Chapter 7Plane Kinetics of Rigid BodyJumptofirstpage
Jump to first page Chapter 7 Plane Kinetics of Rigid Body
Moment of inertia of a rigid body about an axisymvaZ-KEmiViG22m;12Zr;IGmVGXm;(riiGo22ri1G1mveZ(miriG)10122X112my22Ig is the moment of inertia about G.Jumptofirstpage
Jump to first page Moment of inertia of a rigid body about an axis. 2 2 2 / 2 2 2 / 2 2 2 2 1 2 1 ( ) 2 1 2 1 ( ) 2 1 2 1 2 1 2 1 . . G G i i G i G i i G i G i i G i G m v I m v m r m v m r K E m v m v = + = + = + = + y x mi i r G r G i G r/ IG is the moment of inertia about G
Example : Moment of inertia of a cylinder aboutthe axis.I=Zm,r?=[r2dmIf p is the density, dm = p(2元rL)dr, m = πa’LpI = ( r2p(2元rL)dr=2元pL[r3drrLdr1πpLa21ajma?2Jumptofirstpage
Jump to first page Example : Moment of inertia of a cylinder about the axis. dr r L 2 a 4 0 4 0 3 2 2 2 2 2 1 πρ 2 1 4 1 2π 2π d ρ(2π )d If ρ is the density, d ρ(2π )d , π ρ d m a La L r L r r I r rL r m rL r m a L I m r r m a a i i i = = = = = = = = =
Parallel axis theorem :I。=m,R, R, = m,(h+R/G)(h+R,/G)=Zm,h? +22m,(h.R/c)+Zm,R/G1= mh? +2h.m,R/c +m,R/Gi0y=mh2 +0+ Igm;R.R,IGGhI。= IG + mh?0xJumpto firstpage
Jump to first page x y mi G Ri /G Ri h O G i G i i G i i i i G i i G i i i i i i G i G i i i i O i i m h I m h h m R m R m h m h R m R I m R R m h R h R = + + = + + = + + = = + + 0 2 2 ( ) ( ) ( ) 2 2 / / 2 2 / / 2 / / Parallel axis theorem : Io = IG + mh2
ExampleaxisIo = Ic +mh?a= ma /2 + mhhJump to firstpage
Jump to first page 2 2 2 ma /2 mh IO IG mh = + = + Example axis h a