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In the last class, we have seen that electrical
quantities and mechanical quantities can be
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equivalanced, and let us start by just recapsulating
where we were. We saw that in
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mechanical domain, the basic coordinate is
position for displacement for from some the
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term position
and it is electrical equivalent would be charge,
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the rate of change of
position is velocity. So, velocity is equivalent
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to the rate of change of charge which is
current, then acceleration is rate of change
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of current DIDT, force is equivalent to
electromotive force. What else, mass is equivalent
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to inductance, in the sense that mass
has the property of resisting change in it
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is inertial status, inductance has the property
of
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resisting any change in current.
Then you have spring constant K, that is equivalent
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to 1 by c 1 upon capacitance good,
they damping constant are frictional coefficient
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is equal to resistance. Finally, the two
things I forgot to talk about in the last
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class, in mechanical domain there are two
types of
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energy the kinetic energy and the potential
energy. So, let see what they are equivalent
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to
the kinetic energy
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is how much half m v square, half m, m is
proportional to...
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.Student: m.
M is equivalent to l, v is proportional to…
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Student: i.
I, so half l i square it becomes equivalent
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to what is half l i square, the energy stored
in
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the inductor. So, kinetic energy is equivalent
to the energy, so if in a circuit there are
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three inductors, the kinetic energy or equivalent
of the kinetic energy, in that circuit
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would be the total energy stored in the three
energies, potential energy then would be
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equivalent to the energy stored in the capacitors.
So, this sort of computes the story of the
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equivalences, it is not easy or common sense
to
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see that the energy stored in the inductor
is actually has the property of a kinetic
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energy,
and the energy stored in the capacitor has
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a property of potential energy. But, it should
you know that, that was essential content
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of the past class, last day’s class. So
let us start
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from there, we said that the Newton’s method
can be written simply as force is equal to
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mass into acceleration.
.
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So, we will write it as the mass into acceleration
in the left hand side, because that is the
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derivative part. So, M q double dot is mass
into acceleration is equal to, equal to what
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whatever the force is, now this force we said
has two components, one something that is
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.impressed from outside, either it can be
some externally applied force or it can be
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a force
applied by the other elements of the system.
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For example, if it is sliding against something
two bodies are sliding against each other,
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there is one body experiences a force applied
by the other body, which is also part of the
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system. For that body that this frictional
force will have to be included, if this body
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is
connected to another body by means of a spring,
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then because of the spring it the other
body pulls with some force, so that is also
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included here.
So, all these forces are to be included, so
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this has to include the externally applied
force,
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it has to include the forces between elements
which are connected by springs. Forces
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between elements, which are connected or interacting,
which with the help of frictional
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elements, also and which is crucial the constraint
forces. So, let us just recapsulate what
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we said about the constraint forces, most
of the motions of bodies are constant in some
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way.
As we showed the they example, that if you
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have a pendulum like this then it is bob is
constraint to move on the surface on a sphere
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and that happens, because of the constraint
applied a force on the body. So, this force
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then has to been included, if you have a
surface like this and you release some kind
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of a body here and it slides down, why does
it
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follow this particular equation, because it
is acted on by a force that is the constraint
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force.
So, when you write the Newton in equation
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then, this constraint forces have to be
included clear, that is why the Newton in
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equation we will have to written as plus F
c this
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is the constraint forces. And if you have
n number of such bodies, for each body then
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it
will have to be written with the subscript,
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for each body you will have the equation like
this for all the n bodies you will have the
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equation like that clear. Now, we need to
say if
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you more things about the constraints.
Constraints can be of a few different kinds
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see these constraints, let us how can we
mathematically express the character of this
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constraints. What kind of constraint is this,
here we can express the constraint as some
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kind of a function, some kind of a eligible
equation this one can we not. So, it was originally
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a three dimensional space and we can
then expressed a equation which will restricted
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it to a lower dimensional space clear.
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.So, that requires one equation to be written
what is the form of that equation, it will
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normal be of this form some kind of a f of
if there are many such variables. So, you
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will
write x 1, x 2, x 3 and so on so forth is
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equal to 0. In this case the position is given
by x
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1, x 2, x 3 and a specific equation relating
this x 1, x 2, x 3 equal to 0 gives the constraint
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equation. In general this might also with
this function might also defined on time,
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imagine, imagine that this it is moved imagine
this is being moved, so in general this
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might also contain time.
Now, any constraint that can be expressed
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in this form is note as holonomic constraint,
remember this one is called holonomic constraint.
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Some holonomic constraints do not
need this term t that is it is independent
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of time, like this and some holonomic constraints
will need additionally this concept of time,
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the dependence on time, but both are
holonomic constraints. If you easily see that
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the holonomic constraints actually reduce
the dimensionality of the system.
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For example, here the bob is constrained to
move on the surface of the sphere and
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therefore, we can as we will see we can define
a new coordinate system on the surface of
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the sphere and that is actually what we do.
For a pendulum it will be stupid to write
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in
terms of x, y, z, it will far more logical
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to write in terms of theta and phi, spherical
coordinate system why, because what we are
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actually doing is that we are writing in
terms of a lesser, smaller number of coordinates,
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constrained to the constraint surface
remember that.
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So, holonomic constraints are actually our
advantage, holonomic constraints are
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advantage we will have to learn how to take
advantage of that in a systematic way. In
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case of the pendulum just by looking at it
by applying our common sense, we saw that
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no, no we do not need to use x, y, z we can
better use theta phi. But, that is observation
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there has to be some kind of systematic way
of going about it that is what we will learn.
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But, let us a natural question is can there
be a non holonomic constraint a constraint,
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but
non holonomic nature yes, that there can be
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something that cannot be expressed in this
form. Imagine, in your hostel do you play
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carrom the what is are constrained, because
they have remain within that boundary, but
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can you express their equation by means of
like this, know still it is a constraint.
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.That kind of a constraint, which is expressible
as an inequality not a equality, here is a
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equality right. But, a body constrained move
within the boundaries of the carrom board
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is given by a inequality, this point that
point, this point let to be 0 and that point
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let it be
l. So, x coordinate is constraint within 0
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and l it is inequality, that is the non holonomic
constraint, billiard is a non holonomic constraint.
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If you have a container and molecules moving
within that, they are constraint by a
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known non holonomic constraint. Because, they
have to be within that and even very
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simple situation can give rise to non holonomic
constraint let me imagine, very
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interesting example.
.
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Suppose you have got a line and somebody that
can move on this line, this is another
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body that can move on this line, one dimensional
problem not a difficult thing at all. So,
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it is the q coordinate of this body and that
body and they are free to move are their
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motion constraint by anything yes of course,
they cannot occupy the same position, their
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body is they cannot occupy the same position.
Now, imagine if I plot there, what will be
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the configuration space in this case,
configuration space we introduce that concept
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in the last class. The minimum number of
coordinates position coordinates, that we
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need in order to express the complete position
of status of this system, this system contains
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two bodies. So, you need two position
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.coordinates and the configuration space will
be then q 1 and q 2, q 1 is the position of
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this one and q 2 is the position of that one
from something.
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And then in this q 1, q 2 space these bodies
will just move around their motion; that
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means, in the q 1, q 2 space the motion of
these two will be given by some kind of a
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trajectory. Notice, I am not talk about the
trajectory of this one given by some orbital
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placed over that one given by another I am
talking about then in the configuration space
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the total positional status of this system
consisting on these two bodies given by a
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point a
point.
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Now, that point can move around like this
which would then mean, suppose it in here
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which would then mean that the position of
this fellow q 1 is here minus something and
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position of that fellow q 2 is plus something.
So, this is a configuration point is it clear,
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that here we are not talking about individual
positions, but their collective positions,
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the
position coordinates of the whole thing represented
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graphically in the configuration
space.
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Now, notice this line then becomes a forbidden
line, this line means what q 1 and q 2 are
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the same 45 degree line, q 1 and q 2 are the
same means that is the forbidden state they
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cannot take the same position, which means
that in the configuration space, this
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configuration point can either wonder in this
side or in this side, but cannot cross. So,
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it
can go like this no problem fine, but it cannot
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cross, because crossing would mean that
the at some point of time, the two occupy
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the same position.
That means, within the configuration space
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then the configure is the point is then
constrained what kind of constraint is that,
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it is a inequality constant it cannot take
this
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value, either it is here or here. So, it is
also a non holonomic constraint fine, so the
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non
holonomic constraints as you can see that
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does not reduce the system dimension
holonomic constraints do. Unfortunately, most
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of the engineering systems have some
kind of holonomic constraints, so it will
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be more important for us to understand or
to be
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utilize the advantages of holonomic constraints.
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..
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So, let us pay some more attention to that,
first in case of the simple pendulum, if you
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express in terms of x, y, z, what is the holonomic
constraint equation, it is constant to lie
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on the sphere how does we expressed as, it
will be x 1 square plus x 2 square plus x
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3
square is equal to x square assuming this
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to be the body origin, it will be trivial
to move
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the origin to this point, because that would
be a simple first step that anybody would
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do.
So, this minus l square is equal to 0 and
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we expressing as some function of x’s equal
to 0
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in this point. Now, suppose this point is
moved as a function of time then what, then
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also
at every position suppose it is in here, then
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also from that position it will be a surface
of a
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sphere again it moves here, again from that
position it will be a ..
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So, essentially what is happening is that,
this if it is moved like this, then also it
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is a
constraint.
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But, what kind of constraint can we express
that, this is a x direction suppose in that
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case
you will say that is x 1 plus that function
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of time, this square plus x 2 square plus
x 3
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square minus l square is equal to 0, that
will be the constraint equation. Notice that,
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this
is expressed as f x j comma t equal to 0,
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this is expressed not as this and this is
expressed
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as f x j comma t is equal to 0. Actually they
have some names you will find in textbooks
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that this has the name that has a name, this
name is schleronomic constraint and this
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name is do not be scared by the names I will
not shoot you if you forgot this names.
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.But, still why I am writing it because, in
books if you come across this words you should
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know what they are, then not fruit’s, hanging
in the tree is that you can pluck and eat,
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they are specific names or specific type of
constraints you should know that is all. So,
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if
you come across these words you should be
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able to say that these are specific types
of
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constraints. But, you will forget if this
type what is the name, there is no problem
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about it
because everybody cannot remember everything
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I know.
So, these terms were coined these are no coming
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from Latin and still these are the terms
used. I will go by assuming that you will
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forget these names and I am not take that
as a
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big thing seen, whatever it is the point is
these are the names when come across them
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in
the text books if you know what they are.
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So, we have seen that there are two types
of
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constraints in the main, holonomic constraints
and non holonomic constraints.
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Holonomic constraints offer the advantage
that you can if you are clever enough, reduce
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the system dimension in case of non holonomic
constraint you can do that. So, if you
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have non holonomic constraints you are force
to use the whole set of coordinates for
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each body 3 coordinates x, y, z. So, normally
you if there are 3 bodies, then how many
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coordinates will be there 9 coordinates, if
it is constant by non holonomic constraint,
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you
will have use all the 9 coordinates.
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But, if there is one holonomic constraint,
then the system dimension reduces by 1, if
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there are 2 holonomic constraints, the system
dimension reduces by 2, if there are 3
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holonomic constraints, so goes on reducing.
Imagine here you have the equation as this
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that is one equation, that makes the dimension
1 less from 3 to 2, but imagine that if it
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is
a planar pendulum, then there is one more
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constraint equation that is say y is equal
to 0
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or x 2 is equal to 0.
Say x 2 is equal to 0 is what is a constraint
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equation then, it is also in this form x 2
is
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equal to 0 geometrically what are you doing,
here this equation one constraint equation
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defines some kind of a sub space. The second
constraint equation give us another sub
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space and then the ultimate constraint is
what, the intersection between the two. Since,
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it
is the intersection between the two, then
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the system dimension reduces by the number
of
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holonomic constraints.
So, the system dimension reduces by a number
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of holonomic constraints and that;
obviously, is a major advantage. Now, the
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point is that in the classical Newton method
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.there is no systematic way of utilizing this
advantage, let me illustrate by doing it for
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this
simple pendulum, if you write the equation
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for the simple pendulum by the Newtonian
way x, y, z do it.
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.
207
00:25:13,329 --> 00:25:32,370
Write the equation you have the simple pendulum,
here we will take the theta, here there
208
00:25:32,370 --> 00:25:46,639
would be a tension working t, here there would
be m g working. So, x coordinate, y
209
00:25:46,639 --> 00:25:58,130
coordinate and z coordinate, if we draw the
free body diagram these were as shown, the
210
00:25:58,130 --> 00:26:04,769
D' Alembert’s way then we will have to do
two free body diagrams along the x
211
00:26:04,769 --> 00:26:11,070
coordinate along the y coordinate.
Along the x coordinate what will be the equation,
212
00:26:11,070 --> 00:26:24,399
in this direction what is the pull theta
is here T sin theta T sin theta, that will
213
00:26:24,399 --> 00:26:27,359
be counted balanced by mass into acceleration
in
214
00:26:27,359 --> 00:26:36,499
that direction m x double dot. So, that is
the free body diagram in the x direction,
215
00:26:36,499 --> 00:26:41,380
in the y
direction here is a pull working that is T
216
00:26:41,380 --> 00:26:56,379
cos theta, so T cos theta, in this direction
m g.
217
00:26:56,379 --> 00:27:06,039
So, you have the m g minus T cos theta working
and naturally there will be a component
218
00:27:06,039 --> 00:27:15,769
of the acceleration mass into acceleration
m y double dot, that will be the free body
219
00:27:15,769 --> 00:27:20,259
diagram simply equation, you have the equations
given.
220
00:27:20,259 --> 00:27:43,700
So, m d 2 x d t 2 is equal to T sin theta
m d 2 y d t 2 is equal to m g minus T cos
221
00:27:43,700 --> 00:27:53,809
theta
and z direction m d 2 z d t 2 is equal to
222
00:27:53,809 --> 00:28:01,480
0, there is nothing working in the direction.
So,
223
00:28:01,480 --> 00:28:16,119
the Newtonian state of equations are like
this, how many equations did you need 3, how
224
00:28:16,119 --> 00:28:22,450
.many coordinates did you need 3 and also
T becomes included in these equations, do
225
00:28:22,450 --> 00:28:29,309
know how much is you do not know really and
as it swings the T varies all time.
226
00:28:29,309 --> 00:28:35,470
So, here is something that is very difficult
to know, but it has gone into the formulation,
227
00:28:35,470 --> 00:28:42,609
so, that is the problem of the Newtonian method
that the constraint force gets into the
228
00:28:42,609 --> 00:28:50,549
formulation. And in case of a situation like
this will you able to write down what the
229
00:28:50,549 --> 00:28:59,529
constraint forces very difficult, so it was
realized soon after Newton that something
230
00:28:59,529 --> 00:29:02,249
has
to be done about the constraint forces, we
231
00:29:02,249 --> 00:29:06,509
cannot write down the equations properly or
you can really solve the equations, if the
232
00:29:06,509 --> 00:29:14,309
constraint force is get it to the system equations
that is one difficulty.
233
00:29:14,309 --> 00:29:15,309
.
234
00:29:15,309 --> 00:29:25,379
Second difficulty is that if you have say
many bodies interacting by means of something
235
00:29:25,379 --> 00:29:37,570
like. So, this is a frictional element or
the another spring element, so like this,
236
00:29:37,570 --> 00:29:44,080
then all
these mutual interactions should be consider,
237
00:29:44,080 --> 00:29:49,349
on this body not only the force due to
gravity. But, also all the forces to be consider
238
00:29:49,349 --> 00:29:56,980
and they are vectors, can you see even if
the system is, so simple only three bodies
239
00:29:56,980 --> 00:30:01,639
with this write it down the differential equation
would be mesh.
240
00:30:01,639 --> 00:30:06,850
Because, all these have to be written in vectorial
terms, all these will have different
241
00:30:06,850 --> 00:30:12,830
directions. And then the Newton’s equation
itself will be a vectorial equation will be
242
00:30:12,830 --> 00:30:20,690
tough and I am sure if I give you this problem
in your exam you will fail that is why, we
243
00:30:20,690 --> 00:30:23,419
need to do something about these forces. What
are these forces, .
244
00:30:23,419 --> 00:30:28,519
.out of these there was a problem with this
one, out of this also a problem with this
245
00:30:28,519 --> 00:30:33,970
one,
because these are all vector forces.
246
00:30:33,970 --> 00:30:40,389
Something needs to be done do that something
needs to simplify that and thirdly the
247
00:30:40,389 --> 00:30:45,510
Newtonian technique does not offer any direct
way of reducing the system dimensions.
248
00:30:45,510 --> 00:30:53,479
These three, where recognized as a major practical
difficulties with direct application of
249
00:30:53,479 --> 00:30:58,350
the Newton’s method true, Newton gave the
basic idea of writing the differential
250
00:30:58,350 --> 00:31:02,739
equation.
And after we had the Newtonian approach that
251
00:31:02,739 --> 00:31:08,370
is when we started being able to solve the
equation for the mass for the example, it
252
00:31:08,370 --> 00:31:12,739
was possible to write down the differential
equation for the motion of the marsh, calculate
253
00:31:12,739 --> 00:31:18,039
the initial condition solve it and then
predict where it will be it actually was that.
254
00:31:18,039 --> 00:31:23,120
So, all these major advantages came because
of the Newton’s law, but these were the
255
00:31:23,120 --> 00:31:26,059
practical difficulties, we ultimately need
to write
256
00:31:26,059 --> 00:31:34,139
down the differential equations.
So, the within a 100 years after Newton the
257
00:31:34,139 --> 00:31:40,789
solution of this practical problems came and
they are very valuable when we try to write
258
00:31:40,789 --> 00:31:45,190
down the differential equations for
engineering systems, so let us try to learn
259
00:31:45,190 --> 00:31:52,279
how this practical difficult to where the
work,
260
00:31:52,279 --> 00:31:54,070
that will need a bit of oriented to mathematics.
So, do not gets .
261
00:31:54,070 --> 00:32:01,769
scared about that most of you are more or
less conversant with math's, in fact at home
262
00:32:01,769 --> 00:32:04,349
with math's I will say, so do not get scare
about that.
263
00:32:04,349 --> 00:32:14,390
So, let us recognize the first problem, the
first problem was that there were constraint
264
00:32:14,390 --> 00:32:20,989
forces that were troublesome. Here, we have
to write down the equations using the
265
00:32:20,989 --> 00:32:28,099
constraint forces and that is a mesh, if there
are many constraint forces then it will be
266
00:32:28,099 --> 00:32:41,379
even more mesh. About a century after Newton
people like Lagrange, D' Alembert,
267
00:32:41,379 --> 00:32:48,899
Hamilton they proved this technique.
Then noticed, that there is a specialty of
268
00:32:48,899 --> 00:32:58,379
the constraint forces, they do more work have
you noticed. . The constraint force is in
269
00:32:58,379 --> 00:33:03,309
this direction and this
fellow is moving in this direction perpendicular
270
00:33:03,309 --> 00:33:08,659
direction and the constraint force is
doing no work. So, instead of writing the
271
00:33:08,659 --> 00:33:15,639
equations in terms of forces, if I write the
equation in terms of the work done, we are
272
00:33:15,639 --> 00:33:21,320
true we can straight away get read out the
constraint forces can be...
273
00:33:21,320 --> 00:33:22,320
..
274
00:33:22,320 --> 00:33:27,269
So, the next stage was to write down the equations
in terms of the work, work done by
275
00:33:27,269 --> 00:33:38,419
the constraint forces will be 0. Now, that
give rise to a very solid way of write it
276
00:33:38,419 --> 00:33:41,239
down
the equations, but let us first start with
277
00:33:41,239 --> 00:33:58,820
the Newton’s laws by rewriting the Newton’s
laws. The Newton’s equation would be m for
278
00:33:58,820 --> 00:34:08,220
each mass point from a coordinate system,
every mass point will have a r, a vector another
279
00:34:08,220 --> 00:34:16,300
vector and r, so r 1, r 2 third mass point
r
280
00:34:16,300 --> 00:34:26,750
3 and all that.
So, for each one m j, r j double dot this
281
00:34:26,750 --> 00:34:33,030
r j is the vector how to I normally in print
we
282
00:34:33,030 --> 00:34:42,859
mark that by a bold space. But, let us put
a small arrow over right to mean that it is
283
00:34:42,859 --> 00:34:50,560
a
vector is equal to F j which is the vector
284
00:34:50,560 --> 00:34:58,690
plus F c j, for each mass point we will have
a
285
00:34:58,690 --> 00:35:06,460
equation like this. And then, we can add them
up to obtain equations of this form I will
286
00:35:06,460 --> 00:35:11,599
put this one here to keep the right hand side
only with the constraint force, so that we
287
00:35:11,599 --> 00:35:15,190
can
eliminate that later we will write it as add
288
00:35:15,190 --> 00:35:23,819
them up all the equations.
So, sigma j is equal to 1 to say N, N number
289
00:35:23,819 --> 00:35:41,650
of bodies m j r double dot j this term minus
this minus F j this is equal to the total
290
00:35:41,650 --> 00:35:49,930
number of constraint forces add it together.
So,
291
00:35:49,930 --> 00:36:02,089
that is the Newton’s law and we are now
trying to get read of this diagram, we have
292
00:36:02,089 --> 00:36:07,380
just
mention that the constraint force now does
293
00:36:07,380 --> 00:36:12,230
not work is it always true I will come back
to
294
00:36:12,230 --> 00:36:13,230
this.
295
00:36:13,230 --> 00:36:14,230
..
296
00:36:14,230 --> 00:36:26,550
Imagine this situation, where is the constraint
force, the tension and the pulling does it
297
00:36:26,550 --> 00:36:30,789
do
know what know it does. For this one if it
298
00:36:30,789 --> 00:36:37,660
is moves by q amount, then it does work and
this fellow also it work, but then overall
299
00:36:37,660 --> 00:36:44,050
taking the system into concentration we cancel
off can you see that. So, it is not always
300
00:36:44,050 --> 00:36:46,710
true that the constraint force does not do
any
301
00:36:46,710 --> 00:36:53,710
work, but overall it will sum up to 0 and
that is why we needed it to sum up, we need
302
00:36:53,710 --> 00:36:57,059
the
sum, because of that we get need to get read
303
00:36:57,059 --> 00:37:01,500
of the constraint forces.
So, the constraint force in some situations
304
00:37:01,500 --> 00:37:09,520
may do some work, but ultimately when
summed over all the bodies in the system that
305
00:37:09,520 --> 00:37:24,130
gives it just one situation. Suppose, a body
is sliding down the surface, the constraint
306
00:37:24,130 --> 00:37:32,690
force is acting like that it is always sliding
down in a direction that is orthogonal to
307
00:37:32,690 --> 00:37:39,030
the constraint surface, constraint force and
therefore, you are happy that there is no
308
00:37:39,030 --> 00:37:48,500
work done by it or suppose there is a friction.
Then, the constraint force will not act in
309
00:37:48,500 --> 00:37:50,480
the orthogonal direction, but whether it will
act
310
00:37:50,480 --> 00:37:57,589
in a that kind of a direction.
In that case what, you might argue that now
311
00:37:57,589 --> 00:38:03,380
the constraint force is doing work, yes and
that is why in such situations we will break
312
00:38:03,380 --> 00:38:08,119
up the constraint force into a component
orthogonal and a component in the direction
313
00:38:08,119 --> 00:38:14,910
of the motion. And the one in the direction
of motion is caused by the friction as we
314
00:38:14,910 --> 00:38:18,320
will include it in this term, because it caused
by
315
00:38:18,320 --> 00:38:23,520
friction it is not really a constraint force,
the constraint force is the one that is orthogonal
316
00:38:23,520 --> 00:38:26,660
to it and it does work clear.
317
00:38:26,660 --> 00:38:32,340
.So, even if this body is applying a force
on this body which is in that direction, we
318
00:38:32,340 --> 00:38:34,549
will
break it up into the constraint force part
319
00:38:34,549 --> 00:38:37,210
and the non constraint force part and the
non
320
00:38:37,210 --> 00:38:41,410
constraint force part was the one that is
due to friction will be included as in the
321
00:38:41,410 --> 00:38:46,150
given
forces .. So, we are still happy constraint
322
00:38:46,150 --> 00:38:53,480
forces it will do no work,
there are situations for example, where is
323
00:38:53,480 --> 00:39:02,790
that like this it is by moved ((Refer Time:
38:58)) is it now true that the constraint
324
00:39:02,790 --> 00:39:10,920
force is not doing any work.
Imagine carefully, here the point of suspension
325
00:39:10,920 --> 00:39:17,480
is being moved like this and the fellow is,
is it always true that the fellow is moving
326
00:39:17,480 --> 00:39:34,450
in a direction that is orthogonal to the string,
why not be. Then, what then that is the mathematical
327
00:39:34,450 --> 00:39:45,130
nice city that Lagrange thought off,
he is saying that at any specific position
328
00:39:45,130 --> 00:39:54,339
of my such point of suspension, at any specific
position this bob has some admissible motion.
329
00:39:54,339 --> 00:40:00,089
That means, if it is like this it can only
move in that direction which is orthogonal,
330
00:40:00,089 --> 00:40:04,430
that is the admissible motion, admissible
displacement.
331
00:40:04,430 --> 00:40:12,890
So, in a given position this bob always knows
always has a specific admissible direction,
332
00:40:12,890 --> 00:40:18,440
admissible motion. So, even if it actually
motion moves in a direction that is not always
333
00:40:18,440 --> 00:40:28,099
orthogonal, but it has a specific admissible
motion and that admissible motion always is
334
00:40:28,099 --> 00:40:36,230
orthogonal at every point of time is orthogonal
to the constant force. So, he said take a
335
00:40:36,230 --> 00:40:44,420
camera and freeze it, if you freeze it at
every point of time you will see that individual
336
00:40:44,420 --> 00:40:47,360
at
every point of time the admissible motion
337
00:40:47,360 --> 00:40:53,860
is orthogonal to the constraint.
So, take freeze shot at every moment of time
338
00:40:53,860 --> 00:41:00,820
is that it is happening, so when we talk of
this kind of admissible motion. That means,
339
00:41:00,820 --> 00:41:05,910
at every point what is the direction it could
move, that is admissible motion whether it
340
00:41:05,910 --> 00:41:12,349
does move or not that is the different issue,
but it could move in that direction and that
341
00:41:12,349 --> 00:41:17,680
is something that is called admissible
displacement, in some books you will find
342
00:41:17,680 --> 00:41:23,600
the word for virtual displacement, but virtual
often give rise to some confusion among this
343
00:41:23,600 --> 00:41:26,829
rule, so I prepared to use the term
admissible displacement
344
00:41:26,829 --> 00:41:36,769
So, what is the technical meaning that at
any specific point of time given a specific
345
00:41:36,769 --> 00:41:44,450
constraint what is the possible direction
of it is motion, so that is the admissible
346
00:41:44,450 --> 00:41:45,989
motion
admissible displacement.
347
00:41:45,989 --> 00:41:46,989
..
348
00:41:46,989 --> 00:41:56,570
And that is given by the symbol delta, so
if r is the position coordinate then delta
349
00:41:56,570 --> 00:42:06,160
r is it is
admissible motion for the j body delta r j.
350
00:42:06,160 --> 00:42:17,150
So, this is...
Student: .
351
00:42:17,150 --> 00:42:33,780
No, yes, but admissible motion is related
to instantaneous position, so if this fellow
352
00:42:33,780 --> 00:42:37,000
is
moving the . also moving that is, but then
353
00:42:37,000 --> 00:42:41,930
fix it at any point of time
here is a string and therefore, this is the
354
00:42:41,930 --> 00:42:47,599
direction of the constraint force. In this
position
355
00:42:47,599 --> 00:42:55,970
in which direction can the bob move, bob can
always move in orthogonal direction.
356
00:42:55,970 --> 00:43:01,059
Student: .
Yes, yes that is not the actual case, but
357
00:43:01,059 --> 00:43:03,450
every point of time that is true, every point
of
358
00:43:03,450 --> 00:43:11,980
time if you freeze it, it can only move in
that direction. Again at some other point
359
00:43:11,980 --> 00:43:15,500
of time
let it move again freeze it, it is here, but
360
00:43:15,500 --> 00:43:19,349
it can only move in the direction that is
orthogonal to the direction of the string.
361
00:43:19,349 --> 00:43:22,150
Student: .
Yes it is moved.
362
00:43:22,150 --> 00:43:23,440
Student: Yes.
363
00:43:23,440 --> 00:43:31,260
.Yes it is move true that is why it is a conceptual
nice city it is not immediately visible, he
364
00:43:31,260 --> 00:43:37,799
says that at every point of time take time
instance separately fix the shot. Then, look
365
00:43:37,799 --> 00:43:41,769
at
the bob, look at the string and say which
366
00:43:41,769 --> 00:43:44,769
direction could it move, then you will find
that
367
00:43:44,769 --> 00:43:51,140
at the moment it can only move in the direction
that is orthogonal. So, even though it is
368
00:43:51,140 --> 00:43:56,119
not moving in the orthogonal direction from
this concept of nice city you see that at
369
00:43:56,119 --> 00:44:01,330
every moment of time, individually take a
move by moment it is always moving in the
370
00:44:01,330 --> 00:44:08,619
direction that is orthogonal.
So; that means, this delta r j has to be taken
371
00:44:08,619 --> 00:44:13,500
at every instant of time and that is not same
for this instant and that instant, it is different
372
00:44:13,500 --> 00:44:24,010
where was it here. .
So, we where is here from the Newton’s equation
373
00:44:24,010 --> 00:44:29,640
in order to talk in terms of the work
done, what you have to do, you have to this
374
00:44:29,640 --> 00:44:38,830
is the force, that is the force just multiplied
by the admissible displacement that is what.
375
00:44:38,830 --> 00:45:05,380
So, multiply both sides by the admissible
displacement what you get is times this is
376
00:45:05,380 --> 00:45:16,109
the
admissible displacement is equal to by delta
377
00:45:16,109 --> 00:45:32,690
this delta r j. So, you are multiplying it
by
378
00:45:32,690 --> 00:45:38,930
just the delta r j which is the admissible
displacement, here is the force, here is the
379
00:45:38,930 --> 00:45:44,930
displacement and therefore, this term is what,
this term is also work. But, the right hand
380
00:45:44,930 --> 00:45:53,460
side is 0, because here we have multiplied
the for the rth body, the constraint force
381
00:45:53,460 --> 00:45:55,280
times
the admissible displacement that is always
382
00:45:55,280 --> 00:45:59,590
0.
So, this in the right hand side is 0, we have
383
00:45:59,590 --> 00:46:10,470
completely got read of the constraint forces
from the formulation. You might argue with
384
00:46:10,470 --> 00:46:15,340
that we still have a troublesome quantity,
here we have got read of this, but we have
385
00:46:15,340 --> 00:46:17,789
brought in this admissible displacement let
us
386
00:46:17,789 --> 00:46:22,519
we have do something about it, yes we will
do something this about it sure. But, then
387
00:46:22,519 --> 00:46:25,680
we
have we have been able to get read of the
388
00:46:25,680 --> 00:46:27,890
more troubles some fellow and then we will
do
389
00:46:27,890 --> 00:46:36,630
something about this.
Now, do you have time yes another 10 minutes,
390
00:46:36,630 --> 00:46:42,880
let us get to the next conceptual stage,
keep this equation written in your copy. This
391
00:46:42,880 --> 00:46:46,180
left hand side equal to 0 that is what we
are
392
00:46:46,180 --> 00:46:49,270
derived.
Student: .
393
00:46:49,270 --> 00:47:03,220
No we have added up for each one, it is 0
n number of 0's added up gives 0.
394
00:47:03,220 --> 00:47:09,160
.Student: .
Yes, so for each one it is 0 and for n number
395
00:47:09,160 --> 00:47:16,240
of 0s added together you get 0, so in the
right hand side you have 0. But, then remember
396
00:47:16,240 --> 00:47:21,420
it was a conceptual quantity it is
something that is not actual motion, remember
397
00:47:21,420 --> 00:47:28,920
this is not actual motion, this is the virtual
or in books you will find virtual displacement
398
00:47:28,920 --> 00:47:33,799
I prefer to call it admissible displacement,
you will find in some books also admissible
399
00:47:33,799 --> 00:47:38,599
displacement, so at every point of time this
is the admissible displacement.
400
00:47:38,599 --> 00:47:51,440
Now, what was the next problem, the next problem
was that holonomic constraints
401
00:47:51,440 --> 00:47:57,190
offered you an advantage that can reduce the
number of equations, but there was no
402
00:47:57,190 --> 00:48:04,930
systematic direct way of doing that when it
comes to the Newton’s law. So, we needed
403
00:48:04,930 --> 00:48:15,060
to
do something about it, now if say normally
404
00:48:15,060 --> 00:48:17,780
the configuration space will be some trice
n
405
00:48:17,780 --> 00:48:24,329
dimensional, but in order to facilitate our
you know visual concept.
406
00:48:24,329 --> 00:48:25,329
.
407
00:48:25,329 --> 00:48:36,630
Let us draw a three dimensional configurational
space and the constraint equation gives
408
00:48:36,630 --> 00:48:48,369
what a surface in this three dimensional space
clear it will gives a surface. For example,
409
00:48:48,369 --> 00:48:54,400
the motion of the pendulum gave a spherical
surface on which it have move, similarly for
410
00:48:54,400 --> 00:48:58,819
a different situations it will be different,
but always it will be a surface. Because,
411
00:48:58,819 --> 00:49:02,000
a
surface is what has the n minus 1 dimension,
412
00:49:02,000 --> 00:49:12,789
so it will be a surface suppose it has a
surface like
413
00:49:12,789 --> 00:49:18,770
this can you see a surface like this, there
is no reason to for it to be a straight
414
00:49:18,770 --> 00:49:21,400
surface or something like that.
415
00:49:21,400 --> 00:49:29,289
.But, then the motion in the configuration
space is always constant to the surface,
416
00:49:29,289 --> 00:49:35,900
therefore the next conceptual step is to define
a new coordinates system in this smaller
417
00:49:35,900 --> 00:49:43,420
surface, smaller dimensional surface what
will you look like, it will look like we will
418
00:49:43,420 --> 00:49:46,810
say
on this let this be origin and let this be
419
00:49:46,810 --> 00:49:56,860
the one coordinate and let that be the other
coordinate. And then, suppose in the actual
420
00:49:56,860 --> 00:50:04,900
position is somewhere here, actual position
somewhere here, we will say fine since this
421
00:50:04,900 --> 00:50:12,049
actual position is always on the constraint
surface, we can resolve it into components
422
00:50:12,049 --> 00:50:17,809
in this directions.
And therefore, it has this much of this new
423
00:50:17,809 --> 00:50:24,930
coordinate and that much of this new
coordinate, that defines the position clear.
424
00:50:24,930 --> 00:50:29,470
So, we are now taking the next conceptual
step
425
00:50:29,470 --> 00:50:35,080
of going from one set off coordinates to another
setup off coordinates in a smaller
426
00:50:35,080 --> 00:50:45,970
number of coordinates clear. So, initially
you have the x 1, x 2, x 3 coordinates, but
427
00:50:45,970 --> 00:50:48,940
now
we are going into a smaller number of coordinates
428
00:50:48,940 --> 00:50:59,019
let that we call q 1, q 2, q 3, fine.
So, the number of q’s would be that many
429
00:50:59,019 --> 00:51:02,109
less than the number of x’s as the number
of
430
00:51:02,109 --> 00:51:10,460
holonomic constraints, we have established
that. So, we are going from the x coordinate
431
00:51:10,460 --> 00:51:24,609
system to the q coordinate system fine and
that would be given by some kind of a
432
00:51:24,609 --> 00:51:39,420
transformation equation, where the x k can
be expressed as functions of q 1, q 2 q 3
433
00:51:39,420 --> 00:51:50,319
to q
n and possibly also time. So, that is the
434
00:51:50,319 --> 00:51:54,950
coordinate transformation equation, imagine
that
435
00:51:54,950 --> 00:52:05,730
in case of a equation ..
Like this in this case the new set off coordinates
436
00:52:05,730 --> 00:52:11,829
q’s are the theta and phi, because that
uniquely specifies the position on the constraint
437
00:52:11,829 --> 00:52:21,599
surface, original set off coordinates was
x 1, x 2, x 3 x, y, z. Now, here we are trying
438
00:52:21,599 --> 00:52:27,849
to express x 1 in terms of theta and phi,
x 2
439
00:52:27,849 --> 00:52:33,549
in terms of theta and phi, x 3 in terms of
theta and phi and that is what have written
440
00:52:33,549 --> 00:52:35,900
here,
in general it will be as possible in this
441
00:52:35,900 --> 00:52:44,090
form. And since, each of the bodies are given
by
442
00:52:44,090 --> 00:52:50,920
some kind of r coordinate r we are writing
in terms of r the radial vector.
443
00:52:50,920 --> 00:53:00,930
So, this will be thrice end time, but we can
simplify it by writing an equations r, r 1
444
00:53:00,930 --> 00:53:09,150
will
be then a similar function of q 1, q 2, q
445
00:53:09,150 --> 00:53:17,099
n, t. Similarly, r 2 the position of the second
body
446
00:53:17,099 --> 00:53:51,410
will be and to r N, Nth body I want to understand
what we are doing, so each body was
447
00:53:51,410 --> 00:53:57,720
given in the old coordinate system by an r
now, this r is expressed in terms of the new
448
00:53:57,720 --> 00:54:02,830
coordinate system q 1, q 2, q 3 and may be
it can also be function of time, if it is
449
00:54:02,830 --> 00:54:05,440
a
holonomic constraint there were the surface
450
00:54:05,440 --> 00:54:08,400
is also time variable, then it will be a
function of time.
451
00:54:08,400 --> 00:54:13,759
.So, in general I have written like that,
but that might be difficult to conceptualized,
452
00:54:13,759 --> 00:54:17,690
so
you might say that r 1 is expressed as a function
453
00:54:17,690 --> 00:54:25,890
of the new coordinates q 1, q 2, q 3, and
q n.
454
00:54:25,890 --> 00:54:26,890
.
455
00:54:26,890 --> 00:54:42,150
Then, we have this equation in general in
short we can write r j is equal to r j q i
456
00:54:42,150 --> 00:54:52,320
comma t
simple, short expression of what we have written
457
00:54:52,320 --> 00:54:59,059
.. The jth radial
vector from the origin to the jth mass point
458
00:54:59,059 --> 00:55:10,150
is expressed in terms of the new coordinate
system and time. If it that is, so then how
459
00:55:10,150 --> 00:55:21,299
do you express r dot j, because r dot j was
representing the velocity we need to convert
460
00:55:21,299 --> 00:55:30,710
the velocities also.
So, r dot j will be d r j d t, then it has
461
00:55:30,710 --> 00:55:39,579
to be written as a chain rule it would be
sum of i is
462
00:55:39,579 --> 00:55:49,619
equal to 1 to the number of n is the let me
write somewhere, ((Refer Time: )) the number
463
00:55:49,619 --> 00:55:59,980
of bodies were capital N. So, number of configuration
coordinates was thrice N, number
464
00:55:59,980 --> 00:56:08,789
of holonomic constraints where h this is small
n, the number of new coordinates that we
465
00:56:08,789 --> 00:56:17,980
need. And that is what we are writing in terms
of that 1 to small n and then it will be
466
00:56:17,980 --> 00:56:49,509
partially derivative of r j delta q i q dot
i plus delta r j chain rule fine, we will
467
00:56:49,509 --> 00:56:53,220
stop here
and then we will continue with this the next
468
00:56:53,220 --> 00:56:54,220
class.
469
00:56:54,220 --> 00:56:54,220
.