理论力学第十一章英文ppt共37页

t2
t2
t2
IR d t Fd t F d t Ii
t1
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t1
t1
The unit of impulse is the same as that of momentum.
Ns(km g2)/skg m/s.
§12.3 Theorem of momentum
1. Theorem of momentum for one particle:
and the velocity of the center of mass of the i-th rigid body .
are mi , vc.i For the whole system we get then
Px mivCix mixCi
P
mivCi
Py
In terms of projections on cartesian axes we have
Px MvCx MxC , Py MvCy MyC , Pz MvCz MzC
3)Momentum of a system of rigid bodies: Assume that the mass
The position of the
center of mass c is (Mmi)
rC M m iri oM rrC m iri
Frrc o x m ciycjzck,wgeot
x C M m ix i,y C m M iy i,zC M m izi
m [ (1lsi4n5 5lco s2l)i(1lco 4s5 5lsin )j]
2
2
2
2
m [l (12532)i(1251)j]
2 2 2 10 2 2 2 10
2. Impulse:
The product of a force and the action time of the force is called impulse. Impulse is used to characterize the accumulated effect on a body of a force acting during a certain time interval.
1) Force F is a constant vector. IF(t2t1)
2) ForceF is a variable vector (Include magnitude and direction.)
The elementary impulse is dI Fdt
The impulse is
d (mv)F0 dt
mvC
If Fx ,0then mv x is a const and the motion of the particle along the axis X is an inertial motion.
t2
m2vxm1vx Ix Fxdt0
t1
m2xv m1x vmx vC
t2
I F dt
t1
t2
t2
t2
IxF xd,tIyF yd,tIz F zdt
t1
t1
t1
3) The impulse of a resultant force is equal to the geometric sum of the impulses of all component forces:
③ Projection form:
ddt(mvx) Fx ddt(mvy) Fy ddt(mvz) Fz
t2
m2vx m1vx Ix Fxdt
t1 t2
m2vy m1vy Iy Fydt
t1 t2
m2vz m1vz Iz Fzdt
t1
④ The law of conservation of the linear momentum of a particle If F 0,then mv is a constant vector and the particle is an inertial motion.
2) The momentum of a system of particles is defined as the vector
equal to the geometric sum of the momenta of all the particles of the
system:
Pmivi .
P m ivi m i d dr td d tm iri
rc

miri
M
miri M rC
P m ivi M vC
The momentum of a system is equal to the product of the mass of the whole system and the velocity of its center of mass.
d d ( m i tv i)F i(i)F i(e ).BF u ii t0 an w g de et
dP
dt
(e)
Fi
the momentum theorem of a system of particles.
The derivative of the linear momentum of a system of particles with respect to time is equal to the geometric sum of all the external forces acting on the system.
2.Theorem of momentum of a system of particles For any particle i in the system, we have dd(tmivi)Fi(i)Fi(e). For the whole system of particles, we have
theorem for one particle.
① Differential form: d(mv)Fd tdI
The differential of the momentum of a particle equals the elementary
impulse of the force acting on it.
m, vC112l
.
For the slide block mass = m,vC3 2. l
For the rod AB: mass=m,vC2 25lAB 25lPC2 25l,AB)
Pm vC1m vC2m vC3
m [ (vC1sijnvC2co svC3)i j (vC 1co svC 2sin )j]
F i ( i ) 0 ; m O ( F i ( i ) ) 0 om r x ( F i ( i ) ) 0 。
§12.2 Momentum and Impulse
1. Momentum
1) Momentum of a particle.The product of the mass of a particle and its velocity is called the momentum of a particle. It is a time-dependent vector with the same direction as the
Bem c a a m u d v s F esd o (m v ) F
dt
dt
The derivative of the momentum of a particle with respect to time is
equal to the force acting on the particle. This is the momentum
§12.1 The Center of Mass of a System of Particles
1. The center of mass. The center of mass of a system of particles is called center of mass. It is an important concept representing the distribution of mass in any system of particles.
Chapter 12: Theorem of Momentum
§12.1 The center of mass of a system of particles §12.2 Momentum and impulse §12.3 Theorem of momentum §12.4 Theorem of motion of the center of mass
2. External forces and internal forces of a system of particles
External forces are the forces exerted on the members of a system by particles or bodies not belonging to the given system Internal forces are the forces of interaction between the members of the same system. As far as the whole system of particles is concerned, the geometrical sum (the principal vector) of all the internal forces of a system is zero. The sum of the moments (the principal moment )of all the internal forces of a system with respect to any center of axis is zero, too.
The rods OA and AB are homogeneous and of mass m. The mass of the slide block at B is also m. Determine the momentum of the system when j=45º.
Solution: For the rod OA mass =
,
.
Pz
mivCiy mivCiz
mi yCi mi zCi
, , .
Example
In the mechanism shown in the figure, OA rotate with a constant angular
velocity . Assume that OA=L,OB=L.
② Integral form.
t2
mv2 mv1 FdtI
t1
In a certain time interval, the change of the momentum of a particle
is equal to the impulse of the force during the same interval of time.
velocity, the unit of which is kgm/s. pmivi
Momentum is a physical quantity measuring the intensity of the mechanical motion of a material body. For example, the velocity of a bullet is big but its mass is small. In the case of a boat it is just opposite.