Newton's first law tells us about the motion
of isolated bodies.
By an "isolated body," we mean one on which the net force
is 0, one that's isolated from all interactions
to as great a degree as possible.
Newton's first law states that an isolated body
moves in a straight line at constant velocity
and will continue to do so as long as it remains undisturbed.
Note that that constant velocity might be 0.
So for example, a body at rest-- an isolated body at rest
will remain at rest if left undisturbed.
An isolated body moving at constant velocity
will continue to move at a constant velocity
as long as it remains undisturbed.
It turns out that it's always possible to define
a coordinate system in which an isolated body moves
at constant velocity, perhaps 0.
Such a coordinate system is called an "inertial coordinate
system."
So another way of stating Newton's first law
is that inertial coordinate systems exist.
Now, it's worth pointing out that not
all useful coordinate systems are inertial.
The ones that aren't we call "non-inertial coordinate
systems."
As an example, imagine that I am standing in an elevator that's
accelerating upward.
If we defined a coordinate system that
moves with the elevator, that coordinate system
is accelerated with respect to an observer who is at rest.
In that accelerated coordinate system,
an isolated body would not move at a constant velocity.
So there are applications where using
non-inertial coordinate systems is convenient.
However, in this course, we will concentrate
on inertial coordinate systems in which isolated bodies move
at a constant velocity.
What if the body is not isolated?
If a force acts on a body, then it
will accelerate in proportion to that force.
In particular, for a point like constant mass,
Newton's second law tells us that the vector force
is equal to the mass times the vector acceleration.
So in other words, the acceleration caused by a force
is proportional to that force and the constant
of proportionality is the mass.
And there are two important points
I want to make about Newton's second law.
The first is that forces always involved
real physical interactions.
Something must be acting on the object.
If nothing is acting on the object,
if the object is isolated, then we
know-- from Newton's first law-- that an isolated body never
accelerates in an inertial frame.
It moves at constant velocity.
So the only way to have an object
move with a velocity that's changing--
that is, to have an acceleration--
is to have a force acting on it.
And Newton's second law tells us exactly how that works.
The second point I want to make is
that this statement of Newton's second law, F equals ma,
is actually a special case.
It's for the special case of a constant point like mass.
More generally, Newton's second law is written as the force
is equal to the time derivative of the momentum, p.
Now, we'll talk about momentum later in this course in more
detail, but I'll just tell you that the momentum,
for a point like mass, is defined
as the mass times the velocity.
Now, for a point mass that's constant,
these two equations are exactly identical.
If you take the derivative of m times v where m is a constant,
you just get m times the vector a.
And you get the first equation back.
So it's just a fancier way of writing the same thing.
It doesn't convey any new information.
So why do we talk about this as being the more general form?
It's because this second form of Newton's second law
can be generalized to a system of particles or a system
where mass is flowing and the mass of something is changing.
You can't describe that by F equals ma,
but it's always true that, for a given system, F equals dp/dt.
And we'll see how that works in more complicated mass flow
problems later in the course.