Download 15-1. principle of linear impulse and momentum

Chap 15 Kinetics of a particle impulse and
momentum
15-1. principle of linear impulse and momentum
(1) Linear impulse I ,N S
The integral I =∫F dt is defined as the linear impulse which measure the
effect of a force during the time the force acts .
 t2 
I   F (t )dt
t1
(2) Linear momentum L , kg/s
The form L=m v is defined as the linear momentum of the particle .
(3) Principle of linear impulse and momentum the initial momentum of the
particle at time T1 plus the vector sum of all the time integral t1 to t2 is
equivalent to the linear momentum of the particle at time t2 .
t2
mv1   
t1

Fdt  mv2
Equation of motion for a particle of mass m is ∑F =m a = m dv /dt
∑F dt = m dv
Integrating both sides to yield .

t2
t1

v2

Fdt   mdv  mv2  mv1
v1
mv1   
t2
t1

Fdt  mv2
In x,y,z, components
m(v x )1    Fx dt  m(v x )
t2
t1
2
m(v y )1    Fy dt  m(v y ) 2
t2
t1
m(v z )1    Fz dt  m(v z ) 2
t2
t1
15.2 Principle of linear impulse and momentum for a
system of particles.
Equation of motion for a system of particle is

dv


 Fi   mi ai   mi dti


  Fi dt   mi dvi

t2
t1

or

 Fi dt 

( vi ) 2
( vi )1

m
d
v
 i i
 ( vi ) 2
 mi vi 
( vi )1





m
v

m
v
 i i 2  i i 1
   

 mi vi
1
t2
t1
 i  ...........1


Fi dt  m v
The location of the mass G of the system is
i
2


mrG   m r
i i
m

drG
dt
m   m 
i
dr
 m
i
i dt
or


vG   m vi ............2
i
Substitute Eq (2) to (1) to get






t
m vG    t Fdt  m v


G
2
1
1

2
15.3 Conservation of linear momentum for a system
1. Internal impulses
The impulse occur in equal but opposite collinear pairs.
t2 
t2
t2
 t1 Fdt  t1 f12dt  t1 f 21dt
2. Non impulsive forces
The force cause negligible impulses during the very short time
period of the motion studied .
(1)weight of a body.
(2)force imparted by slightly de formed spring having a
relatively small stiffness ( Fs = ks ).
(3)Any force that is very small compared to other longer
impulsive forces.
3.Impulsive forces
The forces are very large , act for a very short period of time and
produce a
significant charge in momentum.
Note :when making the distinction between impulsive and
nonimpulsive
forces it is important to realize that it only applied during a
specific time.
Conservation of linear moment.
The vector sum of the linear moments for a system of particles
remain constant throughout the time period t1 to t2
15.4 Impact
Impact
(1) Definition. Two bodies collide with each other during a very
short internal of time which causes relatively large impulsive
forces exerted between the bodies.
(2) Types.
(A). Central impact. The direction of motion of the mass
center of the two colliding particles is along the line of
impact.
(B). Oblique impact. The motion of one or both the particles
is at angle with the line of impact.
Central Impact
To illustrate the method for analyzing the mechanics of impact ,
consider
the case involving the central of 2 smooth particles A and B.
mAvA1
mAvA2
V
∫Pdt
-∫Pdt
B
A
Require
vA1>vA2
A
Effect of A on B
B
Effect of B on A
AB
Maximum deformation
Deformation impulse
Before impact
conservation of momentum of the system of panicles A and B
mAv A1  mbvb1 m A v A2  mB vB 2
Principle of impulse and momentum for panicle A (or B)
(a) Deformation please for A
 Pdt
mAv A   Pdt  mAv
(b) Restitution place for A
mv   Rdt  mvA 2
deformation impulse is
 Pdt  m v
A A
 m Av
Restitution impulse is
 Rdt  m v  m v
A
A A2
Define
 Rdt  m v  m v
 Pdt m v  mv
A
A A2
A A1
v  vA2
e
vA1  v
Similarly by considering particles B We have
e
v  v A2
v A1  v
(v A1  v)e  v  v A 2
1.v A1e  v A 2  (1  e)v
(v  v B1 )e  v B 2  v
2.(1  e)v  vB 2  vB1e
v A1e  v A2  vB 2  vB1e
vB 2  v A 2
e 
v A1  vB1
=relative velocity just after impact
=relative velocity just before impact
In general
0  e 1
(1)e  1
Elastic impact
 Pdt   Rdt
(1)e  0
Plastic impact
 Pdt  0
3. Oblique impact
Plastic Impact (e = 0):The impact is said to be inelastic or plastic when e = 0.
In this case there is no restitution impulse given to the particles
(∫Rdt = 0), so that after collision both particles couple or stick
together & move with a common velocity.
Oblique Impact. When oblique impact occur between2 smooth
particles, the particle move away form each other with velocities
having unknown direction as well As unknown magnitudes.
Provided the initial velocities
are known,4 unknown are present in the problem.
vA2
vB2
mAvAy2
Line of impact
mAvAx1
∫Fdt
+
=
mAvAx1
vA1
vB1
mAvAy1
Plane of contact
mBvBy2
mBvBx1
∫Fdt
+
=
mBvBx2
mBvBy1
15-5 angular momentum
1. angular momentum
Moment of the particle’s linear momentum about point 0 or other
point.
2. scalar formulation
Assume that the path of motion of the particle lies in x y-plane.
Magnitude of angular momentum Ho is
Ho= Ho = d(mv)
Angular momentum=moment of momentum
Ho=r x mv= i
j
k
rx
ry
rz
mvx mvy mvz
15-6 Moment of a force and Angular momentum
Equation of motion of a particle is ( m=constant )


 F  mV
By performing a cross-product multiplication of each side of this
equation by the position vector r , We have the moments of forces

About point O
F
 


r  F  r x(mV )
  

 Ho  r xmv   Mo


or  Mo  Ho

 Mo
 

 Ho  r x(mv )

d  
 Ho  (r xmv )
dt  

 r xmv  r xmv
   
 m(r xv )  r xmv
Resultant
moment
about pt.o of
all force
Time rate change
of angular
momentum of the
particle about pt.o
Recall
 
F  L