THEORYMECHANICS902Chapter12 Theorem of moment of momentumCollege of Mechanical and Vehicle EngineeringWangXiaojun
College of Mechanical and Vehicle Engineering WangXiaojun THEORY MECHANICS Chapter12 Theorem of moment of momentum
Chapter12Theoremofmomentof momentum>12.1 Moment of momentum ofa particle and its system> 12.2 The moment of inertia of a rigid body against an axis12.3theoremofmomentofmomentum 12.4 Differential equation of a rigid body rotating about a fixed axis> 12.5 Theorem of moment of momentum of a system of particlesrelativetothecenterofmass 12.6 Differential equations of motion in a rigid body plane
Ø 12.1 Moment of momentum of a particle and its system Ø 12.2 The moment of inertia of a rigid body against an axis Ø 12.3 theorem of moment of momentum Ø 12.4 Differential equation of a rigid body rotating about a fixed axis Ø 12.5 Theorem of moment of momentum of a system of particles relative to the center of mass Ø 12.6 Differential equations of motion in a rigid body plane Chapter12 Theorem of moment of momentum
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12.1Moment of momentum of aparticleand its system1Moment of momentum of a particleTALet the momentaofa particle M be mv,and themo(mv)vector diameter of the particle relativeto thefixedmiparticle O be r.MAy0Moment of momentum of particle to particle OLo(mi) = r× mim。(mv) perpendicular △OMA , it's going to be twice the area △OMA , and thedirection is determined by the right-hand ruleMoment of momentum of a particle on a fixed axis:L,(mi) =[i。(mi)]It is the number of generations, whose sign can be determined by the right hand ruleMomentofmomentumis instantaneous.Theinternationalunitof momentofmomentum is kg-m / s
1、Moment of momentum of a particle x y z O M A r mv m (mv) O Let the momenta of a particle be , and the vector diameter of the particle relative to the fixed particle be . M mv O r 12.1 Moment of momentum of a particle and its system L mv r mv O ( ) perpendicular , it's going to be twice the area , and the direction is determined by the right-hand rule. m (mv ) O OMA OMA Moment of momentum of particle to particle O: Moment of momentum is instantaneous. The international unit of moment of momentum is z O z L (mv ) L (mv ) Moment of momentum of a particle on a fixed axis: kg m /s 2 It is the number of generations, whose sign can be determined by the right hand rule
12.1 Moment of momentum of a particle and its system2, Moment of momentum of a system of particlesNparticlesmass m, volecity , radius vectorconstitutetheparticleiparticle system:MomentofmomentumofaparticleThemomentof momentum ofasystem with respect to a fixed particleparticle system to any fixed particle isthe vector sum of themoment ofLo = Emo(m,j,) = Er, ×m,imomentumofeachparticleinthesystemtothefixed particleMomentofmomentumofaparticle system on a fixed axisThemoment of momentum ofaparticle system for any fixed axis isthe algebraic sum of the moment ofL, =[Emo(m,y)] = Em,(m,,)momentumofeachparticleinthesystem for the fixed axis
12.1 Moment of momentum of a particle and its system N particles constitute the particle system: particle i O O i i i i i L m m v r m v ( ) Moment of momentum of a particle system with respect to a fixed particle Moment of momentum of a particle system on a fixed axis ( ) ( ) z O i i z z i i L m m v m m v The moment of momentum of a particle system to any fixed particle is the vector sum of the moment of momentum of each particle in the system to the fixed particle The moment of momentum of a particle system for any fixed axis is the algebraic sum of the moment of momentum of each particle in the system for the fixed axis mass , volecity , radius vector ir mi i v 2、Moment of momentum of a system of particles
12.1 Moment of momentumof aparticleandits systemThe moment of momentum of a translational rigid bodyRigid body: mass M, velocity of center of mass isymass m, , volecity,particle i,X?Lo=Er xmy, =(Em,r)xvcEmr = Mr司Lo =rc×MvThe moment of force of a translational rigid body to any fixed particle is equalto the moment of force of a particle with mass concentrated at the center of masstothe fixed particle
12.1 Moment of momentum of a particle and its system 1、The moment of momentum of a translational rigid body x y z C i Cv i v O O i i i i i C L r mv m r v ( ) O C C L r Mv The moment of force of a translational rigid body to any fixed particle is equal to the moment of force of a particle with mass concentrated at the center of mass to the fixed particle C v Rigid body: mass M, velocity of center of mass is i mi i v particle , mass ，volecity i i C m r Mr
12.1Momentofmomentum of particleand system of particle2、The moment of force of a rotating rigid body on a rotating shaftOThe angular velocity of the rigid body around thefixed axis Z is @m.yDistance fromparticle i :Timass mithe shaft业L, = Em,(m;v,)=Em;V;ri=(Em;r:)aJ, =Zm,r?Themoment ofinertiaof therigidL, =J,bodywithrespecttotheZaxisThe moment of force of a rotating rigid body on a fixed axis is equal to theproduct of the moment of inertia of the rigid body on the axis and the angularvelocity ofthe rigid body
12.1 Moment of momentum of particle and system of particle 2、The moment of force of a rotating rigid body on a rotating shaft z i ir i i m v z z i i i i i L m (m v ) m v r Lz J z The moment of force of a rotating rigid body on a fixed axis is equal to the product of the moment of inertia of the rigid body on the axis and the angular velocity of the rigid body particle i： i r Distance from the shaft m i mass ， 2 z i i J m r The moment of inertia of the rigid body with respect to the Z axis ( ) 2 i i m r The angular velocity of the rigid body around the fixed axis Z is
12.1 Moment of momentum of a particle and its systemExamplelThehomogeneous disk canberotated about its axis0O, with a rope wrapped around it and a weight A hung at theOend of the rope. If the moment of inertia of the disk to the axisis J, radius is r, angular velocity is @ , mass of the weight is m,and there is no relative slip between the rope and the originaldisk,the moment of momentum of the system to the axis Oismvcalculated.Solution: Lo = Lblock + Lplate= mvr + Jo= mr + Jo = (mr2 + J)oLo turn of theta is counterclockwise
r O A mv Example1 The homogeneous disk can be rotated about its axis O, with a rope wrapped around it and a weight A hung at the end of the rope. If the moment of inertia of the disk to the axis is J, radius is r, angular velocity is , mass of the weight is m, and there is no relative slip between the rope and the original disk, the moment of momentum of the system to the axis O is calculated. Solution： LO Lblock Lplate turn of theta is counterclockwise LO 12.1 Moment of momentum of a particle and its system ( ) 2 2 mr J mr J mvr J
12.2 The moment of inertia of a rigid body against an axis1, The concept of moment of inertiaThe moment of inertia of the rigid body towards the axis Z is defined as the sumof the sum of the masses of all particles on the rigid body multiplied by thesquare of the perpendicular distance from the particle to the axis Z. namelyJ,=Em;rFor a rigid body with a continuous mass distribution, the above formula can bewritten in integral form[rdm=illustrate:The moment of inertia is related not only to mass, but also to the distribution of massIn the SI system of units, the unit of inertia is:kg - m?★ When referring to the moment of inertia of a rigid body, it is necessary to specify themoment of inertiaforwhichaxis
1、The concept of moment of inertia The moment of inertia of the rigid body towards the axis Z is defined as the sum of the sum of the masses of all particles on the rigid body multiplied by the square of the perpendicular distance from the particle to the axis Z. namely 2 z i i J m r For a rigid body with a continuous mass distribution, the above formula can be written in integral form Jz r dm 2 The moment of inertia is related not only to mass, but also to the distribution of mass. In the SI system of units, the unit of inertia is: . When referring to the moment of inertia of a rigid body, it is necessary to specify the moment of inertia for which axis. 2 kg m 12.2 The moment of inertia of a rigid body against an axis illustrate:
12.2 The moment of inertia of a rigid body against an axis2 The moment of inertia of a regular shaped homogeneous rigid body2.1、The moment of inertia of a homogeneous thin rod71Thez-axis passingthroughthe centerof mass22and perpendicular to the axis of the barMxOdx :dmdxxdx11Mmass: M length: l2MI1221z2.2、The moment of inertia of a thin ringRThe z-axis passing through the center of massand perpendiculartothetorus planemass: M , radius:RJ, = Em;r? =(Zm,)R? = MR
2、The moment of inertia of a regular shaped homogeneous rigid body 2.1、The moment of inertia of a homogeneous thin rod O z 1z 2 l 2 l x x dx dx : dx l M dm 2 2 2 2 12 1 x dx Ml l M J l z l 12.2 The moment of inertia of a rigid body against an axis The z-axis passing through the center of mass and perpendicular to the axis of the bar 2.2、The moment of inertia of a thin ring z R 2 2 2 Jz miri ( mi )R MR The z-axis passing through the center of mass and perpendicular to the torus plane mass：M ，radius：R mass：M length: l