University Physics I - Classical Mechanics, 2019

6.1. FORCE 115
however, somewhat more general, so it is technically preferred, even though this semester we will
directly use F = ma throughout.
If you want an example of a physical situation where F = dp/dt is not equivalent to F = ma,
consider a system where object 1 is a rocket, including its fuel, and “object” 2 are the gases ejected
by the rocket. In this case, the mass of both “objects” is constantly changing, as the fuel is burned
and more gases are ejected, and so the more general form F = dp/dt needs to be used to calculate
the force on the rocket (the thrust) at any given time.
At this point you may be wondering just what is Newton’s first law? It is just the law of inertia:
an object on which no force acts will stay at rest if it is initially at rest, or will move with constant
velocity.
6.1.1 Forces and systems of particles
What if you had, say, three objects (let us make them “particles,” for simplicity), all interacting
with one another? In physics we find that all our interactions are pairwise additive, that is, we can
write the total potential energy of the system as the sum of the potential energies associated with
each pair of particles separately. As we will see in a moment, this means that the corresponding
forces are additive too, so that, for instance, the total force on particle 1 could be written as
F all,1 = F 21 + F 31 = dp 1
dt
(6.7)
Consider now the most general case of a system that has an arbitrary number of particles, and is
not isolated; that is, there are other objects, outside the system, that exert forces on some or all of
the particles that make up the system. We will call these external forces. The sum of all the forces
(both internal and external) acting on all the particles will take a form like this:
F total = F ext,1 + F 21 + F 31 + ...+ F ext,2 + F 12 + F 32 + ...+ ...= dp 1
dt + dp 2
+ ... (6.8)
dt
where F ext,1 is the sum of all the external forces acting on particle 1, and so on. But now, observe
that because of Newton’s third law, Eq. (6.5), for every term of the form F ij appearing in the sum
(6.8), there is a corresponding term F ji = −F ij (you can see this explicitly already in Eq. (6.8)
with F 12 and F 21 ), so all those terms (which represent all the internal forces) are going to cancel
out, and we will be left only with the sum of the external forces:
F ext,1 + F ext,2 + ... = dp 1
dt + dp 2
+ ... (6.9)
dt
The left-hand side of this equation is the sum of all the external forces; the right-hand side is the
rate of change of the total momentum of the system. But the total momentum of the system is