《理论力学 Theoretical Mechanics——动力学》（双语版）第十三章 动量矩定理 Moment of Momentum Theorem

Theoretical Mechanics Chapter13; Moment of Momentum Theorem

1 Theoretical Mechanics

理论力学

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Chapter 13: Moment of Momentum Theorem 813-1 Moment of momentum 813-2 moment of momentum theorem D813-3 Differentialequations for the rotation of a rigid body around a fixed-axis t813-4 Moment ofinertia of a rigid body with respect to an axis s13-5 moment of momentum theorem for a system with respect to its center of mass, differentialequations of plane motion of a rigid body Exercises

3 §13–1 Moment of momentum §13–2 moment of momentum theorem §13–3 Differential equations for the rotation of a rigid body around a fixed-axis §13–4 Moment of inertia of a rigid body with respect to an axis §13–5 moment of momentum theorem for a system with respect to its center of mass, differential equations of plane motion of a rigid body Exercises Chapter 13: Moment of Momentum Theorem

第十三章动量矩定理 §13-1动量矩 §13-2动量矩定理 §13-3刚体定轴转动微分方程 §13-4刚体对轴的转动惯量 d§13-5质点系相对于质心的动量矩定理, 刚体平面运动微分方程 习题课

4 §13–1 动量矩 §13–2 动量矩定理 §13–3 刚体定轴转动微分方程 §13–4 刚体对轴的转动惯量 §13–5 质点系相对于质心的动量矩定理， 刚体平面运动微分方程 习题课 第十三章 动量矩定理

Dynamic Theorem of momentum The change of the momentum of a particle (or of a system of particles) is the result of external forces(the principal vector of an external force system) Theorem of motion of the center of mass The motion of the center of mass is the result of external forces ( the principal vector of an external force system) When the center of mass coincides with a certain point of a fixed axis, then the momentum of a rotating body is al ways zero because Vc=0. But in this case the system is still subjected to the action of external forces. The moment of momentum theorem establishes the dependence between the change of the moment of momentum of a particle or a system with respect to a center(or a fixed axis) and the torque given by all external forces acting on the particle or the system with respect to the same center (or axis)

5 Theorem of momentum: The change of the momentum of a particle (or of a system of particles) is the result of external forces (the principal vector of an external force system). Theorem of motion of the center of mass: The motion of the center of mass is the result of external forces (the principal vector of an external force system). When the center of mass coincides with a certain point of a fixed axis, then the momentum of a rotating body is always zero because Vc=0. But in this case the system is still subjected to the action of external forces. The moment of momentum theorem establishes the dependence between the change of the moment of momentum of a particle or a system with respect to a center (or a fixed axis) and the torque given by all external forces acting on the particle or the system with respect to the same center (or axis)

学 质点 动量定理:质点系动量的改变→外力(外力系主矢) 质心运动定理:质心的运动—→>外力(外力系主矢) 若当质心为固定轴上一点时,v=0,则其动量恒等于零, 质心无运动,可是质点系确受外力的作用。动量矩定理建立了 质点和质点系相对于某固定点(固定轴)的动量矩的改变与外 力对同一点(轴)之矩两者之间的关系

6 质点 动量定理： 质点系 动量的改变—→外力（外力系主矢） 若当质心为固定轴上一点时，vC=0，则其动量恒等于零， 质心无运动，可是质点系确受外力的作用。动量矩定理建立了 质点和质点系相对于某固定点（固定轴）的动量矩的改变与外 力对同一点（轴）之矩两者之间的关系。 质心运动定理：质心的运动—→外力（外力系主矢）

Dynamic 813-1 Moment of momentum Moment of momentum of a particle with respect to a center O is (mv)=r× mi it is a vector Moment of momentum of a particle with respect to an axis Zis m (mv)=mo(mv,)it is an algebraic qu mo(mv)=240AB B m(m)=±2△OB mmv The definition o of the sis he sign of the moment of momentum with respect to an axis is the same as that of the y moment of a force with respect to an axis. Looking from the positive end of the axis, it's positive if it is counterclockwise, and it is negative if it is clockwise 7

7 §13-1 Moment of momentum Moment of momentum of a particle with respect to a center O is , it is a vector. Moment of momentum of a particle with respect to an axis Z is , it is an algebraic quantity. m mv r mv O ( )=  ( ) ( ) z O xy m mv =m mv mO (mv) =2OAB m (mv) 2 OA'B' z =   The definition of the sign of the moment of momentum with respect to an axis is the same as that of the moment of a force with respect to an axis. Looking from the positive end of the axis, it’s positive if it is counterclockwise, and it is negative if it is clockwise

学 §13-1动量矩 质点的动量矩 质点对点O的动量矩:m(mV)=F×m失量 质点对轴z的动量矩:m2(m)=m0(mn)代数量 B z、mv m(m)=2A04B m2(m)=±2△O4B mo(mv) 正负号规定与力对轴矩的规定相同 对着轴看:顺时针为负 ArMy 逆时针为正

8 §13-1 动量矩 一．质点的动量矩 质点对点O的动量矩： 矢量 质点对轴 z 的动量矩： 代数量 m mv r mv O ( )=  ( ) ( ) z O xy m mv =m mv mO (mv) =2OAB m (mv) 2 OA'B' z =   正负号规定与力对轴矩的规定相同 对着轴看：顺时针为负 逆时针为正

Dynarnics The relation between the moment of momentum of a particle with respect to an axis and a center is mo(mv):=m(mv Moment of momentum measures the intensity of rotation of a body rotating around a fixed center or an axis at any instant. kg. m/s 2. Moment of momentum of a system of particles. The moment of momentum of a system with respect to a center O is 1o=∑m(m2v1)=∑×mn1v The moment of momentum of a system with respect to an axis Zis L=∑m:(m)=ol Calculation of moment of momentum of a rigid body. 1)Rigid body in translational motion Lo=mo(mv=rc xmv (ΣXm=m×=xm)L=m(m) The moment of momentum of a rigid body in translational motion with respect to a fixed center(axis) is equal to the moment of momentum of the center of mass of the rigid body with respect to that center(axis)

9 The relation between the moment of momentum of a particle with respect to an axis and a center is 2. Moment of momentum of a system of particles. The moment of momentum of a system with respect to a center O is The moment of momentum of a system with respect to an axis Z is O O i i i mi vi L =m (m v )=r    z z i i LO z L =m (m v )= kg·ｍ2 /s. Moment of momentum measures the intensity of rotation of a body rotating around a fixed center or an axis at any instant. m (mv) m (mv) O z = z Calculation of moment of momentum of a rigid body. 1) Rigid body in translational motion O O C C C L =m (mv )=r mv ( ) i i i i i C C mvC r m v =m r v =r  ( ) z z C L =m mv The moment of momentum of a rigid body in translational motion with respect to a fixed center (axis) is equal to the moment of momentum of the center of mass of the rigid body with respect to that center (axis)

力单 质点对点O的动量矩与对轴z的动量矩之间的关系 Imo(mv))=m(mv 动量矩度量物体在任一瞬时绕固定点(轴)转动的强弱kgm2/s 质点系的动量矩 质系对点O动量矩:L=∑m0(m1,)=∑xm1 质系对轴z动量矩:L:=∑m(m)=[一 刚体动量矩计算: 1.平动刚体 o=mo(mvc)=×mc (∑Xm21=∑mF×WC=1xmC) =m (mvo) 平动刚体对固定点(轴)的动量矩等于刚体质心的动量 对该点(轴)的动量矩。 10

10 质点对点O的动量矩与对轴z 的动量矩之间的关系： 二．质点系的动量矩 质系对点O动量矩： 质系对轴z 动量矩： O O i i i i i L =m (m v )=r m v   z z i i LO z L =m (m v )= kg·ｍ2 动量矩度量物体在任一瞬时绕固定点(轴)转动的强弱 /s。 m (mv) m (mv) O z = z 刚体动量矩计算： 1．平动刚体 O O C C C L =m (mv )=r mv ( ) i i i i i C C C r m v =m r v =r mv ( ) z z C L =m mv 平动刚体对固定点（轴）的动量矩等于刚体质心的动量 对该点（轴）的动量矩