Physical Sciences 2
Reading for Thursday, Sept. 17
1
Forces and Momentum
At this point, we’ve talked a great deal about
isolated
systems.
As you know, the total
momentum of any isolated system—no matter how complex—is constant:
For an isolated system,
d
dt
!
p
tot
(
)
=
0
Although this fact about isolated systems is useful in those limited cases when we can treat a real
system as approximately isolated, nearly all interesting systems are
not
isolated.
Indeed, most
real systems exhibit both
internal
and
external
interactions.
As you recall, any system that
participates in external interactions (interactions between an object
inside
the system and an
object
outside
the system) is
not isolated
:
system
surroundings
boundary
internal
external
external
We now must consider how a system will behave if it is
not
isolated.
Newton proposed that the
time rate of change of the momentum of a system is equal to the
net external force
on that
system.
That is, he proposed:
For a nonisolated system,
d
dt
!
p
tot
(
)
=
!
F
ext
!
(The
∑
symbol, which represents a sum, is a reminder that we are considering the
net
external
force—i.e., the vector sum of all external forces—and not just any individual external force.)
These
forces
, which change the momentum of a system, arise from the
interactions
between
various objects.
External forces arise from external interactions; internal forces arise from
internal interactions.
This proposal, that
a net external force causes a change in momentum
,
is the heart of Newton’s system of mechanics and is indeed the original statement of Newton’s
Second Law.
Forces on a Single Object: Newton’s Second Law
Usually in Newtonian mechanics, we will choose our “system” to consist of a single
object; any system of more that one object can be broken down into several simpler systems,
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Physical Sciences 2
Reading for Thursday, Sept. 17
2
each of which has only one object.
For a single object, of course,
!
p
=
m
!
v
.
(From our previous
discussion, we know that the velocity
!
v
in this expression is the velocity of the
center of mass
of
the object.)
Thus, if our system consists of a single object, we can write:
d
!
p
dt
=
!
F
ext
!
You should read this equation as “The time rate of change of the momentum of an object is equal
to the net external force acting on that object.”
This expression is close to Newton’s original
statement of the second law:
Lex II: Mutationem motus proportionalem esse vi motrici impressae et fieri
secundum lineam rectam qua vis illa imprimitur.
Law II: The rate of change of motion (momentum) of a body is
proportional to the force acting on the body and is in the same
direction
.
Although this equation may look unfamiliar, we can cast it in a familiar form if we turn it around
and remember that we’re dealing with objects whose mass is essentially constant:
!
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 Fall '10
 LOGANMCCARTY
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