PHYS 2001 Lecture Notes - Lecture 16: Momentum, Net Force, Pulmonary Artery
7.2 Conservation of Linear Momentum
We talked last time how Newton’s Laws naturally show up in our discussions of
momentum.
It turns out we can write all of Newton’s Laws in terms of momentum:
1. Law of Inertia: A body at rest will remain at rest, and a body in motion will
continue in straight line motion unless acted upon by some external force.
constant=p
!
2. Force equals the rate of change of momentum.
3. Action Equals Reaction:
2112 FF !=
t
p
t
p
!
!"
=
!
!
#21
21 pp !"=!#
*Changes in the momentum are of equal magnitude and in opposite directions!
What are the consequences of this???
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Let’s just consider the simplest interacting system that we can. Such a system
will consist of two particles, m1 and m2, interacting with each other.
m1
m2
Assume for now that the two particles form an isolated system from the rest of the
universe, and the only forces they experience are their mutual forces.
In other words, there are no external forces.
Let the net force on particle 1 by 2 be F.
FThen by Newton’s 3rd Law, we know that the
net force on particle 2 is –F.
-F
t
p
F!
!
=1
t
p
F!
!
="2
By Newton’s 2nd Law.
Now let’s add these two equations together.
t
p
t
p
FF !
!
+
!
!
="+21
)(
0=
0
)( 21 =
!
+!
"
t
pp
,0=
!
!
"
t
P
where , the total momentum.
21 ppP +=
Thus, does not change in time. In other words, it is a constant of the
motion!
21 ppP +=
This is the Law of Conservation of Momentum.
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It is a direct consequence of Newton’s 3rd Law.
In general:
=
"
"
ext
F
t
P!
!
In words, this says that the rate of change of the total
momentum in a system is equal to the sum of the
external forces acting on the system.
But, if the , then and P is a constant of the motion, i.e. it does
not change in time!
0=
!ext
F
!
0=
!
!
t
P
!
In the absence of external forces, the total momentum final of a system
has to be equal to the total momentum initial of the system.
fo PP =
This leads to equations like:
2121 2121 ffoo vmvmvmvm =
This is exactly analogous to what we did with energy
conservation when which leads to:
0WNC =
.
fo EE =
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Document Summary
We talked last time how newton"s laws naturally show up in our discussions of momentum. Force equals the rate of change of momentum: action equals reaction: *changes in the momentum are of equal magnitude and in opposite directions! Let"s just consider the simplest interacting system that we can. Such a system will consist of two particles, m1 and m2, interacting with each other. Assume for now that the two particles form an isolated system from the rest of the universe, and the only forces they experience are their mutual forces. m2. In other words, there are no external forces. Let the net force on particle 1 by 2 be f. Then by newton"s 3rd law, we know that the net force on particle 2 is f. Now let"s add these two equations together. m1. ,0= where , the total momentum. pp p. In other words, it is a constant of the motion! This is the law of conservation of momentum.
