力单 两个基本定理: 在理论力学中,我们关心的主要是由于碰撞冲量的作用而 使物体运动速度发生的变化。因此,动量定理和动量矩定理就 成了研究碰撞问题的主要工具。 1、用于碰撞过程的动量定理冲量定理。 设质点的质量为m,碰撞开始时的速度v,结束瞬时的速 度矿,碰撞冲量S,不计普通力的冲量,则质点动量定理 的积分形式为: mu-my=s (19-1) 16
16 两个基本定理： 在理论力学中，我们关心的主要是由于碰撞冲量的作用而 使物体运动速度发生的变化。因此，动量定理和动量矩定理就 成了研究碰撞问题的主要工具。 1、用于碰撞过程的动量定理——冲量定理。 设质点的质量为 m，碰撞开始时的速度 ，结束瞬时的速 度 ，碰撞冲量 ，不计普通力的冲量，则质点动量定理 的积分形式为： v u S mu−mv=S (19-1)
Dynamies For a system of n particles, dividing the impulses of impact acting on the i-the particle into an external impulse of impact and an internal impulses of impact S( e) we haves(i mm=S+50、(=12,…m) Adding these n equations and using ES=0 (internal impulses of Impact al ways appear pair wise ) we get ∑m一∑m=∑ Si) This is the theorem of impulse i=1 (19-2) Let the total mass of the system of particles be M,ucand v are the velocities at the end and the beginning of the impact Applying the theorem of the motion of the center of mass to the, to the equation(19-2)we get Mc-Mc=∑S (19-3)17
17 For a system of n particles , dividing the impulses of impact acting on the i-the particle into an external impulse of impact and an internal impulses of impact ,we have (e) Si (i) Si . ( 1,2, , ) ( ) ( ) m u m v S S i n i i e i i − i i = i + = Let the total mass of the system of particles be M, and are the velocities at the end and the beginning of the impact .Applying the theorem of the motion of the center of mass to the , to the equation (19-2)we get C u C v − = (e) C C Si Mu Mv （19-3） Adding these n equations and using (internal impulses of impact always appear pair wise)we get 0 ( ) = i Si − = = = ( ) 1 1 e i n i i i n i mi ui m v S （19-2） This is the theorem of impulse
力单 对于有n个质点组成的质点系,将作用于第i个质点上的 碰撞冲量分为外碰撞冲量S)和内碰撞冲量S),则有: m-m形1=S(2)+S(i=1,2,…,m) 将这n个方程相加,且∑S=0(内碰撞冲量总是成对出现的,故 2m-∑m=∑S0冲量定理(192) 设质点系总质量M,u和v分别为碰撞结束和碰撞开始 时质心的速度,则利用质心运动定理,上式可写成: Mc-Mc=∑ (193) 18
18 对于有n个质点组成的质点系，将作用于第 i 个质点上的 碰撞冲量分为外碰撞冲量 Si (e) 和内碰撞冲量 Si (i) ，则有： ( 1,2, , ) ( ) ( ) m u m v S S i n i i e i i − i i = i + = 将这n个方程相加, 且 Si (i) = 0 (内碰撞冲量总是成对出现的),故 − = = = ( ) 1 1 e i n i i i n i mi ui m v S 冲量定理 （19-2） 设质点系总质量M， 分别为碰撞结束和碰撞开始 时质心的速度，则利用质心运动定理，上式可写成： C C u 和v − = (e) C C Si Mu Mv （19-3）
Dynamics The change in the momentum of a system of particles during an mpact to t vecto ternal impul acting on the system. The projection forms of the expressions(19-1), (19-2 )and(19 The difference is that the impulses of the conventional forces are neglected here The angular momentum theorem applied to an impact. Theorem of the moment of the impulse. We know from assumption(2)that the position of a particle does not change From expression(19-1)we then have F×m-Fxmv=r×S Because rxmv=lo1, rxmu=lo2, represent the angular moment a of the particle with respect to point O at the beginning and at the end of the impact rx S=mo(s)is the moment of the impact impulse s with respect to the point 0, 02 olmo(S) 19 (194)
19 The change in the momentum of a system of particles during an impact is equal to the vector sum of all the external impulses acting on the system. The projection forms of the expressions (19-1),(19-2) and (19- 3) are the same as in the case of the ordinary momentum theorem. The difference is that the impulses of the conventional forces are neglected here. 2. The angular momentum theorem applied to an impact. Theorem of the moment of the impulse. We know from assumption (2) that the position of a particle does not change. From expression (19-1) we then have: r mu − r mv = r S. Because , represent the angular moment a of the particle with respect to point O at the beginning and at the end of the impact . is the moment of the impact impulse with respect to the point O 1 2 , O O rmv =l rmu =l r S m (S ) = O s ( ) l O2 −l O1 =mO S （19-4）
学 碰撞时质点系动量的改变等于作用在质点系上所有外碰 撞冲量的矢量和。 式(19-1)、(19-2)和(19-3)都写成投影形式,形式上与普 通的动量定理相同,所不同的是在这里都不计普通力的冲量。 2、用于碰撞过程的动量矩定理—冲量矩定理 由假设(2)知,碰撞过程中,质点的矢径F保持不变, 则由(19-1)式,有: Fxm-Fxmv=r×S 而FXm=b0,Pxm=l2;l1和2为碰撞始末时质点对 O点的动量矩。FXS=m0(S是碰撞冲量S对O点的矩,所以 02-lo1=mo(s) (19-4) 20
20 碰撞时质点系动量的改变等于作用在质点系上所有外碰 撞冲量的矢量和。 由假设(2)知，碰撞过程中，质点的矢径 保持不变， 则由(19-1)式，有： r 式（19-1)、(19-2)和(19-3)都写成投影形式，形式上与普 通的动量定理相同，所不同的是在这里都不计普通力的冲量。 2、用于碰撞过程的动量矩定理——冲量矩定理 r mu −r mv = r S 而 ； 为碰撞始末时质点对 O点的动量矩。 是碰撞冲量 对O点的矩，所以： 1 2 , O O rmv =l rmu =l O1 O2 l 和l r S m (S ) = O s ( ) l O2 −l O1 =mO S （19-4）