Newton's Second Law of Motion
Newton's 2nd law is really
a mathematical statement of how "force"
relates to the change in an object's motion. In part, it acts as a defining
statement for the properties termed "force" and "mass."
We already know that mass is a measure of the degree of inertia an object possesses,
and we intuitively understand what force is  but we don't really have independent
definitions of them or procedures for measuring them. The second law relates
them to each other, and in so doing, acts as a joint operational definition
of them.
Mathematically, Newton's
2nd law says F =
Δp/Δt.
In words, we might say that force, F, is equal to the change
in momentum, Δp, with the corresponding
change in time, Δt. This brief mathematical
statement contains quite a bit of information.
First, what is momentum?
By definition, p = mv, where p
is the property called momentum, m is the object's mass, and v
is its velocity. [In this equation, and above, you see that some of the letters
in the equation are in bolditalic while others are just italic.
The items designated in bolditalic are called vectors;
they have mathematical properties of size and direction. The items in italic
are scalars  regular numbers denoting size only. So mass, m,
is a scalar defined only by its size while velocity, v, is a vector
designating the size of an object's speed and which direction it is traveling
in.]
Newton recognized that the change in an object's motion with an applied force
depended also on its mass (degree of inertia). This is a common sense idea to
us; if we push with a certain amount of force on a toy car and with the same
amount of force on a real car, the toy car will have a much greater change in
its motion. Momentum incorporates inertia (mass) with an object's motion (speed)
to produce a composite that relates to force.
The relationship with force
is to the rate at which momentum changes over time! [The Δ symbol in the
equation means "change
in."] So, force F produces a change in an object's momentum
p over time, where the size of the force defines the rate of
change and the direction of the force defines the direction of the change. If
the force's direction lines up with the object's original direction of travel,
the object's speed will increase (force in the same direction as the object's
velocity) or decrease (force in the opposite direction to the object's velocity).
But if the force points in some other direction, it will cause the object to
change course as well.
As long as the mass part
of the momentum does not change, the change in momentum can be expressed
as Δp = Δ(mv)
= m(Δv). Because we define acceleration,
a, as the change in velocity with time a = Δv/Δt,
Newton's second law can also be written as F = ma.
This is a more familiar expression to anyone who had a physics course before.
[It is interesting to note that since the acceleration a relates
to Δv, and Δv
is not zero if the direction of v changes even when its size stays
the same, an object accelerates if its direction of travel changes while it
is moving at constant speed.]
Newtons 2^{nd} law can also be understood to describe the action of
many forces acting on a single object, or many forces acting on many objects.
Under these circumstances, the force F is understood to represent
the overall, or net, force and the momentum change Δp
to be the overall, or net change  over all of the objects involved.
Continue to Newton's Third Law.
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