工程力学全英文Engineering Mechanics (21)

p
pi
mi
v i
In rectangular coordinates,
px py
mi
v ix
mi viy
pz
mi
v iz
vix
dxi dt
viy
dyi dt
v iz
dzi dt
xC
mi xi m
yC
mi yi m
zC
mi zi m
Center of mass
dp dt
F (e) i
Differential form of the for a system of particles
px
mi
v ix
mi
dxi dt
d dt
(mi xi )
d dt
(
mi
xi
)
d dt
(mxC
)
m dxC dt
Principle of Impulse and Mox
py
m
dyC dt
pz
m
dzC dt
px py
mvCx mvCy
pz mvCz
or p mvC
dxC
dt
v Cx
dyC dt
v Cy
dzC dt
v Cz
The momentum of a system of particles is equal to the product of the total mass of the system and the velocity of the mass center of the system.
d(mvz dt
)
Fiz
Differential form
or
mmvv22yx
mv1x mv1y
Six Siy
mv2
z
mv1z
Siz
where
Six
t2 t1
Fixdt
Siy
t2 t1
Fiydt
Siy
t2 t1
Fiz dt
Integrating form
Principle of Impulse and Momentum
This equation is the differential form of the principle of impulse and momentum for a particle. It is actually the original form of Newton’s second law.
Principle of Impulse and Momentum
Principle of impulse and momentum for a system of particles
Consider a system of particles. For any arbitrary particle i, it is subjected to a resultant
p2 p1 mv2 mv1 Si
This equation is the integrating form of the principle of impulse and momentum for a particle. It can be stated as: the change of momentum of a particle during a finite time interval is equal to the impulse of the acting forces during the same interval.
with
mass
m
and
velocity
v
at
an
instant
is
defined
as
p mv
The momentum of a particle is a vector quantity that acts in the same direction as the velocity vector.
Principle of Impulse and Momentum
dp dt
Fi
Multiply both sides of the equation by dt and integrate from t1 to t2,
p2 p1
t2 t1
Fi
(t)dt
the impulse of force Fi for the time interval t1 to t2, denoted as Si
Principle of impulse and momentum for a system of particles
The momentum of a system of particles is defined as the vector sum of the momenta of all particles in the system
Principle of Impulse and Momentum
The principle of impulse and momentum can be expressed in rectangular coordinates as:
d(mvx
dt d(mv
y
dt
) )
Fix Fiy
internal force and a resultant external force.
d(mi
v i
)
dt
F (i) i
F (e) i
For the whole system,
(i 1, 2, n)
0
d(mivi ) dt
F (i) i
F (e) i
d dt
(mi
v i
)
F (e) i
Principle of Impulse and Momentum
Principle of Impulse and Momentum
Principle of impulse and momentum for a particle
Momentum of a particle The momentum of a particle
Rephrasing of Newton’s second law
dp dt
dmv
m
dv
dt
dt
ma
Fi
The rate of change of the momentum of a particle is equal to the resultant force acting on the particle.