EA3: Systems Dynamics
Momentum Methods
Sridhar Krishnaswamy
1
V IMPULSE AND MOMENTUM
V.1 Linear Impulse:
Newton’s laws say that if a net force
F
acts on a body, then:
F
=
d
p
dt
(5.1)
where
p
=
m
v
is the linear momentum of the body, where m is the mass and
v
is the
velocity of the body.
Integrating the above with respect to time:
F
dt
t
1
t
2
∫
=
d
p
p
1
p
1
∫
=
p
2

p
1
(5.2)
The left side is called the
linear impulse
due to the force
F
over the time interval t
1
to t
2
:
I
F
=
F
dt
t
1
t
2
∫
(5.3)
Remarks:
(i)
The units of linear impulse in SI are N.s.
(ii)
Often, such as during impact of bodies, it is not possible to measure the force of
impact, but it is possible to obtain an
average
measure of the force by measuring
its momentum before and after the impact through:
F
av
=
1
t
2

t
1
I
F
=
1
t
2

t
1
F
dt
t
1
t
2
∫
=
1
t
2

t
1
p
2

p
1
{
}
(5.4)
(iii)
The statement (5.2) says that the
linear impulse imparted to an object is equal to
the resulting change in linear momentum of the object
.
(iv)
If there is no net force acting on a body, clearly its linear momentum is
unchanged.
Figure 5.1
: Some representative impulsive forces.
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EA3: Systems Dynamics
Momentum Methods
Sridhar Krishnaswamy
2
(v)
As we did when we discussed Newton’s laws, we can extend these ideas to a
system
of N particles. Let the
ith
particle have a mass m
i
, and be located at
position
r
i
moving with velocity
v
i
with respect to some chosen coordinate
system. Let us consider the forces on the ith particle in two parts:
f
ij
is the force on
the ith particle exerted by the jth particle in the system, and
F
i
is the force exerted
on the ith particle by something external to the system.
f
ij
j
∑
+
F
i
=
d
p
i
dt
(5.5)
There are N such equations, one for each particle. Suppose we sum all these N
equations together:
i
∑
f
ij
j
∑
+
F
i
i
∑
=
d
p
i
dt
i
∑
(5.6)
Clearly the first term on the left side is zero (from Newton’s third law
f
ij
=
f
ji
). Integrating the above with respect to time, we have:
F
i
i
∑
t
1
t
2
∫
dt
=
P
2

P
1
(5.7)
where the linear momentum of the total system is given by:
P
=
p
i
=
i
=
1
N
∑
m
i
v
i
i
=
1
N
∑
.
(5.8)
(vi)
If there is no net external force acting on a system of particles, then the above
says that the linear momentum of the system is conserved.
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 Fall '08
 KRISHNASWAMY
 Force, Kinetic Energy, Momentum, Sridhar Krishnaswamy

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