ENT252 - DYNAMICS
Chapter Objectives
To develop the principle of linear impulse and momentum for a particle
To study the conservation of linear momentum for particles
To analyze the mechanics of impact
To introduce the concept of angular impulse and momentum
To solved problems involving steady fluid streams and propulsion with variable
mass

Equation of motion for a particle of mass m:

F
 ma
 m dv dt

Rearranging:
 t
2 t
1
F dt or
 m v
1  v
2 dv principle of linear impulse & momentum
(time integration of the equation of motion)
Eq. 1
 2 t t
1
F dt
 mv
2
 mv
1
linear particle’s
impulse linear
momentum
Linear Momentum
- 1 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS
 Vector of L = mv in Eq. 1 is referred to as the particle’s linear momentum

Magnitude of mv units of mass-velocity
Linear Impulse

Integral I =

F dt in Eq. 1 is referred to as the linear impulse

Measures the effect of a force during the time the force acts.

Magnitude: force-time

If force – expressed as a function of time, impulse = direct evaluation of the integral

If force – constant direction during the time period t
1
to t
2
, impulse = area under the curve of force vs. time:
I

2  t t
1
F dt

If force – constant in magnitude & direction:
I

2  t t
1
F c dt

F c
( t
2
 t
1
)
 shaded rectangular area
Principle of Linear Impulse & Momentum mv
1
  t
2  t
1
F dt
 mv
2

Initial momentum of the particle at t
1
plus the vector sum of all the impulses applied to the particle during the time interval t
1
to t
2
is equivalent to the final momentum of the particle at t
2
.
- 2 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS
Scalar Equation

The scalar x, y, z components of the previous equation are: m ( v x
)
1
   t
2 t
1
F x dt
 m ( v x
)
2 m ( v y
)
1
   t
2 t
1
F y dt
 m ( v y
)
2 m ( v z
)
1
   t
2 t
1
F z dt
 m ( v z
)
2
Procedure for Analysis

To solve a linear impulse and momentum: a) FBD - establish the x, y, z inertial frame of reference
- draw the particle’s FBD – account for all the forces that produce impulses on the particle
- establish – direction & sense of the particle’s initial & final velocities b) Principle of Impulse & Momentum
- apply the principle: mv
1
  t
2  t
1
F dt
 mv
2
- if motion occurs on x-y plane – the 2 scalar component equation can be formulated by:
1. resolving the vector components of F from FBD
2. using the data on the impulse and momentum diagrams
See Example:

15.1

15.2

15.3
- 3 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS

F i
  m i dv i dt
Sum of external forces

Multiplying both sides by dt and integrating between the limits t = t
1
, v i
= (v i
)
1
& t = t
2
, v i
= (v i
)
2
,
 m i
( v i
)
1
  t

2 t
F i dt
The initial linear momenta of the system added vectorially to the impulses of all the external forces acting on the system during the time period t1 to t2 are equal to the system’s final linear momenta.
  m i
( v i
)
2

Location of mass centre G of the system: m r
G
  m i r i
,

Taking the time derivatives: m v
G m

  m i v i
 m i
Total linear momentum of the system of particles plus (vectorially) the external impulses acting on the system of particles during the time interval t
1
to t
2
equal to the aggregate particle’s final linear momentum.
- 4 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS

When the sum of the external impulses acting on a system of particles is zero:
 m i
( v i
)
1
  m i
( v i
)
2 conservation of linear momentum

In other form:
( v
G
)
1

( v
G
)
2

Applied when particles collide or interact.

Nonimpulsive forces: - causing negligible impulses
- including any force that is very small compared to other larger (impulsive) forces

Impulsive forces: - forces which are very large & act for a very short period of time; produce a significant change in momentum
- normally occur due to an explosion or the striking of one body against another
See Example:

15.4

15.5

15.6

15.7

15.8
- 5 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS

Impact occurs when two bodies collide with each other during a very short period of time causing relatively large (impulsive) forces to be exerted between two bodies.

E.g.: striking of a hammer on a rail golf club on a ball

Central impact – the direction of motion of the mass centers of the two colliding particles is along a line passing through the mass centers of the particles

Oblique impact – when the motion of one or both of the particles is at angle with the line of impact
Analysis of Central Impact

Particle have the initial momenta, ( v
A
)
1
> ( v
B
)
1

During the collision – the particles will undergo a period of deformation – equal but opposite deformation impulse

P dt

At maximum deformation – both particles move with common velocity v

Then a period of restitution occurs (particles will either return to original shape or remain deform) – restitution impulse

R dt pushes particles apart from another, where

P dt >

R dt
 Just after separation, particles’ final momenta = ( v
B
)
2
> ( v
A
)
2
- 6 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS

To determine the velocities, apply conservation of momentum for the system of particles by referring to the figure above:
+ m
A
(v
A
)
1
+ m
B
(v
B
)
1
= m
A
(v
A
)
2
+ m
B
(v
B
)
2

To solve for final velocities (initial values of particles will be known in most cases), consider deformation phase:
+ m
A
(v
A
)
1
-

P dt = m
A
v

For restitution phase:
+ m
A
(v
A
)
1
-

R dt = m
A
( v
A
)
2

Ratio of the restitution impulse to the deformation impulse = coefficient of restitution, e: e



R dt
P dt
 v
( v

A
( v
)
1
A

) v
2

For particle B: e



R
P dt dt

( v v
B

(
)
2 v
B

)
1 v

If remove unknown v:
+ e

( v
B
)
( v
A
)
1
2

( v
A

( v
B
)
)
1
2
- 7 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS
Coefficient of Restitution

In general, value of e = between zero and one:
1.
Elastic Impact (e = 1) : If the collision between the two particles is perfectly elastic, deformation impulse

P dt = equal and opposite to restitution impulse

R dt
2.
Plastic Impact (e = 0) : No restitution impulse given to the particles (

R dt = 0); after collision, both particles couple or stick together and move with common velocity.
Procedure for Analysis
(Central Impact)

Use the following two equations:
1.
The conservation of momentum applies to the system of particles,
 mv
1
=
 mv
2
2.
The coefficient of restitution, e = [( v
B
)
2
– ( v
A
)
2
] / [( v
A
)
1
– ( v
B
)
1
], relates the relative velocities of the particles along the line of impact, just before and just after collision.
Procedure for Analysis
(Oblique Impact)

In determining ( v
Ax
)
2
, ( v
Ay
)
2
, ( v
Bx
)
2
and ( v
By
)
2
, consider these four equations:
1.
Momentum of the system is conserved along the line of impact, x axis, so that
 m(v x
)
1
=
 m(v x
)
2
2.
The coefficient of restitution, e = [( v
Bx
)
2
–
( v
Ax
)
2
] / [( v
Ax
)
1
– ( v
Bx
)
1
], relates the relative velocity components of the particles along the line of impact (x axis)
3.
Momentum of particle A is conserved along the y axis, perpendicular to the line of impact, since no impulse acts on particle A in this direction.
4.
Momentum of particle B is conserved along the y axis, perpendicular to the line of impact, since no impulse acts on particle B in this direction.
- 8 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS
See Example:

15.9

15.10

15.11
 Angular momentum of a particle about point O = “moment” of the particle’s linear momentum about O.

Also being referred to as the moment of momentum.
Scalar Formulation

If a particle is moving along a curve lying in the x-y plane, magnitude of Ho :
( H o
) z
= ( d )( mv )

Common unit = kg m
2
/ s

Direction – right-hand rule
Vector Formulation

If the particle is moving along a space curve, angular momentum Ho :
H o = r x m v

In Cartesian components: i
Ho
 r x mv x j r y mv y k r z mv z
- 9 -
ZOL BAHRI / SHAZMIN ANIZA

ENT252 - DYNAMICS

- 10 -
ZOL BAHRI / SHAZMIN ANIZA