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