Post on 10-Jun-2018
Chungnam National University
Kinetics of a Particle: Impulse and Momentum
Linear momentum
Linear impulse
mºL v
dtº òI F
( )d dm mdt dt
= = =å LF a v
2
12 1 2 1
t
tdt m m= - = -åò F L L v v
Newton’s 2nd law: The resultant of all forces acting on a particle is equal to its time rate of change of linear momentum.
Chungnam National University
Principle of Linear Impulse and momentum
2
11 2
t
tm dt m+ =åòv F v
2
1
2
1
2
1
1 2
1 2
1 2
( ) ( )
( ) ( )
( ) ( )
t
x x xt
t
y y yt
t
z z zt
m v F dt m v
m v F dt m v
m v F dt m v
+ =
+ =
+ =
åòåòåò
2
12 1 2 1
t
tdt m m= - = -åò F L L v v
Chungnam National University
Linear Impulse and momentum: Examples
Chungnam National University
Chungnam National University
Principle of Linear Impulse and momentum for a system of particles
ii i i
dmdt
+ =å å å vF f
2
11 2( ) ( )
t
G i Gtm dt m+ =åòv F v
( )ii i
d mdt
+ =vF f for particle i
For system of particles
2
121
( ) ( )t
i i i i itm dt m+ =å å åòv F v
G i i
G i i
m m
m m
=
=åå
r r
v vim m=åwhere
For rigid body
Chungnam National University
Conservation of Linear Momentum for a system of particles
21( ) ( )i i i im m=å åv v
1 2( ) ( )G G=v v
If the resultant force on a particle is zero during an interval of time, 2
121
( ) ( )t
i i i i itm dt m+ =å å åòv F v
G i im m=åv vFor rigid body, since
When is the resultant force on a particle zero during an interval of time? (1) Particles collide or interact.
(2) External impulse is negligible, when the time period is very short2
11 2
t
A A A Atm dt m+ - =åòv F v
2
11 2
t
B B B Btm dt m+ =åòv F v
1 1 2 2A A B B A A B Bm m m m+ = +v v v v
Chungnam National University
Conservation of Linear Momentum: example
Chungnam National University
Conservation of Linear Momentum: example
1 1 ( )proj proj Block Block proj Block proj Blockm m m m m m+ = + = +v v v v v
Chungnam National University
Impact
Definition of impact: collision between two bodies is characterized by the generation of relatively large contact forces that act over a very shot interval of time.
Chungnam National University
Central Impact
1 1 2 2( ) ( ) ( ) ( )A A B B A A B Bm v m v m v m v+ = +
Law of conservation of linear momentum is valid. Why?Internal impulse of deformation and restitution cancel, during collision
How many unknowns in the above equation?
Chungnam National University
Central Impact
1( )A A Am dt m- =òv P v
Deformation PeriodFor particle A
For particle B
1( )B B Bm dt m+ =òv P v
1( )A A Adt m m= -òP v v
1( )B B Bdt m m= -òP v v
-
Chungnam National University
Central Impact
2( )A A Am dt m- =òv R v
Restitution Period
For particle A
For particle B
2( )A A Adt m m= -òR v v
2( )B B Bm dt m+ =òv R v
2( )B B Bdt m m= -òR v v
-
Chungnam National University
Coefficient of Restitution
2
1
( )( )
A
A
Rdt v vev vPdt-
= =-
òò
2
1
( )( )
B
B
Rdt v vev vPdt
-= =
-òò
2 2
1 1
( ) ( )( ) ( )
B A
A B
v vev v
-=
-
Coefficient of restitution: the ratio of the restitution impulse to the deformation impulse.
For particle A
For particle B
(1)
(2)
Eliminate v using eqs. 1 & 2
relative velocity of separationerelative velocity of approach
- =
Chungnam National University
Central Impact
1 1 2 2( ) ( ) ( ) ( )A A B B A A B Bm v m v m v m v+ = +
e=1: restitution impulse = deformation impulse
No energy loss – perfectly elastic
e=0: plastic impact 100% energy loss 2 2( ) ( )A Bv v v= =
Summary of Central impact problem
2 2
1 1
( ) ( )( ) ( )
B A
A B
v vev v
-=
-
Chungnam National University
Impact example
Chungnam National University
Impact example
Chungnam National University
Oblique ImpactX direction
1 1 1( ) ( ) cosAx Av v q=1 1 1( ) ( ) cosBx Bv v f=
Y direction1 1 1( ) ( ) sinAy Av v q=
1 1 1( ) ( ) sinBy Bv v f=
2 2 2( ) ( ) cosAx Av v q=
2 2 2( ) ( ) cosBx Bv v f=
2 2 2( ) ( ) sinAy Av v q=
2 2 2( ) ( ) sinBy Bv v f=unknowns
Chungnam National University
Oblique ImpactX direction
1 1 2 2( ) ( ) ( ) ( )A Ax B Bx A Ax B Bxm v m v m v m v+ = +
2 2
1 1
( ) ( )( ) ( )
Bx Ax
Ax Bx
v vev v
-=
-
Y direction
1 2( ) ( )A Ay A Aym v m v=
1 2( ) ( )B By B Bym v m v=
Chungnam National University
Angular Momentum
( ) ( )( )oH d mv=
The moment of the linear momentum L about O is defined as the angular momentum Ho of particle P about O.
o
o x y z
x y z
m
i j kr r rmv mv mv
= ´
=
H r v
H
Unit: kg(m/s)m=kg(m/s2)ms=Nms
Chungnam National University
Relation Between Moment of a Force and Angular Momentum
( )
o
o
m
md m m mdt
=
= ´ = ´
= ´ = ´ + ´
åå åF v
M r F r v
H r v r v r v
&
&
& & &
oo=åM H& =åF L&
The resultant moment about the fixed point O of all forces acting on a particle is equal to its time rate of change of angular momentum of the particle about O.
Chungnam National University
System of Particles
( ) ( ) ( )i i i i o´ + ´ =ir F r f H&
( ) ( ) ( )i i i i i o´ + ´ =å å år F r f H&
For the particle i
oo=åM H&
For system of particles
The sum of the moments about point O of all the external forces acting on a system of particles is equal to the time rate of change of the total angular momentum of the system about point O.
Chungnam National University
Angular Impulse and Momentum Principles
2
1
2
1
2 1
1 2
( ) ( )
( ) ( )
t
o o ot
t
o o ot
dt
dt
= -
+ =
åòåò
M H H
H M H
Principle of angular impulse and momentum
ooo
ddt
= =å HM H& oodt d=åM H
2 2
1 1
( )t t
ot tangular impulse dt dtº = ´ò òM r F
2
11 2( ) ( )
t
o o otdt+ =å åòH M H
For a particle
For system of particles
Chungnam National University
2
1
2
1
1 2
1 2( ) ( )
t
t
t
o o ot
m dt m
dt
+ =
+ =
åòåò
v F v
H M H
2
1
2
1
2
1
1 2
1 2
1 2
( ) ( )
( ) ( )
( ) ( )
t
x x xt
t
y y yt
t
o o ot
m v F dt m v
m v F dt m v
H M dt H
+ =
+ =
+ =
åòåòåò
Vector formulation
Scalar formulation (2D case)
Impulse and Momentum Principles
Chungnam National University
Conservation of Angular Momentum
1 2( ) ( )o o=H H
When the angular impulses acting on a particle are all zero during the time t1 to t2, then
1 2( ) ( )o o=å åH HFrom t1 to t2, the particles angular momentum remains constant.
Conservation of angular momentum of a system of particles
Chungnam National University
Conservation of Angular Momentum: example
Chungnam National University
Conservation of Angular Momentum: example