810.3 The angular momentum of a system of particles and moment of inertia of rigid body Example 2: There is a light thin staff of mass m and length l. find the moment with respect to different axis. Solution: dm Ⅰ=[r2dm 2 =「x2dm 2 32 L L 3 mL 8 10.3 The angular momentum of a system of particles and moment of inertia of rigid body dm L Ⅰ=[r2dm=[x2dm=「x2dx 11L1 L L303
11 Example 2: There is a light thin staff of mass m and length L. Find the moment with respect to different axis. x m I r m d d 2 2 ∫ ∫ = = o x dm x 2 L 2 − L 2 3 3 2 3 2 2 12 1 3 8 8 1 2 2 3 1 d mL L L L m L L x L m x L m x L L =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + − = = ∫− Solution: §10.3 The angular momentum of a system of particles and moment of inertia of rigid body L dm o x x 3 2 0 2 2 2 3 1 3 0 1 d d d mL L x L m x L m I r m x m x L = = = = = ∫∫∫ §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 Some rotational inertias la)Slender ro (b) slender rod c)Rectangular plate. OOOO le) Hollow cylinder (f)Solid cylinder (g) Thin-walled hollow (h) Solid sphere Summary of the previous sections I. The orbital angular momentum of a particle Define:L=r×p=rxmν For a circular orbital motion of a particle Rotational inertia L=mro=o 2. The time rate of angular momentum for a =rX TX dt 点 Defil total =rX total
12 §10.3 The angular momentum of a system of particles and moment of inertia of rigid body Some rotational inertias Summary of the previous sections 1. The orbital angular momentum of a particle 2. The time rate of angular momentum for a single particle For a circular orbital motion of a particle ω ω r r r L = mr = I 2 total total d d d d τ r r r r r r = × = r × F = t p r t L Define: total Ftotal r r r r τ = × x y z m θ r r p r o ⊥r L r p⊥ r r Ftotal r θ o m L r p r mv r r r r r Define: = × = × Rotational inertia r r p r o m ω r