8 10.2 The time rate of change of angular momentum and torque 3. Dynamics of circular orbital motion of a single particle L=mr20=lo r×F dt d d total (o= dr Example 1: P3 10.5 Can not be used Example 2: P433 10.6 in noncircular orbital motion 8 10.3 The angular momentum of a system of particles and moment of inertia of rigid body 1. The angular momentum of a system of particles L=∑L=∑x=∑x CM +r! P L=∑Gw+xm ×∑m+∑xm(m+) =xm+mm+∑m可 6
6 3. Dynamics of circular orbital motion of a single particle ω ω r r r Q L = mr = I 2 total total d d d d τ r r r r r r = × = r × F = t p r t L Example 1: P433 10.5 Example 2: P433 10.6 α ω τ ω r r r r I t I I t ∴ = = = d d ( ) d d total Can not be used in noncircular orbital motion. §10.2 The time rate of change of angular momentum and torque §10.3 The angular momentum of a system of particles and moment of inertia of rigid body 1. The angular momentum of a system of particles = ∑ = ∑ × = ∑ × ii i i i i i i i L L r p r m v r r r r r r ⎩ ⎨ ⎧ = + ′ = + ′ i CM i i CM i v v v r r r r r r r r r Q θ pi r o rCM r ir r mi ir r C ′ ( ) ( ) i i i i CM i i i CM i i i i CM i i i CM i i i i i i CM i r m v r m v r m v r m v r m v v L r r m v = × + ′× + ′× ′ = × + ′× + ′ = + ′ × ∑ ∑ ∑ ∑ ∑ ∑ r r r r r r r r r r r r r r r ∴
810.3 The angular momentum of a system of particles and moment of inertia of rigid body First term: FcM x2m, ",=FCM XMVCM Second term ∑水mFCM=∑mx下m= Mcmv=0 le position vector of center of mass with respect to the center of mass Third term: ∑ r×mv is the vector sum of angular momentum of all particles with respect to the center of mass. 33中mHm出e then L=×Mc+∑可xm可 orbital 十 2. Spin angular momentum of oN a rigid body about a axis through the center of mass ∑xm可 =0X ⊥ i∥ 十 ∴Lm=∑m+)x(D×n)
7 ×∑ = × i rCM mivi rCM MvCM r r r r First term: ∑ ′× = ∑ ′× = ′ × = 0 CM CM CM i i CM i i i i r m v m r v Mr v r r r r r r Second term: The position vector of center of mass with respect to the center of mass i i i ir m v r r Third term: ∑ ′× ′ is the vector sum of angular momentum of all particles with respect to the center of mass. §10.3 The angular momentum of a system of particles and moment of inertia of rigid body Lorbital Lspin L r Mv r m vi i i CM CM i r r r r r r r = + = × +∑ ′× ′ then 2. Spin angular momentum of a rigid body about a axis through the center of mass i v ′ r o ω r z o′ mi ir′ r ⊥ ′ ir r i // r′ r i i i i L r m v r r r Q = ∑ ′× ′ spin ⊥ ⊥ ′ = × ′ ′ = ′ + ′ i i i i i v r r r r r r r r r r ω // ( ) ( ) spin // ⊥ ⊥ ∴ =∑ ′ + ′ × × ′ i i i i i L m r r r r r r r r ω §10.3 The angular momentum of a system of particles and moment of inertia of rigid body
810.3 The angular momentum of a system of particles and moment of inertia of rigid body ∑m而X(x) +∑m×(xh) n..uol iL +∑mh The vector is an involved vector summation. The rotation of an oddly shaped object about any axis of rotation is beyond the scope of this course 8 10.3 The angular momentum of a system of particles and moment of inertia of rigid body 3. The moment of inertia or rotational inertia of a rigid body about a fixed axis through 乡 center of mass ∑mhX(x)=∑mrO1 O fOthe rigid body is symmetry r about the axis; ② the axis is fixed. This term has no effect. Then ∑ (a×r1) ∑m 8
8 §10.3 The angular momentum of a system of particles and moment of inertia of rigid body ∑ ∑ ∑ ∑ ⊥ ⊥ ⊥ ⊥ ⊥ + ′ = − ′ + ′ × × ′ = ′ × × ′ i i i i i i i i i i i i i i i m r m r r m r r L m r r ω ω ω ω r r r r r r r r r 2 // spin // ( ) ( ) i v ′ r o ω r z o′ mi ir′ r ⊥ ′ ir r i // r′ r The vector is an involved vector summation. The rotation of an oddly shaped object about any axis of rotation is beyond the scope of this course. i v ′ r o ω z r o′ mi ir′ r ⊥ ′ ir r i // r′ r 3. The moment of inertia or rotational inertia of a rigid body about a fixed axis through center of mass ∑ ⊥ ∑ ⊥ ′ × × ′ = − ′ i i i i i i i i m r r m r r r r r r // (ω ) ω If 1the rigid body is symmetry about the axis; 2the axis is fixed. This term has no effect. Then ∑ ∑ ⊥ ⊥ ⊥ = ′ = ′ × × ′ i i i i i i i m r L m r r ω ω r r r r r 2 spin ( ) §10.3 The angular momentum of a system of particles and moment of inertia of rigid body
810.3 The angular momentum of a system of particles and moment of inertia of rigid body Define:cw=∑mh This is the moment of inertia or rotational inertia of a rigid body about a fixed axis through center of mass The angular momentum 0 MO=C∑m s10.3 The angular momentum of a system of particles and moment of f inerti a of rigid body 4. The moment of inertia of various rigid bodies (a) point particle r-a distance from the axis of rotation (b) Collection of point particles I=∑ ri1 --the perpendicular distance of each mass m from the axis of rotation (c) Rigid body of distributive mass 几L d
9 i v ′ r o ω z r o′ mi ir′ r ⊥ ′ ir r i // r′ r ∑ ⊥ = ′ i CM i i I m r 2 Define: This is the moment of inertia or rotational inertia of a rigid body about a fixed axis through center of mass The spin angular momentum of a rigid body ω ω r r r ( ) 2 spin = = ∑ ⊥ i CM i i L I m r §10.3 The angular momentum of a system of particles and moment of inertia of rigid body 4. The moment of inertia of various rigid bodies (a) Point particle 2 I = mr r –a distance from the axis of rotation (b) Collection of point particles = ∑ ⊥ i i I m ri 2 --the perpendicular distance of each mass mi from the axis of rotation ri⊥ §10.3 The angular momentum of a system of particles and moment of inertia of rigid body (c) Rigid body of distributive mass I r dm2 = ∫ ⊥
810.3 The angular momentum of a system of particles and moment of inertia of rigid body -the perpendicular distance of each mass dm from the axis of rotation The element of mass: adl linear density: a dm= ds surface density: o pdv volum density P 8 10.3 The angular momentum of a system of particles and moment of inertia of rigid body Examplel: 5 particles are connected by 4 light staffs as shown in figure. Find the moment of the system with respect to the axis through point A, and perpendicular to the paper plane. Solution:I=∑m ●4 I=2m12+3m(2) n +(4m+5m)(√2l) =32ml 5m 10
10 --the perpendicular distance of each mass dm from the axis of rotation r⊥ dm = λdl linear density：λ σdS surface density：σ ρdV volum density：ρ The element of mass: §10.3 The angular momentum of a system of particles and moment of inertia of rigid body l l l l A m 2m 3m 4m 5m 2 2 2 2 32 (4 5 )( 2 ) 2 3 (2 ) ml m m l I ml m l = + + = + Example1: 5 particles are connected by 4 light staffs as shown in figure. Find the moment of the system with respect to the axis through point A, and perpendicular to the paper plane. Solution: 2 = ∑ i⊥ i i I m r §10.3 The angular momentum of a system of particles and moment of inertia of rigid body