since, a = Rα , I = MR 2 2 1 2 mg 2 mgR = MR ( a / R ) + mRa → a = M + 2m 2
3. Symmetrical versus asymmetrical bodies The system in fig10.11 is “unbalanced”. l is the same for each particle, so L = l1 + l 2 are not parallel to ω . If the connecting rod were not rigidly fasted to the vertical shaft near o, it would tend to move until angle β became 900, in which position the system would then be symmetrical about the shaft. For a symmetrical rotating body(for example, fig.10.7),there is no bearing wobble(摇摆), and the shaft rotates smoothly.
ω
F F
I
ω
C：For a body at rest initially, if one part of it rotates, it will result in an opposite rotation of another part of it. D: Some examples Spinning skater, springboard diver, rotating bicycle wheel, etc.
+
m M
Solution:Considering M, m to be X the “body”,
∵ ∑ τ ext = 0
The angular momentum is conservative. Choosing the IRF and the positive direction of coordinate system as shown.
10-2 Systems of particles 1. A system of N particles:
L = l1 + l
2 N
+ ⋅⋅⋅+ l
N
=
∑
N
l
n
n =1
d L = dt
d (∑ l n )
n =1
dt
=
∑
N
n =1
d ln = dt
∑τnFra bibliotekTorques→ (1) from internal forces; (2) from external forces. Because the torque resulting from each action-reaction force pair is zero, only the torque from external forces remains.
∑τ
ext
=
dt
Solution: Three external forces mg, M g , N Only mg exerts a torque about the origin. τz=mgR, Lz=Iω +(mv)R d L z, we have Applying τ =
∑
ext , z
ext
dL , if = dt
∑τ
ext
dL = 0 ( = 0) → L = constant or Li = L f dt
Both the conservation of linear momentum and the conservation of angular momentum are valid for a wide range of systems, they hold in both relativistic limit and in the quantum limit. Any component of the angular momentum will be constant if the corresponding component of the torque is zero. For a body rotating about a fixed axis (say, the z axis), if τext,z=0, L=constant→ Ii ω i = If ωf ( Lz= Iω )
∑τ
ext
dL = dt
The external torque acting on a (1 0 - 9 ) system of particle is equal to the time rate of change of the total angular momentum of the system.
10-3 Angular momentum and angular velocity l ,ω 1. Single particle
l = r× p
Analogy: v, p → ω , l
p is always parallel to v ,
but l (with respect to the origin o ) is not always parallel to ω
must be angular impulse. A large angular impulse must be exerted on the object with large angular momentum.
Example: Find the acceleration of block by dL
Chap. 10 Angular momentum Key terms: angular momentum(角动量) conservation of angular momentum(角动量守恒) gyroscope(回转仪) procession(进动) nutation (章动)
Chap. 10 Angular momentum 10-1 Angular momentum of a particle 1. The angular momentum of a particle: l = r × p = r × mv Its magnitude is l=rmvsinθ, its direction: according to the right-hand rule. 2. Relation between torque and angular momentum:
2. Analogy between the way a force changes linear momentum and the way a torque changes angular momentum
F = F⊥ + F F → ΔP , F ⊥ → ΔP⊥ P → P + ΔP + ΔP⊥
Note：1）For general motion, the law is also true when the body rotates about an axis that is through its center of mass. 2）The law is valid not only for rigid body but also for non rigid body. Three cases： A：Both I and ω keep constant，uniform rotation.（常 平架上的回转仪） B：I changes，but Iω keep constant，then ω will change.
补充知识:刚体做定点运动时对定点的角动量的计算
类似刚体转动惯量的推导,
Ixx, Iyy, Izz , 分别称为刚体对x轴、y 轴、z轴的转动惯量, Ixy,Iyz, Izx 称为惯量积, 统称为惯量系数.