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8
Momentum and impulse
8.1 Linear momentum
LO
=
c
R, (m,
oR,
)
We have shown in Chapter 3 that, for any system
of particles or rigid body,
(3.15)
Integrating this equation with respect to time
=
wxrn,R,2
Since
c
rn, R:
is
Io,
the moment of inertia of
the body about the zaxis, the total moment
of
momentum for this case is
F
=
IMifc
Lo
=
low
(82)
gives
[f
Fdl= [2Mi;;dt
=
~i~~~i~~
(8.1)
The integral J:Fdt is known as the impulse and
In Chapter 6 we showed that, for a rigid body in
(6.11)
1
general plane motion,
MG
=
IGh
is a vector quantity. Because
c
m,i,
=
MiG
=
the linear momentum
gives
we can write
[
r MGdt
=
[
f
IG
hdt
=
IGO~

6J1
(8.3)
impulse
=
change in linear momentum
or, symbolically,
that is,
I=
AG
moment of impulse
=
change in moment of
momentum
8.2 Moment of momentum
From Fig. 8.1 we see that the moment
momentum about the zaxis of a particle which is
KG
=
ALG
If rotation is taking place about a fixed axis
then equation 6.13 applies which, when inte
grated, leads to
M,dt
=
Iowdt
=
Io%
lowl
(8.4)
or
Ko=ALo
8.3 Conservation of momentum
If
we now consider a collection of particles or
effects from bodies outside the system, then
moving
On
a circu1ar path, radius
R1,
about the
rigid bodies interacting without any appreciable
zaxis is
=
R, (m, OR,
1
Crn,Fl
=
o
(8.5)
For a rigid body rotating about the zaxis with
angular velocity
w,
the total moment of momen
tum is
so
that
c~,~,
=
constant
i.e. linear momentum is conserved.
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Momentum
and
impulse
Extending equation 6.12a for a system of
bodies,
c
ZG
h
+
TGMaGe
=
0
(8.6)
c
o
+
2
rG
MvGe
=
constant
(87)
Integrating with respect to time gives
which is an expression of the conservation of
moment of momentum. The term ‘angular
momentum’ is often used in this context but is not
used in this book since the term suggests that only
the moments of momentum due to rotation are
being considered whereas, for example, a particle
moving along a straight line will have a moment
of momentum about a point not on its path.
8.4
Impact of rigid bodies
We can make use of these conservation principles
very effectively in problems involving impact. In
many cases of collision between solid objects the
time
of contact is very small and hence only small
changes in geometry take place during the contact
period, although finite changes in velocity occur.
As an example, consider the impact of a small
sphere with a rod as shown in Fig. 8.2. The rod is
initially at rest prior to the impact, so that
u1
=
0
w1
=
0;
u2, v2
y
are the velocities after
impact.
Figure
8.2
Conservation of linear momentum gives
mvl
=
Mu2
+
mv2
(8.8)
and conservation of moment of momentum about
an axis through G gives
mvla
=
IG(r4?+mv2a
(8.9)
So far we have two equations, but there are three
unknowns. To provide the third equation we shall
make some alternative assumptions:
i)
the rigidbody kinetic energies are con
served, or
ii) the two objects coalesce and continue as a
single rigid body.
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 Fall '06
 EDeSturler
 Equations

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