>> What up, what up, you all, welcome back.
We're talking about the coupled
pendulum problem.
Remember in the description of this
problem, we have these two pendulums
that are fixed to a stable support structure.
Each pendulum consists of an
inextensible massless rod.
We measured the length of each rod in the
two pendulums, those lengths are the same.
At the bottom free end of
each rod, we attach a mass.
The mass of the first one is M 1,
the mass of the second one is M 2.
Then we take extension spring and connect each
end of the extension spring to either mass.
When we're ready, we put
one of the masses in motion,
and then watch the entire system move
back and forth in interesting ways.
The coupled pendulum problem is to
predict the exact location of each center
of mass along the ruler, given only
information about the length of the pendulum,
the masses themselves, and
also the spring constant.
In previous videos, I claimed that we can
transform the problem of predicting the location
of each center of mass into a standard
eigenvalue problem in the form:
a times x equals lambda times x; where
we're searching for both the scalar lambda
and the vector x. That process
of translating the real problem
into a mathematical ideal problem
is known as mathematization.
For the coupled pendulum problem,
we're going to have five steps
with some sub steps to mathematize that problem.
The first thing that we're going to do is
study the dynamics of a single pendulum
that is not connected to anything else.
In other words, when studying a very hard
problem, we break that problem into pieces.
The first steps that we focus on
should make simplifying assumptions
so that we can get our head around the
individual components that we're looking at.
With that in mind, our first simplification is
just to study the dynamics of a single pendulum.
Once we've done that, we're going to formulate
a modeling framework for the coupled system
of masses by introducing
proper variables and notation.
Once we get our head around how
each pendulum moves individually,
we're going to start introducing
notation in the system to express
that study using mathematical variables.
At this stage in the game, we're going to
combine our understanding of the dynamics
of the simple pendulum with the mathematical
variables that we generated in step two,
and we're going to look at net forces on each
mass using something called a free body diagram.
These free body diagrams give us a
visual representation of net forces.
That's really powerful because there's a famous
law in physics known as Newton's second law
that says the net force on any object is
equal to the mass times the acceleration.
But acceleration is the second derivative
of displacement, or movement in position.
Which means we can relate the net forces
we got from step three to some set
of coupled ordinary differential equations
that describe the motion of our system,
which is what we do in step four.
In other words, in this stage, we're going to
have a set of equations that involve functions
and derivatives, whose solution is the
description of the position of each mass.
Once we have those differential equations
describing the motion of each mass,
as a system coupled together,
we're going to transform
that into a standard eigenvalue problem by
creative use of matrix and vector notation.
These five steps will play a crucial role
of translating our original coupled pendulum
problem, which is a real-world problem
that we can measure, into the
corresponding standard eigenvalue problem.
In the next video we'll get
started with step one.
I'll see you there.
