Up until now, we have analyzed the motion of objects
that we treated as so-called point masses, where literally
all of the mass of the object was treated
as lying at a single point.
This is obviously an unrealistic idealization.
Real objects have spatial extent and come
in various sizes and shapes, some regular
and some highly irregular.
The mass distribution within these extended objects
might be smooth and uniform or highly non-uniform and lumpy.
Moreover, sometimes, we will be interested in simultaneously
analyzing a large system of particles or objects.
To deal properly with such situations,
it is necessary to introduce a new physical quantity called
the momentum, which is a vector that we traditionally denote
with the symbol P. For a single point mass,
the momentum is the product of the mass and the velocity
vector.
The form of Newton's second law that we have used so far,
F equals ma, is actually only true for the special case
of a point mass.
The more general form of the Newton's Second Law
is that F equals dp/dt, that is that the force is
equal to the time derivative of the momentum vector.
This is applicable to either a single particle
or to a system of particles.
This more general form provides us
the way to address extended objects
or multi-particle systems.
We will once again appeal to a calculus-like argument,
breaking down an extended distribution
into a large number of small elements,
treating each element separately, and then summing
the results.
As part of this process, we will introduce the concept
of the center of mass, a sort of mass weighted average position
for an extended object or system.
Momentum is also the first of several special measurable
quantities that we will discuss that obey special powerful
rules called conservation laws.
The other two quantities are energy and angular momentum.
Conservation laws are principles to provide that the given
measurable quantity remains unchanged during the evolution
of a system with time as long as certain requirements are
met, essentially the requirement that a system remains isolated.
With this restriction in place, the possible evolution
of motion of a particle or system of particles
is greatly restricted, simplifying its calculation.
We will see how this can be used to our advantage
in understanding what will happen.