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Greetings again. We started discussing about
the oscillators. We discussed what the simple
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harmonic oscillator is and we pretended that
the only force which is acting on the oscillator
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is a restoring force, which is proportional
to the displacement.
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In other words, we ignored damping, because
quite often, there are other forces which
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act. And in such situations when the system
is treated as an ideal system, then the system
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is conservative - the total energy remains
unchanged; the average kinetic energy is equal
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to the average potential energy, but this
is not true when damping is present.
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Now, in this class, we will discuss the damped
harmonic and we will discuss what is damping.
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In real situations, damping is not always
a nuisance, because quite often you need it
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, like if you are driving fast and you want
to stop, you do need brakes. So, it is not
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always malice; it is not always a bad thing.
You need damping in various devices - in galvanometers
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for example. And one really needs to understand
what this damping is and how it works.
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So, this is our equation of motion for a simple
harmonic oscillator. And this sample harmonic
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oscillator means - We have written the spring
constant and the inertia, but it could be
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an LC circuit or it could be any other physical
phenomenon which displace displays simple
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harmonic oscillations in which the potential
is a quadratic function of the independent
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degree of freedom.
Now, in a damped oscillator, we consider such
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cases in which in addition to this restoring
force which is minus kx - in addition to this,
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there is some other force which we had not
earlier taken into account. And this force
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- a very common form that is used for this
force is one which is proportional to the
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velocity and given as minus c times the velocity
or this is a very common expression used for
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the damping force.
The equation of motion will contain the restoring
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force minus kx and it will have this additional
term. There is a position dependent force
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and there is a velocity dependent force. And
it is for this reason that very often one
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is led to believe that if a force is velocity
dependent, it is not conservative. Now, that
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is not quite correct; that is not how a conservative
force or a non conservative force is identified;
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because, there are some velocity dependent
forces which can in fact be conservative - the
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electromagnetic force is one such example;
because you have got the v cross b force coming
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in the Lorentz force and it is of course a
conservative force.
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So, the criterion of a conservative force
is not that it is independent of velocity,
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but that the line integral that it generates
is independent of the path. So that is the
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criterion that should be used and not anything
to do with just the velocity; that means - a
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velocity dependent force, in this particular
case, is a non conservative force, but not
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always. So anyhow, we consider here damping
to be represented by this minus cx dot term
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and this is the equation of motion that you
need to solve.
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So, you arrange, rearrange these three terms,
bring all of them to one side and divide by
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the mass. So you get acceleration plus twice
gamma x dot gamma is this ratio c over 2 m;
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because, I have divided every term by m plus
omega 0 square; because, this is the relation
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between k and omega 0. So, that relation I
have used. This is the differential equation
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of motion that we now have to solve.
Now, one must ask at this point before we
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go any further - that ok, what is meant by
conservative force? Real forces, in nature,
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are made up of fundamental forces and then
what is it that is dissipative? Because, when
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you talk about dissipation, when you talk
about friction, when you talk about a force
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being non conservative, when you talk about
energy being lost, what is the cause of this
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loss? What is being lost and why?
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Now, the net interaction between any two objects
in nature is some superposition of the fundamental
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interactions. And the fundamental interactions
are nuclear, strong and weak, which we really
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do not have to worry about in day to day life
and most of physical phenomena. The electromagnetic
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interaction of course, yes. Or gravity, One
can also talk at a certain level - the electro
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weak unification; but it does not matter -means
- whether you talk about the unified field
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theory or the grand unification or whatever;
in any case, that is some superposition of
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the fundamental forces of which the nuclear
forces do not contribute anything to most
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of the physical phenomena that we observe
in nature.
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The gravitational interaction of course, is
conservative; so is the electromagnetic interaction.
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Because the electromagnetic interaction, as
I said, which is represented by the Lorentz
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force, which is charge times e plus v cross
B. At the v cross B term, is also conservative
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interaction; because it generates a path integral
for the work done which is independent of
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path.
So, if all the fundamental interactions are
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conservative, the better; because, you and
I do not create or destroy energy. So that
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they do not bring build temples for you and
me -right? We do not create energy; we do
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not destroy energy. All the fundamental interactions
are conservative and any net interaction is
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a superposition of these fundamental interactions.
So, how can there be any dissipation at all?
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How can there be any thing that you might
call as energy non conservation?
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So, what is non conservation of energy? So,
what is an origin of dissipation is the question
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that we must really address. Whatever I like
to point out is that, this comes only because
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of the manner in which we set up the equation
of motion -means, ultimately when you solve
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a mechanical problem, what you do is - you
identify some system -that ok, I am going
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to solve the mechanical problem for a certain
system. There is a certain mechanical system
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- it could be this; I look at it. Alright
and say that I want to study the mechanical
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state of this system and how this state evolves
with time.
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This mechanical state of the system is described
by its position and velocity. These two parameters
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are required. And how these two parameters
will change with time is what I want to study.
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When I discover that, I would have solved
the mechanical problem -right? So, define
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my mechanical system to be this; nothing else.
But this system is on the table and when I
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trace the mechanical evolution of this system
- if I am dragging it along the table. right
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then I have not included the table in my analysis,
but the system is interacting with the table.
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So, the whole issue is centered around this
particular - you know - identity of the system
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for which you are solving the mechanical problem.
If the mechanical problem is being solved
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for this system and nothing else and you do
not consider anything else, then this is my
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pre-specified mechanical system. And I am
not keeping track of everything else that
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this system is interacting with.
There are these unspecified degrees of freedom
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and if I keep track of all the degrees of
freedom and take all the interactions into
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account, I will not have to deal with non
conservation.
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Now, if I set up the equation of motion for
this system alone without keeping track of
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these other degrees of freedom, but I still
want to take their effect into account, what
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I am going to do is to bundle up the effects
of the interactions of this system with everything
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else. So, I am going to study the response
of this to a primary force which could be
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the push of by my hand - that is the primary
force and that is the only interaction I am
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going to be concerned with. And I am going
to ignore the interaction of this object with
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everything else - with the table and so on.
But I still want to bundle up those effects
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into my analysis - not in terms of the pair
interactions between every particle of this
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object and every particle of the table and
the air and the atmosphere and so on and so
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forth.
Then I can do it in some approximate manner
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by pretending that the net result of all of
this is an additional force, which is minus
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C times of velocity. Why is it minus c times
the velocity and why not minus c times velocity
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square? Yes, there may be some features which
go like that. It is not coming from first
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principles.
So, in a large number of cases, minus c times
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v turns out to be a good approximation. It
is not going to be so in every situation;
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it is not coming from any fundamental principle
and it works in a good number of cases and
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that is the kind of situation that we are
going to work with. So, this is what causes
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dissipation; because, we have not kept track
of all the details. And having ignored the
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details, we are then condemned to an approximation;
we are not now condemned do an analysis only
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in an approximate manner; we are not going
to get exact solutions. We will therefore
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lose some of the energy of the system; because,
this the energy of the system is not that
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of this piece alone; there is the energy of
the other unspecified degrees of freedom which
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we are not keeping track of. So, some energy
will be lost to the other unspecified degrees
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of freedom and this is the cause of dissipation.
It comes from essentially the unspecified
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degrees of freedom. This is the origin of
friction in real situation - in real physical
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situation. Friction is not a fundamental force;
fundamental forces are only gravity, electromagnetic
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forces, nuclear strong and nuclear weak -that
is; nothing like a friction force.
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Now, this is now our equation of motion. Omega
0 and gamma are connected to the intrinsic
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properties of the system. c is coming from
our approximation c or b. I think I have used
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b over here and c over here - sorry about
that; it is the same thing and What we do
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is - we seek a solution to this differential
equation in this form. Now, note the expression
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here. We seek a solution in this form which
is A e; to the e to the power qt, where q
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is independent degree of freedom. And then
we ask the question - what conditions would
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result on q, which is what we are trying to
determine; because, the description of q and
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q dot, as time evolves, is what we are after.
That is our mechanical problem - right?
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We are asking - what conditions will result
on q and q dot if such a form is admitted
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as a solution? Now, why do we seek this form?
There is a good reason to do that. One is
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that if you look at the form e to the q t,
then the function e to the q t is already
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like e to the plus or minus i omega t. Then
we have met for the simple harmonic oscillators.
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So we know that the solution is going to be
different, but we expect it to be somewhat
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similar. So we expect a solution which will
have some similar form and that is similarity
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is happily accommodated in this. And that
is part of the reason that you seek a solution
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in this form.
The other reason is - you can also see from
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the form of this solution that you can meet
- you can expect very easily to accommodate
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a situation in which, as t goes to infinity,
x will go to 0 and the system will come down
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- be arrested in its equilibrium position,
which is what you intuitively you expect the
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system to do in the presence of damping. So
there are good reasons for why you seek a
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solution in this form. And then, of course,
we want to look for the most general solution
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and we do know that we are dealing with a
second order differential equation; it should
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have two linearly independent solutions; there
should be two and not more than two constants
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and these will have to be determined from
certain initial conditions.
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This obviously is not the most general solution.
It does not even have two constants right.
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So this only gives us the form and not the
actual solution. Now, using this form, what
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do we get? What we do is to take the derivative.
The first derivative x dot and the second
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derivative x double dot. When you do that
and you plug this first derivative and the
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second derivative back at this equation, you
get Aq square e to the qt - these exponential
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functions are so easy to take derivatives
off and then you get this Aq square e to the
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qt from this term. And then you get twice
gamma Aq e to the power qt from this term
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and omega 0 square x, which is Ae to the qt.
And then Ae to the qt is common to all the
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three terms; so you can actually strike it
out; because the right hand side is 0. You
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can divide the whole equation by Ae to the
power qt; strike it out. And then you are
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left with a quadratic equation in q and this
is a consequence of the second order differential
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equation for x.
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From differential calculus, we have reduced
the problem to algebra. Iinstead of dealing
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with a first order derivative and a second
order derivative and having a second order
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differential equation, all we have to do is
a quadratic equation in q which we know how
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to solve. It has got two roots.
This is the quadratic equation for q; it has
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got two roots. These are the two roots and
if you take, a few admit both; because both
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are valid solutions to the quadratic equation.
Then you have the general solution, which
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is A 1 e to the power q 1 t and A 2 e to the
power q 2 t. So, this becomes your general
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solutions. Now, it has got two constants;
these two are independent and you get a general
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solution.
You can get A 1 and A 2. These two unknowns
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are to be determined. But they can be determined
by putting - by finding what is x at t is
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equal to 0 and x dot at t equal to 0. So,
if you plug in the values of x and x dot at
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t equal to 0, you can determine the two unknowns
and your problem is solved.
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So, essentially, the whole problem of a damped
oscillator is now done. It is only the details
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that we have to worry about. And these details
are quite interesting; because the details
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depends on the nature of these two roots;
because these two roots have got values - one
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is minus gamma plus this square root factor
and the second is minus gamma minus this square
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root factor.
So, the nature of the solution will depend
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on how this gamma, which is the damping constant,
compares with this intrinsic natural frequency
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omega 0. If gamma is equal to omega 0, then
this square root factor will vanish. And gamma
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the relation between gamma and omega 0 can
be one of equality and the only other two
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possibilities are that gamma can be either
greater than omega 0 or gamma can be less
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than omega 0. Once you admit these three possibilities
- gamma greater than omega 0, gamma equal
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to omega 0 and gamma less than omega 0, you
would have covered all the possibilities.
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And these are the details which govern the
dynamics of a damped oscillators.
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So, we will now consider the case when gamma
is greater than omega 0. This case gives us
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a damped oscillator which is in fact, called
as an over damped oscillator and I will explain
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why it is called as an over damped oscillator.
So, let us consider this case - the first
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case is when gamma is greater than omega 0
and in this case, square root of gamma square
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minus omega square -means - depending on the
proportions of gamma square and omega 0, the
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factor under the square root sign can be either
0 or positive or negative. So, in this case,
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the square root will be a real number whose
magnitude must be less than gamma; because
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it is a square root of gamma square from which
is certain positive quantity is diminished.
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Omega 0 square will have to be positive; omega
0 also is positive - of course. So, from gamma
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square the omega 0 square is diminished and
therefore, this real number must have a value
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which is less than gamma.
And what you are doing is - you have minus
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gamma in both the roots to which you are either
adding or subtracting in number whose magnitude
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is less than gamma. If you subtract, this
becomes even more negative; if you add, it
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still remains negative; because, you are adding
a number whose magnitude is less than gamma.
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So, no matter what both the roots q 1 and
q 2 become essentially negative - both the
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roots - neither of the root has a chance of
being positive.
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Now, what did it means is that, if both q
1 and q 2 are negative, contribution from
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either of these terms - the first term or
the second term can allow x to go to 0 only
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as t goes to infinity. At no finite time can
x ever become 0. In other words, this oscillator
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can never hit the equilibrium in finite time.
It will need infinite time to come back to
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0.
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So, this is the case we are dealing with.
And these are the two solutions - you have
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the position and you have the velocity. If
you take the initial conditions for position,
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you get A 1 plus A 2. If you take the initial
condition for velocity, you get q 1 A 1 plus
195
00:22:27,830 --> 00:22:34,830
q 2 A 2. And from these, you have two equations
and two unknowns - A 1 and A 2. You can solve
196
00:22:35,419 --> 00:22:41,000
them algebraically and get the two unknowns
- A 1 and A 2 and you have your complete solution.
197
00:22:41,000 --> 00:22:48,000
So, you get A 2 as the difference of this
numerator and q 1 minus q 2. And then you
198
00:22:49,250 --> 00:22:56,250
get A 1 from this equation by subtracting
from x at t equal to 0 the value of A 2. So
199
00:22:58,630 --> 00:23:02,529
you can get both A 1 and A 2 and your problem
is essentially solved.
200
00:23:02,529 --> 00:23:09,529
Now, obviously, an overshoot is not possible;
because the system can come to the equilibrium
201
00:23:10,159 --> 00:23:15,960
and it cannot cross that equilibrium point
- that is called as the overshooting which
202
00:23:15,960 --> 00:23:20,870
is to get the equilibrium point and then should
be on it; that cannot happen; because, that
203
00:23:20,870 --> 00:23:26,960
is going to happen only in infinite time.
So, the oscillation is completely killed.
204
00:23:26,960 --> 00:23:31,809
If it cannot cross the equilibrium point,
the question of it coming back - so it is
205
00:23:31,809 --> 00:23:36,970
really not an oscillator at all; because there
is no oscillation.
206
00:23:36,970 --> 00:23:42,960
So, it really does not oscillate; the oscillation
is completely killed, but the equation of
207
00:23:42,960 --> 00:23:47,700
motion is basically governed by the dynamics
of the equation of motion for the oscillator
208
00:23:47,700 --> 00:23:53,309
which is damped - subject to the conditions
that we are dealing with. Such an oscillator
209
00:23:53,309 --> 00:23:58,690
is called as over damped oscillator. So, that
is the origin of the word over damped. This
210
00:23:58,690 --> 00:24:02,159
type of an oscillator is called as over damped
oscillator.
211
00:24:02,159 --> 00:24:09,159
Now, we will deal with the case when gamma
is less than omega 0. These are the three
212
00:24:09,799 --> 00:24:15,750
possibilities that we agreed to discuss - omega
greater than omega 0 which is when we get
213
00:24:15,750 --> 00:24:22,750
the over damped oscillator. Now, we have the
omega gamma less than omega 0 and when gamma
214
00:24:23,049 --> 00:24:30,049
is less than omega 0, gamma square minus omega
0 square will be a negative quantity and we
215
00:24:30,679 --> 00:24:36,080
have its square root. So, this square root
of a negative quantity must be an imaginary
216
00:24:36,080 --> 00:24:42,960
number and this imaginary number is what we
write as i omega.
217
00:24:42,960 --> 00:24:48,940
So, the two roots which are minus gamma plus
or minus this imaginary number - the two roots
218
00:24:48,940 --> 00:24:55,940
are now written as q 1 is equal to minus gamma
plus i omega and the other root is minus gamma
219
00:24:56,059 --> 00:25:03,059
minus i omega - so these are the two roots.
This omega, of course, has the dimensions
220
00:25:04,100 --> 00:25:11,100
of frequency and this omega - its value is
determined by the difference between the damping
221
00:25:16,309 --> 00:25:23,309
coefficient gamma and the natural frequency
omega; the difference between these two, when
222
00:25:23,429 --> 00:25:28,970
you take the difference between their squares
and then take the square root, you get omega.
223
00:25:28,970 --> 00:25:35,809
So, this is determined by the properties of
the natural frequency omega 0 and also the
224
00:25:35,809 --> 00:25:40,799
damping coefficient gamma.
Now, these are this is the complete general
225
00:25:40,799 --> 00:25:47,799
solution, which is A 1 e to the power 1 root
times t and A 2 e to the power the second
226
00:25:48,720 --> 00:25:55,200
root times t; the second root is minus gamma
minus i times omega. So, this is now your
227
00:25:55,200 --> 00:26:02,200
general solution to the oscillator, when gamma
is less then omega 0 and I still have to explain,
228
00:26:02,940 --> 00:26:08,130
why it is called as an under damped oscillator;
I have used the term which we will understand
229
00:26:08,130 --> 00:26:13,070
soon enough .
So, let us write this solution. Now, e to
230
00:26:13,070 --> 00:26:19,519
the minus gamma t is a common factor to both
of these. So, I factor it out; the remaining
231
00:26:19,519 --> 00:26:25,669
part of the solution is this and you can write
this remaining part of the solution in terms
232
00:26:25,669 --> 00:26:31,320
of the cosine and the sine function, instead
of the exponential function, because cos e
233
00:26:31,320 --> 00:26:34,200
to the i theta is just cos theta plus i sin
theta.
234
00:26:34,200 --> 00:26:39,570
So, you can write these exponential functions
in terms of the cosine and the sine functions.
235
00:26:39,570 --> 00:26:46,570
This is the nice way of looking at it; because
when you write it as such - you find that
236
00:26:46,730 --> 00:26:51,389
you, can transform this form in which you
have written the same solutions in terms of
237
00:26:51,389 --> 00:26:58,149
the cosine and the sine functions. By introducing
two parameters B and theta instead of A 1
238
00:26:58,149 --> 00:27:02,990
and A 2. As long as, we do not introduce additional
parameters.
239
00:27:02,990 --> 00:27:09,990
So, these two parameters b and theta, instead
of A 1 and A 2; not in addition to A 1 and
240
00:27:10,429 --> 00:27:16,100
A 2, but instead of A 1 and A 2. So, B and
theta are such parameters, which you can determine
241
00:27:16,100 --> 00:27:23,100
in terms of A 1 and A 2. If you do so, then
nature of the solution becomes very easy to
242
00:27:24,029 --> 00:27:29,049
interpret and it enables you develop some
insight into the form of the solution.
243
00:27:29,049 --> 00:27:35,039
So, that is the advantage in introducing these
two parameters. So, this is how you introduce
244
00:27:35,039 --> 00:27:40,649
these parameters - you define A 1 plus A 2
as B sin theta. And the difference A 1 minus
245
00:27:40,649 --> 00:27:47,649
A 2 times i as B cos theta. And that is these
are the relationships between A 1, A 2 and
246
00:27:50,500 --> 00:27:57,500
B, and theta. In terms of this, the solution
x theta becomes B e to the minus gamma t times
247
00:27:59,460 --> 00:28:03,399
sine omega t plus theta.
Now, this is a general solution in which there
248
00:28:03,399 --> 00:28:10,399
are two parameters B and theta; whereas in
this form the two parameters were A 1 and
249
00:28:10,649 --> 00:28:17,610
A 2. So, this is the general solution there
are two parameters but this form is very easy
250
00:28:17,610 --> 00:28:22,590
to interpret because you immediately see that
this is the sinusoidal motion just like it
251
00:28:22,590 --> 00:28:29,590
is for a simple harmonic oscillator.
So, you can compare it with an ordinary simple
252
00:28:29,919 --> 00:28:34,320
harmonic oscillator. We know that this is
a damped harmonic oscillator. So, it will
253
00:28:34,320 --> 00:28:38,820
have some behavior similar to as simple to
harmonic oscillator and you get a solution
254
00:28:38,820 --> 00:28:44,549
in the same form, which is an oscillatory
sinusoidal function and then there is a phase
255
00:28:44,549 --> 00:28:51,549
shift theta and there is a peculiar feature
here; that there is an amplitude here which
256
00:28:52,870 --> 00:28:58,860
is not constant as it is in the case of a
free oscillator but an amplitude, which has
257
00:28:58,860 --> 00:29:03,960
got this e to the minus gamma t factor. So,
the amplitude is going to progressively diminish
258
00:29:03,960 --> 00:29:10,440
with time and that is the effect of damping.
Because gamma is not 0; if you put gamma equal
259
00:29:10,440 --> 00:29:16,669
to 0 damping will vanish and e to the 0 will
be equal to 1.
260
00:29:16,669 --> 00:29:22,580
So, e to the minus gamma t is the damping
consequence on the amplitude and here you
261
00:29:22,580 --> 00:29:29,580
have a simple harmonic oscillator whose amplitude
diminishes exponentially as time progresses.
262
00:29:30,139 --> 00:29:37,139
So, here you see that this amplitude of this
oscillatory motion diminishes and if you look
263
00:29:39,639 --> 00:29:46,639
at the envelope envelope it will go down exponentially.
So, this is the general function this is oscillatory
264
00:29:47,830 --> 00:29:54,830
it oscillates at a frequency omega which is
determined by omega 0 and gamma.
265
00:29:56,700 --> 00:30:03,700
So, the periodicity of this oscillation is
governed by the natural frequency as we expect
266
00:30:05,630 --> 00:30:12,630
but also by the damping, which has been inserted;
so omega 0 square minus gamma square is responsible
267
00:30:15,070 --> 00:30:22,070
for omega not being equal to omega 0. So,
this oscillator will have oscillations but
268
00:30:22,389 --> 00:30:29,389
not at the original natural frequency of omega
0 but at a slightly different frequency.
269
00:30:29,840 --> 00:30:36,840
Motion is sinusoidal at a frequency omega,
which is less than omega 0. Why less because,
270
00:30:37,610 --> 00:30:44,590
you are taking the square root of omega 0
square but diminished by a factor gamma square.
271
00:30:44,590 --> 00:30:51,590
So, omega will have to be less then omega
0. So, the frequency is less than the natural
272
00:30:52,899 --> 00:30:59,480
frequency, the amplitude decreases exponentially
with time and there is a phase shift because
273
00:30:59,480 --> 00:31:05,870
the argument is not just omega t but phase
shifted by the factor theta.
274
00:31:05,870 --> 00:31:12,870
So, this is our overall picture, this solution
is not really periodic; the motion is not
275
00:31:15,759 --> 00:31:22,690
completely periodic. What is a periodic motion?
Motion is periodic if it exactly repeats itself;
276
00:31:22,690 --> 00:31:28,610
now this cannot repeat itself because amplitude
is diminishing what was true from here to
277
00:31:28,610 --> 00:31:35,610
here is not true from here to the next point.
So, there is some sort of periodicity but
278
00:31:37,809 --> 00:31:44,809
not complete.
Because the amplitudes are not periodic the
279
00:31:45,580 --> 00:31:52,580
zeroes are it goes to the 0 periodically that
is exactly periodic. So, omega is the circular
280
00:31:59,580 --> 00:32:05,580
frequency, which is exact and this zeroes
will repeat exactly at that frequency. So,
281
00:32:05,580 --> 00:32:12,580
many times per unit, per unit time. So, the
zeroes are repetitive but the amplitude is
282
00:32:14,309 --> 00:32:21,309
not the total motion is not periodic amplitude
diminishes exponentially and this omega is
283
00:32:25,860 --> 00:32:31,379
called as a period of the damped oscillator.
284
00:32:31,379 --> 00:32:38,379
So, this is called as under damped oscillator
because motion is damped but it is not damped
285
00:32:42,120 --> 00:32:47,720
the way the over damped oscillation was so
as oppose to this. This is called as under
286
00:32:47,720 --> 00:32:54,159
damped oscillator.
So, now you can ask, what is the number of
287
00:32:54,159 --> 00:33:00,549
oscillations in a small interval of time delta
t because you can see that there is an oscillation
288
00:33:00,549 --> 00:33:06,950
it goes up remains positive, it goes down
remains negative; becomes positive again.
289
00:33:06,950 --> 00:33:12,659
So, the oscillator is actually swinging on
both the positive on the left positive and
290
00:33:12,659 --> 00:33:17,269
the negative sides of the equilibrium points.
So, there is in fact an oscillation at a certain
291
00:33:17,269 --> 00:33:24,269
frequency that frequency is 1 over T and the
number of times it will oscillate in a time
292
00:33:24,909 --> 00:33:31,049
delta t is simply the ratio delta t divided
by the periodic time 1 over T is the frequency
293
00:33:31,049 --> 00:33:36,929
nu and this nu is nothing by the circle of
frequency divided by 2 pi. So, this is the
294
00:33:36,929 --> 00:33:41,149
number of times you will have oscillations
in a time interval delta t.
295
00:33:41,149 --> 00:33:46,820
So, you can get that very easily; you can
also ask over two successive periods, you
296
00:33:46,820 --> 00:33:53,820
go from here to the next point and from here
to the next point and over this period, you
297
00:33:54,679 --> 00:34:01,379
see that the amplitude has fallen and you
can ask by what factor it has fallen. This
298
00:34:01,379 --> 00:34:07,750
is the corresponding factor because this amplitude,
which is B 2 B times this factor at the next
299
00:34:07,750 --> 00:34:11,770
cycle divided by the corresponding factor
in the previous cycle is what I call it as
300
00:34:11,770 --> 00:34:18,770
B 2 over B 1 and this is just B times e to
the power minus gamma t but here the argument
301
00:34:19,399 --> 00:34:24,890
is t plus 1 period where as here it is just
minus gamma t. So, this ratio is nothing but
302
00:34:24,890 --> 00:34:29,470
e 2 the power minus gamma T. So, this is you
know since this is an exponential function
303
00:34:29,470 --> 00:34:33,150
this is called as a logarithmic decrement
factor.
304
00:34:33,150 --> 00:34:40,150
So, this is the factor through which the amplitude
falls over successive periods and you know
305
00:34:50,130 --> 00:34:57,130
the amplitude decrease factor would be 1 over
e because if you let gamma equal to 1 over
306
00:34:57,980 --> 00:35:04,620
NT if gamma is equal to 1 over NT then this
successive ratio it will go as 1 over e. So
307
00:35:04,620 --> 00:35:10,050
this is the factor.
V over here is imaginary right
308
00:35:10,050 --> 00:35:10,650
What is it
Its complex e is complex
309
00:35:10,650 --> 00:35:17,650
It could be its when you plug in the initial
conditions and get the exact values your final
310
00:35:19,710 --> 00:35:26,710
numbers will come out to be exactly real because
you are looking at real quantities.
311
00:35:26,820 --> 00:35:30,820
You have to put in you have to put in initial
conditions we are not determining the value
312
00:35:30,820 --> 00:35:37,050
of B.
B is dependent on A 1 and A 2 but you have
313
00:35:37,050 --> 00:35:44,050
not put the condition that A 1 minus A 2 is
0 is A 1 minus A 2 is the coefficient of the
314
00:35:44,920 --> 00:35:45,610
imaginary part.
315
00:35:45,610 --> 00:35:52,610
It does not matter here means if you look
at this. Nobody were said that A 1 and A 2
316
00:35:57,120 --> 00:36:02,030
are always real. You have a general solution
right, you have a general solution A 1 and
317
00:36:02,030 --> 00:36:08,040
A 2. When you put in all the initial conditions
and the initial condition on x, and the initial
318
00:36:08,040 --> 00:36:12,660
condition on velocity both will be in terms
of real numbers. There will be a certain displacement,
319
00:36:12,660 --> 00:36:18,760
if it is a displacement in position it will
be so many millimeters or centimeters or whatever
320
00:36:18,760 --> 00:36:24,100
it is some length parameter and there will
be some velocity which will have the dimension
321
00:36:24,100 --> 00:36:26,060
of length over time.
322
00:36:26,060 --> 00:36:32,820
So, your final answers whether you deal with
A 1 or A 2 or B or theta they will all turn
323
00:36:32,820 --> 00:36:39,820
out the. So, that is not an issue the only
reason why you use complex numbers is because
324
00:36:45,490 --> 00:36:51,180
when you express it allows you to write your
solution instead of sinusoidal functions as
325
00:36:51,180 --> 00:36:58,180
e to the power i omega t kind of thing i omega
or minus i omega t and that is a very convenient
326
00:36:58,480 --> 00:37:03,360
form because when you take the derivative,
you get the same function, so no matter whether
327
00:37:03,360 --> 00:37:08,870
you take the derivative once or twice all
you have to do is to take the corresponding
328
00:37:08,870 --> 00:37:15,580
powers of the multiplier.
So, converting differential equations to algebraic
329
00:37:15,580 --> 00:37:22,580
equations becomes very simple when you deal
with exponential functions. Essentially, the
330
00:37:24,330 --> 00:37:29,990
only thing you are doing is you are dealing
with two real numbers together.
331
00:37:29,990 --> 00:37:35,830
So, whenever you work with complex numbers
there is nothing imaginary about the physics
332
00:37:35,830 --> 00:37:42,830
or complex number, only gives you the power
to deal with two real numbers at the same
333
00:37:43,650 --> 00:37:49,680
time; one is what you call it as a real part.
The other is what you call as imaginary part,
334
00:37:49,680 --> 00:37:56,000
but the imaginary part is as real as real
part. So, there is nothing imaginary about
335
00:37:56,000 --> 00:38:02,580
the imaginary part. It is just imaginary part
of the complex number by itself, it is a real
336
00:38:02,580 --> 00:38:09,090
number. B is a real number in A plus i v right,
so that is not an issue here.
337
00:38:09,090 --> 00:38:16,090
So, this is your under damped oscillation
because the oscillations are damped and not
338
00:38:16,390 --> 00:38:20,400
completely killed as was the case in the over
damped oscillators that is the reason this
339
00:38:20,400 --> 00:38:27,070
is called as the under damped oscillations.
It has you know two unknowns. So, it is the
340
00:38:27,070 --> 00:38:32,260
solutions is completely general second order
differential equation must have you know two
341
00:38:32,260 --> 00:38:39,260
unknowns and now, we consider the only remaining
case, which is called as critical damping
342
00:38:41,210 --> 00:38:46,590
because the only case that we have not consider
is when gamma is equal to omega 0. We consider
343
00:38:46,590 --> 00:38:51,750
gamma greater than omega 0, we consider gamma
less than omega 0 and now we are going to
344
00:38:51,750 --> 00:38:56,890
deal with the case, when gamma square is equal
to omega 0 square gamma being equal to omega
345
00:38:56,890 --> 00:39:01,970
0 the two roots q 1 and q 2 will be exactly
equal because the differed by this quantity.
346
00:39:01,970 --> 00:39:07,550
And since the difference between the gamma
square and omega 0 square has vanished the
347
00:39:07,550 --> 00:39:12,990
two roots become equal and you really get
only one solution and how can that be, you
348
00:39:12,990 --> 00:39:19,990
need two solutions, you have a second order
differential equation. So, there must be another
349
00:39:20,110 --> 00:39:25,540
solution and you have to look for it; you
also know that this other solution must be
350
00:39:25,540 --> 00:39:32,540
linearly independent of this.
So, what you can do is look for this simplest
351
00:39:35,180 --> 00:39:42,180
linearly independent solution by considering
this least departure from the previous one.
352
00:39:43,840 --> 00:39:50,840
So, it is the previous function was e to the
qt - if the previous one was e to the qt - if
353
00:39:53,850 --> 00:39:58,850
you multiplied by some function of t and this
arbitrary function of t could be some polynomial
354
00:39:58,850 --> 00:40:05,850
function in t, you take only the first term
in t to the 1. So t to the power 1 and e to
355
00:40:09,760 --> 00:40:16,760
the power minus gamma t plus Bt e to the minus
gamma t; if you construct the superposition
356
00:40:18,330 --> 00:40:25,330
of these two linearly independent functions;
you get a general solution.
357
00:40:26,980 --> 00:40:33,980
Because this form Ae to the qt cannot give
you the complete solution. So, you recover
358
00:40:34,330 --> 00:40:40,650
the complete solution by adding to the first
term Ae to the minus gamma t this simplest
359
00:40:40,650 --> 00:40:46,630
departure from this term which is Bt to the
minus gamma t and now you factor out e to
360
00:40:46,630 --> 00:40:52,820
the minus gamma t as common and this is your
solution to what is called as a critical damping
361
00:40:52,820 --> 00:40:59,640
case and you will see what, why it is critical
damping because you see that when t is equal
362
00:40:59,640 --> 00:41:06,640
to minus A or B; when t is equal to minus
A or B, this factor goes to 0 and this system
363
00:41:06,800 --> 00:41:13,800
really reaches an equilibrium and after this
it can return to equilibrium only when time
364
00:41:14,620 --> 00:41:21,620
goes to infinity .So, this system can cross
the 0 but only once, so this is also not quiet
365
00:41:24,010 --> 00:41:28,900
an oscillator because it does not keep oscillating
across the equilibrium point every now and
366
00:41:28,900 --> 00:41:32,810
then as one expects from the term oscillations.
367
00:41:32,810 --> 00:41:37,960
But an overshoot is possible, it does cross
the equilibrium point but the next time it
368
00:41:37,960 --> 00:41:44,960
can come back to the equilibrium is only asymptotically
over infinite period of time. So the amplitude
369
00:41:45,680 --> 00:41:52,680
verses time function you know there will be
no overshoot of the equilibrium you can now
370
00:41:56,580 --> 00:42:03,580
have this case of under damped oscillation
as we have seen this is the under damped oscillation
371
00:42:07,110 --> 00:42:12,500
in which the amplitude diminishes this is
the earlier case that we have discussed . We
372
00:42:12,500 --> 00:42:19,500
have also seen with no damping at all you
do not have any decay in the damping at all.
373
00:42:25,680 --> 00:42:30,070
Notice that there are these factors to keep
track of; there is a damping coefficient over
374
00:42:30,070 --> 00:42:34,840
here there is a frequency omega over here
which is different from the natural frequency
375
00:42:34,840 --> 00:42:41,840
which is reduced by the damping term which
is c over 2m c over 2m c is the ad hoc damping
376
00:42:42,650 --> 00:42:49,090
coefficient that we plugged into the equation
of motion. In the critical damping case we
377
00:42:49,090 --> 00:42:55,140
had one overshoot, but no oscillation the
next time it would come to 0 will be only
378
00:42:55,140 --> 00:43:01,350
after infinite time but there will be one
finite time which is minus A or B when it
379
00:43:01,350 --> 00:43:07,400
can actually go to 0. So, one over shoot is
possible but no real oscillations in that
380
00:43:07,400 --> 00:43:08,590
sense of the term.
381
00:43:08,590 --> 00:43:15,550
So, the next attainment of equilibrium will
be only after infinite time; so, these are
382
00:43:15,550 --> 00:43:22,550
the various cases to put them all together.
These are the various profiles that you see
383
00:43:23,030 --> 00:43:29,490
if you plot amplitude as a function of time
for all the three cases. So, you have got
384
00:43:29,490 --> 00:43:36,490
the over damped oscillator which really would
approach equilibrium point only as t goes
385
00:43:38,590 --> 00:43:43,600
to infinity.
You have the under damped oscillator, which
386
00:43:43,600 --> 00:43:49,310
does go through oscillation in which the amplitude
is not periodic but the zeroes are periodic,
387
00:43:49,310 --> 00:43:53,290
you know what is meant any zeroes of a function:
zeroes of the function are the points of the
388
00:43:53,290 --> 00:43:56,550
independent argument at which the value of
the function goes to 0.
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00:43:56,550 --> 00:44:03,550
So, the zeroes of the function are periodic
and then you have got the critically damped
390
00:44:04,010 --> 00:44:10,780
oscillator in which one overshoot is possible
but the next time it all if it all there is
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00:44:10,780 --> 00:44:17,780
any that the oscillator will come to the equilibrium
point will be after infinite time. So we will
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00:44:19,930 --> 00:44:26,930
take a break here if there any questions I
will be happy to take those, otherwise in
393
00:44:27,080 --> 00:44:34,080
the next class we will now subject .This oscillator
which has got its own intrinsic natural frequency,
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00:44:37,870 --> 00:44:43,530
which is determined by the inertia and the
spring constant all the corresponding electromechanical
395
00:44:43,530 --> 00:44:50,530
analogs whatever they be.
So, the dynamics would be governed by the
396
00:44:53,090 --> 00:44:58,090
intrinsic parameters of the system plus the
damping that maybe there and that damping
397
00:44:58,090 --> 00:45:04,590
is coming from everything that you have ignored.
These are the unspecified degrees of freedom
398
00:45:04,590 --> 00:45:10,790
that you have ignored.
And then in addition to this if you subject
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00:45:10,790 --> 00:45:17,670
it to some additional external force, so there
is not just a restoring force, which is intrinsic
400
00:45:17,670 --> 00:45:24,670
to the system not just the unspecified degrees
of freedom which you had ignored but in addition
401
00:45:25,540 --> 00:45:32,540
to this now you subjected to a known external
force a periodic force. So, this is what gives
402
00:45:33,660 --> 00:45:40,660
us a forced damped oscillator.
So, in our next class we will discuss forced
403
00:45:42,170 --> 00:45:46,830
oscillations in which there will be an external
driving force we will consider an external
404
00:45:46,830 --> 00:45:53,780
force, which has its own periodicity and the
periodicity of this external force need not
405
00:45:53,780 --> 00:46:00,780
be equal to the periodicity of the intrinsic
system. In other words, the frequency of the
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00:46:02,130 --> 00:46:08,690
applied force need not be equal to the frequency
of the intrinsic natural frequency of the
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00:46:08,690 --> 00:46:15,250
oscillator; there is another frequency that
we have met which is the square root of omega
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00:46:15,250 --> 00:46:22,250
0 square minus gamma square, which is the
frequency of the zeroes of the damped oscillator.
409
00:46:23,360 --> 00:46:28,570
The frequency of the damped oscillator is
not equal to the natural frequency its less
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00:46:28,570 --> 00:46:32,590
than that. So, they are going to be three
frequencies that we shall be talking about
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00:46:32,590 --> 00:46:38,320
one is the intrinsic frequency omega 0 , second
is the frequency of the damped oscillator,
412
00:46:38,320 --> 00:46:44,970
which is omega which is less than omega 0,
which is square root of omega 0 square minus
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00:46:44,970 --> 00:46:50,120
square of the damping coefficient, and then
there is a third frequency we will talk about,
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00:46:50,120 --> 00:46:56,860
which is the frequency of the external periodic
force and then what will be the nature of
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00:46:56,860 --> 00:47:03,150
motion and then depending on the phase relationships,
depending on the comparisons between these
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00:47:03,150 --> 00:47:08,990
three frequencies, we get very fascinating
phenomena. Resonances for example and then
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00:47:08,990 --> 00:47:15,590
we will also discuss wave motion, as we go
along that would be in the next class.
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00:47:15,590 --> 00:47:17,640
Actually, we consider critical damping, so
in that case we consider another solution;
419
00:47:17,640 --> 00:47:17,890
Which is another polynomial? So, any arbitrary
function will not work; no we have to consider
420
00:47:17,640 --> 00:47:24,640
another, I mean the next step should be the
solution of
421
00:47:24,680 --> 00:47:31,680
It should be Yes
422
00:47:34,160 --> 00:47:41,160
What we do let me go back to this, this is
essentially a problem in calculus, you have
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00:47:44,440 --> 00:47:50,180
got a differential equation and there is several
methods of solving differential equations.
424
00:47:50,180 --> 00:47:57,180
Mathematicians have developed these physicists
use them or there are techniques which physicists
425
00:47:57,690 --> 00:48:03,060
have also developed to solve differential
equations.
426
00:48:03,060 --> 00:48:09,830
Now, one can solve this differential equation
from first principles using various different
427
00:48:09,830 --> 00:48:16,830
methods and one method may work in one case
and another method may work in another case
428
00:48:19,230 --> 00:48:24,450
and you have to look for the best we have
solving a differential equation.
429
00:48:24,450 --> 00:48:31,450
Physicist look for such ways which are physically
intuitive, our requirement is that the solution
430
00:48:33,440 --> 00:48:39,760
must be exact and must be correct there is
no compromise so far as rigorous concerned
431
00:48:39,760 --> 00:48:45,370
but there is no need to look for the most
complicated way of getting the solution. So,
432
00:48:45,370 --> 00:48:52,370
we look for a way which is intuitively the
easiest physically most appealing and then
433
00:48:54,430 --> 00:49:01,430
ask ourselves is it rigorous, if it happens
to be not rigorous we must correct it, we
434
00:49:02,010 --> 00:49:09,010
must dispense it, and we cannot use it.
So, in this case what we do is we have a solution,
435
00:49:10,800 --> 00:49:17,290
we have already determined the solutions to
be given by these two roots. But we have confounded
436
00:49:17,290 --> 00:49:23,940
with a very unique situation in which, we
have two roots but both are equal. In other
437
00:49:23,940 --> 00:49:28,900
words, we really do not have two roots in
this case and if we do not have two roots
438
00:49:28,900 --> 00:49:35,900
A into the qt and in this case q is equal
to gamma; that will not be q is equal to minus
439
00:49:38,010 --> 00:49:45,010
gamma sorry A e to the minus gamma t cannot
give us the most general solution it has got
440
00:49:47,180 --> 00:49:52,810
only one unknown, so it cannot be the complete
solution to the second order differential
441
00:49:52,810 --> 00:49:58,710
equation it cannot be the complete general
solution and that is what we are interested
442
00:49:58,710 --> 00:50:05,710
in. So, we ask what are the solution should
be mixed. So that I get through the principle
443
00:50:09,970 --> 00:50:15,820
of superposition of two linearly independent
functions get the most general solution to
444
00:50:15,820 --> 00:50:21,530
the differential equation.
It should have two unknowns and the two solutions
445
00:50:21,530 --> 00:50:27,140
must be linearly independent of each other,
you cannot get one from the other, by simply
446
00:50:27,140 --> 00:50:33,670
multiplying it by a constant; you are getting
one from the other by multiplying it not just
447
00:50:33,670 --> 00:50:40,670
by a constant B, but by a function of time
.
Therefore it is linearly independent.
448
00:50:45,050 --> 00:50:52,050
Now, what function of time do you want to
take? You can consider any arbitrary function
449
00:50:54,560 --> 00:51:01,560
and try it out and if you take terms with
other powers of t; you can plug it in nothing
450
00:51:03,970 --> 00:51:10,970
wrong but then, you will get many more constants
because tq will come with its own constant
451
00:51:11,100 --> 00:51:17,680
c t cube d t to the power of 4 e t to the
power of 5; you do not want those many you
452
00:51:17,680 --> 00:51:23,920
want only one.
So, you take the simplest of which will give
453
00:51:23,920 --> 00:51:29,100
you two constants, you do not want more than
two because the second order differential
454
00:51:29,100 --> 00:51:36,100
equation cannot have more than two unknowns.
Because everything in the end must be determinable
455
00:51:37,850 --> 00:51:43,360
in terms of the initial position and initial
velocity, you do not have any more information
456
00:51:43,360 --> 00:51:50,360
available about the system nor do you need;
you neither have not do you need. Which is
457
00:51:53,560 --> 00:52:00,320
why the equation of motion is a second order
deferential equation, so, you take the simplest
458
00:52:00,320 --> 00:52:05,500
function which is the departure from the first
one and the departure cannot be just scaling
459
00:52:05,500 --> 00:52:10,890
by a constant it must include a function of
t. So, you take the first power of t and that
460
00:52:10,890 --> 00:52:15,580
is what gives you A plus B t times the A to
the minus gamma t.
461
00:52:15,580 --> 00:52:22,580
Now, you are satisfied that you have got the
complete general solution. It is not just
462
00:52:22,720 --> 00:52:28,040
a particular solution it is a completely general
solution it consists of a superposition of
463
00:52:28,040 --> 00:52:32,470
two linearly independent functions. There
are two unknowns and the whole physics is
464
00:52:32,470 --> 00:52:39,040
contained in it now that you are satisfied;
that you have a rigorous solution you have
465
00:52:39,040 --> 00:52:44,490
solved the problem.
466
00:52:44,490 --> 00:52:49,510
Any other question good question
We saw the case of over damped oscillations
467
00:52:49,510 --> 00:52:53,780
right , we say that there are no oscillations
are practically speaking, we do not get any
468
00:52:53,780 --> 00:52:58,970
oscillations in this case then why do we call
it an oscillator?
469
00:52:58,970 --> 00:53:05,970
Its semantics it is a mean to call something,
which does not oscillate as an oscillator
470
00:53:08,390 --> 00:53:15,390
you have every right to object that nevertheless
there is some justification because the solution
471
00:53:16,980 --> 00:53:23,980
has come from the differential equation of
motion for an oscillator, with a certain amount
472
00:53:24,440 --> 00:53:31,440
of damping it is coming from a specific condition
of the nature of damping. And will you have
473
00:53:33,140 --> 00:53:38,350
that the combination of that condition is
what prevents the system from oscillating
474
00:53:38,350 --> 00:53:45,350
but if you change that condition or if you
take away that condition you will have an
475
00:53:48,360 --> 00:53:54,380
oscillator. So, if you will please, let me
use the term oscillator anyway. But you are
476
00:53:54,380 --> 00:54:01,380
quiet right there any other question.
Very well, thank you very much and we will
477
00:54:07,470 --> 00:54:11,420
we will take a break here.