MEAM 211
Beyond FBD=IRD
What else can we learn from Newton’s 2nd and 3rd laws?
zLinear momentum and Linear impulse
zAngular momentum and Angular impulse
zApplication
¾Impacts between particles
Dynamics of Particles
[TS, Chapter 3]
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Newton's Laws and Momentum: Linear and Angular Impulse, Conservation of Momentum, Impact, Study notes of Mechanical Engineering
The concepts of linear and angular momentum, impulse, and conservation of momentum as described by newton's laws. It covers the calculation of linear and angular impulse, the principles of conservation of momentum, and the modeling of impacts using poisson's model of restitution and compression impulses.
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MEAM 211
Beyond FBD=IRDWhat else can we learn from Newton’s 2
nd
and 3
rd
laws?
z
Linear momentum and Linear impulse
z
Angular momentum and Angular impulse
z
Application
¾
Impacts between particles
Dynamics of Particles
[TS, Chapter 3]
MEAM 211
Linear Momentum
Newton’s 2
nd
Law
Linear Impulse
m
m
total force
F
m
a
F
m
a d^ dt
L
F
=
linear momentum
L
m
v
F
m
a
∫
∫
=
2 1
2 1
t t
t t
dt
m
dt
a
F
1
2
2 1
(^21)
L
L
a
−
=
=
∫
−
t t
dt
m
LI
m
time t
1
time t
2
F
( t )
LI
is a
vector!!!
MEAM 211
Angular Impulse
1
2
2 1
(^21)
L
L
a
−
=
=
−
t t
dt
m
LI
m
time t
1
time t
2
F
( t )
O
P
r
2 1
2 1
t
t
dt
AI
O
O
t t
O
O
H
H
M
−
MEAM 211
Principles of Conservation of Momentum
Conservation of Linear Momentum
Conservation of Angular Momentum
1
2
2 1
2 1
L
L
a
−
=
=
−
t t
dt m
LI
If
LI
L
2
L
1
1
2
2 1
2 1
t
t
dt
AI
O
O
t t
O
O
H
H
M
−
=
∫
−
If
AI
O
H
O
2
H
O
1
MEAM 211
Example 2: Force/Impulse
mass of the ball = 0.145kg.
mass of the bat = 0.9 kg.
velocity of swing = 30 m/sec
contact time < 1 msec
fastball velocity ~ 100 mph (44.4m/s)
calculate average force, impulse
MEAM 211
z
Linear impulse = Change in Linear momentum z
Angular impulse = Change in Angular momentum z
Impact deals with finite impulses in very, very small time
Impact
[TS, Chapter 3.7]
MEAM 211
Linear Impulse with Infinitesimal time
Large force, infinitesimal timeinterval
z
lim
t tends to zero
Impulse = area under the curve
F
t 1
t 1
t
F
t 1
z
Time interval is (almost)zero
z
Force is (almost)infinitely large
z
But,…impulse is finite
3 key attributes
MEAM 211
When is the “zero time” impulse model appropriate?
Not always…!
MEAM 211 Reality
Collisions have three stages or states
z
Deformation (compression)
¾
Deformation increases
z
State of maximum deformation
¾
Rate of change of deformation is zero
z
Restitution (expansion)
¾
Deformation decreases
δ
D
R
MEAM 211
Collisions in 1 dimension
Change in momentum
z
Let
LI
denote the linear impulse
acting on 2
−
−
=
−
1 1
1 1
v m
v m
LI
−
−
=
2 2
2 2
v
m
v
m
LI
- 1 v
− 1 v
− 2 v
- 2 v
Note all quantities positive
Adding
MEAM 211
How can we model impacts?
Newton’s laws
z
Accelerations are infinite if forces are infinite
Conservation of energy
z
Only conserved for “elastic” collisions
z
Inelastic?
Conservation of momentum
z
For each particle?
z
For the system of particles?
Need an impact model
z
Poisson’s model of restitution and compression(deformation) impulses
z
Newton’s model of approach and separation velocities