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Now let’s look at little more complexity
in it. We looked at one single rigid body
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earlier. Now if I have a look at a system
of planar rigid bodies, how do I find out
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00:01:00,739 --> 00:01:07,030
the total number of degrees of freedom. That’s
very simple. Suppose I have a body here, another
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00:01:07,030 --> 00:01:14,030
body here and I call these two lets say this
is A and this is B. Let’s say this entire
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00:01:14,030 --> 00:01:20,429
thing is called as system of rigid bodies.
I know that the rigid body A will have 3 degrees
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00:01:20,429 --> 00:01:27,429
of freedom and rigid body B will have 3 degrees
of freedom and as we can see here, these two
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00:01:27,679 --> 00:01:32,750
independent of each other and therefore 3
degrees of freedom here. I am just going to
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00:01:32,750 --> 00:01:39,750
call it as dof and 3 degrees of freedom over
here which means I have 6 degrees of freedom
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00:01:42,020 --> 00:01:47,200
for this system of equations.
10
00:01:47,200 --> 00:01:54,200
Now supposing I had one more body over here,
let’s say C. It’s easy for me to find
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out the total number of degrees of freedom
this system constitutes and that is nothing
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00:02:00,140 --> 00:02:06,849
but, I had 3 degrees of freedom more and so
that this entire system will have 3 times
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3 degrees of freedom which means it is equal
to 9 degrees of freedom. Supposing I have
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00:02:16,560 --> 00:02:23,560
n rigid bodies
in a system. I am only talking about planar
rigid bodies here. I know that each one of
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00:02:35,459 --> 00:02:42,459
them will have 3 degrees of freedom. I have
n rigid bodies which are independent of each
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other, these are very important statement.
They have to be independent of each other
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means that I will have 3 n degrees of freedom
that this system of rigid bodies have. This
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00:02:59,799 --> 00:03:06,079
system of rigid bodies which has n rigid bodies
in the system has 3 n degrees of freedom in
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it. So far it is clear there is no problem.
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Please remember if I have to define degrees
of freedom here, I have to introduce what
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is called a fixed frame. In order to illustrate
that lets say I have these two rigid bodies.
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Now let’s say these two rigid bodies are
moving in such a way that one rigid body does
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not see the other body moving. But yet they
are moving with respect to some other frame.
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If you look at it more carefully, we need
to fix a particular frame with which we talk
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about all these things. That particular frame
is the Galilean frame of reference and for
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now we will just take it as a fixed frame
that I define. If I have two rigid bodies,
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I have to introduce the notion of a particular
frame with respect to which I will be talking
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about the motion of the rigid body. This is
x, this is y, I can have rigid body over here
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which is A. This is rigid body B, rigid body
C and so on. So that the system of rigid bodies
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will be there and there motions will be defined
with respect to this fixed frame of reference.
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In this particular case I will have 3 times
3 which means 9 degrees of freedom. What does
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00:04:54,210 --> 00:05:01,210
this mean? If I need to define the motion
of this system of rigid bodies, I need to
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00:05:02,060 --> 00:05:09,060
have 9 different values. Again repeating it,
9 different values independent values that
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I have to use in order to define completely
the motion of this system of bodies. Let’s
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just add a little bit of complexity to this.
Here we assumed that each of these rigid bodies
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00:05:28,340 --> 00:05:35,340
that we have or completely independent of
each other. Supposing I introduce connectedness,
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00:05:35,980 --> 00:05:42,980
I am just going to call it as connectedness.
Let’s say these two rigid bodies don’t
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00:05:47,190 --> 00:05:53,380
move independent of each other but there is
some connection that I have established or
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00:05:53,380 --> 00:06:00,380
it could also be with respect to fixed frame
of reference. In other words unless I say
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that these are completely independent bodies,
if I have some connectedness that I define,
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00:06:09,780 --> 00:06:16,710
the number of values that I have to use to
define the motion will come down. In a sense
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00:06:16,710 --> 00:06:23,210
that way we have maximum of 9 quantities that
we have to use in order to define the motion
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00:06:23,210 --> 00:06:30,210
of this system. Now let’s just look at some
examples here.
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00:06:32,130 --> 00:06:39,130
In order to show connectedness, let me call
them as constraints. Some of the general constraints
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00:06:46,420 --> 00:06:53,420
could be supposing I have a rigid body like
this. I could probably have some kind of connection
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00:06:54,120 --> 00:07:01,120
like this. Whenever I hash like this, it means
that I am fixing it to the fixed frame of
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00:07:05,130 --> 00:07:12,130
reference. This particular point alone I am
fixing it with respect to fixed frame of reference.
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00:07:16,610 --> 00:07:23,610
This is what it means or in other words, if
I have this rigid body and I say this is fixed
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00:07:26,080 --> 00:07:33,080
frame of reference and I am just pinning it.
It’s easy to understand that it has only
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00:07:34,090 --> 00:07:41,090
one degree of freedom here. Instead of doing
this, supposing I have something like this.
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00:07:45,390 --> 00:07:52,390
I am just going to start with certain examples
and then go on with understanding it properly.
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00:07:53,640 --> 00:08:00,640
Supposing I have some other rigid body, two
rigid body A and B and they are connected
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00:08:02,110 --> 00:08:09,110
to each other by a single pin over here. In
a while I will just explain that to you. Its
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00:08:11,060 --> 00:08:16,970
pined like this, one of the ways to find out
the degrees of freedom pertaining to this
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00:08:16,970 --> 00:08:23,970
system of rigid bodies A and B
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00:08:33,200 --> 00:08:40,200
is first to think of no constraint at all.
This as a possibility of 3 degrees of freedom,
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00:08:41,269 --> 00:08:48,079
this is for A and for B again I have 3 degrees
of freedom possible, if there were no constraints
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at all. That’s very simple to understand.
But if I put a constraint how many degrees
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00:08:54,029 --> 00:08:58,829
of freedom will be lost or in other words
what is the minimum set of quantities that
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00:08:58,829 --> 00:09:02,970
I will need in order to describe the motion
of this particular body.
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There are two ways of understanding this.
Let me just explain the first way of understanding
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it. Supposing I take this particular set of
bodies and make sure that this particular
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point is fixed to the fixed frame of reference.
Let me just think of it like this then I know
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00:09:28,200 --> 00:09:35,200
about this particular point, this B can rotate
and about this particular point again let
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00:09:36,520 --> 00:09:43,520
me call this as o. About this particular point
again this A can rotate independent of B.
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00:09:45,190 --> 00:09:52,190
So I have one degree of freedom possible for
the B rigid body to move or rotate and for
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00:09:53,150 --> 00:10:00,150
A rigid body to rotate either this way or
this way independently. I have one independent
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00:10:04,990 --> 00:10:11,710
rotation of B, one independent rotation of
A possible. Mind you I have fixed this particular
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00:10:11,710 --> 00:10:18,710
point o of both the bodies to fixed frame
of reference or in other words if I release
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00:10:19,460 --> 00:10:26,460
this particular point with respect to fixed
frame of reference, I can move with this point
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around.
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This point has two degrees of freedom possible,
it can move in x and y direction in a particular
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way and therefore there are two degrees of
freedom possible that I can move this o with,
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00:10:42,200 --> 00:10:49,200
if I remove the constraint over here and therefore
in total I have… I am going to call this
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as 3 and 4 and I have 4 degrees of freedom
in total that this system of rigid bodies
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00:11:02,020 --> 00:11:09,020
A and B will have. One other way of understanding
is this. Body A can go through 3 degrees of
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00:11:14,960 --> 00:11:21,960
freedom independently, B can go through 3
degrees of freedom independently if this constraint
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was not there. What does this constraint do?
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Supposing I fixed this body A to fixed frame
of reference then this particular point arrest
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00:11:35,089 --> 00:11:42,089
the motion of body from moving, translating
at this point o in x or y direction. Or in
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00:11:43,750 --> 00:11:50,750
other words I have two constraints of point
o, x and y directions and therefore I have
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00:11:59,500 --> 00:12:06,500
6 which can be independent and 2 constraints
that we have added equal to 4 degrees of freedom.
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00:12:11,630 --> 00:12:18,630
This will have a few problems associated with
that. I will ask those questions in a separate
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00:12:18,960 --> 00:12:23,020
clipping.
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00:12:23,020 --> 00:12:30,020
Let us look at some examples so that we will
have some clarity. Look at this, there are
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00:12:32,779 --> 00:12:39,779
3 rigid bodies, rigid body A B C; A and B
are connected at point o1 pinned at o1, B
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00:12:42,750 --> 00:12:49,750
and C are pinned at O2. I am just defining
a fixed frame of reference. Now the question
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is how many degrees of freedom does this system
have? The simplest way of looking at it is
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00:13:00,730 --> 00:13:07,730
A has 3 degree of freedom, B has 3 degrees
of freedom, C has 3 degrees of freedom and
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00:13:11,020 --> 00:13:18,020
total of 9 degrees of freedom if the constraints
o1 o2 are not present or in other words there
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is no pinning occurred here. This pinning
is going to restrain the motion between A
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00:13:26,790 --> 00:13:33,790
and B in both x and y direction which means
I would have lost 2 degrees of freedom pertaining
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to A and B.
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00:13:37,710 --> 00:13:44,260
In a similar way with o2, we would have lost
2 degrees of freedom or in other words you
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don’t need two more quantities in order
to define. The minimum quantity you will need,
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will be 5 degrees of freedom, if you find
the total. This is the very simple method
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of doing it but if we have to look at in a
different fashion there is one more way of
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understanding this and that is this.
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Let’s look at this particular body A. This
has 3 degrees of freedom. Now what we are
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going to do is with respect to the fixed frame
of reference, we will just fix this body A
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and look at what motion body B can have with
this constraint. It’s very simple to understand.
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This body can move, forget about the body
c, now this body can move in only a rotational
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form and therefore this body can have one
degree of freedom.
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Let’s fix A as well as B with respect to
fixed frame of reference and ask the question
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how many ways will this body C move? Very
simple to answer, it can only rotate about
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this point o2 which means it has one degree
of freedom available to it. If you take the
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total, we have 5 degrees of freedom that we
need, minimum of 5 quantities we need in order
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00:15:23,790 --> 00:15:29,480
to define this. The second method is very
simple method and it has a physical understanding
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to how we comprehend. The first one is mathematical.
In the next example make it clear to you that
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there will be a problem if we use the first
method.
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Now let’s look at second problem. This is
clear to us. If I use the method of saying
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that each one of these bodies will have 3
degrees of freedom and if we look at the constraint
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then we can answer the question. We will have
9 degrees of freedom if there is no constraint
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and we have one constraint to over here which
restricts the motion in x and y direction.
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Therefore I should have 9 degrees of freedom
but in a moment I will show you that this
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is a wrong comprehension. Now simple to understand
if I look at the second method which is a
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physical method of understanding.
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00:16:38,360 --> 00:16:44,920
So as we started earlier, if we look at this
body A, this will have 3 degrees of freedom,
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no problem. If I fix this body to the fixed
frame of reference and ask the question how
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many ways can B move? B can move independent
of any other body in one way or in other words
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it is just a rotation possible. From B you
can have one degree of freedom. Now I will
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fix A as well as B and look at C and ask the
question whether there will be moment at all.
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The is answer is yes because this is a pin
and the body C can move independent of A and
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B or in other words this have an additional
degree of freedom to move. If I add the total,
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I will have 5 degrees of freedom possible.
The earlier method, you have to take caution
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in using. The second method is what I would
suggest is the best method to find out what
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00:17:43,600 --> 00:17:49,460
are the degrees of freedom that you can find.
Let’s look at some more examples.
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00:17:49,460 --> 00:17:56,460
Let’s look at some more examples. Let this
is a single rigid body, naturally if I don’t
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have any constraint, unconstraint motion,
I will have 3 degrees possible. If you look
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at it, I am arresting motion of this particular
body A with respect to this direction let
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00:18:13,720 --> 00:18:20,720
me call this as o1 and this is o2. I am arresting
a direction motion here which means there
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is one degree of freedom that is lost. What
A would have had? Two degrees of freedom.
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Since I am doing this also if you look at
this, this is a pin joined or a hinge and
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this arrest the motion of o2 in x as well
as y direction. So in all if I look at this
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particular rigid body and asked the question,
how many degrees of freedom it has? Or in
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other words whatever ways can it move, the
answer is if I do this, it cannot move at
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all. Now in this particular situation it is
so, that it cannot move at all and therefore
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this has zero degrees of freedom.
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The same thing I can answer again by the first
method. The rigid body has three degrees of
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freedom because single rigid body and I have
one constraint here in terms of motion along
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this which is minus 1 and there are two translational
motions arrested which means minus 2 equals
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0. In this particular case, it becomes simple.
Please remember there are some cases where
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you have to take precaution. I will ask you
the questions at the end. Usually we will
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00:19:49,970 --> 00:19:56,970
solve simple problems and ask complex problems
when comes to exams.
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Let’s look at a few more examples so that
you are familiar with this. How about this?
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00:20:02,670 --> 00:20:09,670
Very simple, there are two translations arrested.
One translation arrested here, total of three
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00:20:09,929 --> 00:20:16,040
translations arrested and this body cannot
move in x or y direction. As well as it cannot
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00:20:16,040 --> 00:20:22,160
rotate because about this particular point
connecting these two lines, if I take this
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point and try to rotate, this will arrest
the motion around this direction which means
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00:20:26,450 --> 00:20:33,450
it cannot rotate. Very simple and therefore
this has zero degrees of freedom. In this
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00:20:35,380 --> 00:20:42,380
case again it is easy to show that it will
have zero degrees of freedom by this kind
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of calculation.
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00:20:43,920 --> 00:20:50,920
Let’s move on to one more problem. This
is again very simple one. You have the rigid
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body that is rigidly fixed to the fixed frame.
Mind you when I hash it like this, it means
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I am referring to the fixed frame. If this
were not there, this body would have moved
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00:21:10,410 --> 00:21:16,110
in three different ways, two translation on
rotation but if I fix it like this, it cannot
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00:21:16,110 --> 00:21:21,820
move in the x or y direction. It can also
not rotate about this because it is rigidly
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00:21:21,820 --> 00:21:28,280
fixed like this and therefore it has zero
degrees of freedom. Simple to understand.
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00:21:28,280 --> 00:21:35,280
Let’s move on to the next. In this particular
case, if you notice carefully the motion in
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this direction is arrested. This is a straight
line and this is the fixed frame of reference.
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00:21:53,120 --> 00:22:00,120
In this direction there is a translation that
is arrested. About this particular point if
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00:22:01,960 --> 00:22:08,960
I try to rotate this body, it cannot rotate
and therefore let me say x translational is
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00:22:11,000 --> 00:22:18,000
arrested. One rotation which means the entire
body cannot rotate at all. Only possible movement
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00:22:19,030 --> 00:22:26,030
is in the vertical direction and therefore
it has one degree of freedom.
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00:22:33,030 --> 00:22:40,030
One last example here this is a system of
bodies A and B. Mind you there is a slot on
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00:22:51,690 --> 00:22:58,690
this body A and the two are connected through
a small pin that can slide along the slot.
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00:23:00,540 --> 00:23:06,040
Then I asked the question how many degrees
of freedom can it have? First way can I start
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00:23:06,040 --> 00:23:13,040
with is I can say, I have three degrees of
freedom possible for this rigid body, if I
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00:23:14,350 --> 00:23:21,350
don’t consider anything else. By fixing
A I am going to look at B. This point here
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00:23:26,490 --> 00:23:33,490
can slide and this body B can also rotate.
There are two ways which we can understand.
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00:23:36,440 --> 00:23:38,570
Let’s fix this particular body A.
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00:23:38,570 --> 00:23:45,570
Let’s take this particular pin and another
body and that pin can have one translational
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00:23:46,299 --> 00:23:52,870
along this direction. So one more degree of
freedom and I am going to call that as one
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00:23:52,870 --> 00:23:59,870
more degree of freedom. If I fixed this pin
as well as this A, the body B can rotate about
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00:24:00,890 --> 00:24:05,790
the pin axis. Therefore it can have one more
degree of freedom and therefore this will
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00:24:05,790 --> 00:24:12,790
have 5 degrees of freedom. The only motion
here that is arrested between A and B is the
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00:24:15,040 --> 00:24:22,040
translation which is vertical with respect
to the slotted line that is shown here. I
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00:24:24,510 --> 00:24:31,200
hope it is clear as far as number of degrees
of freedom is concerned. This will help us
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00:24:31,200 --> 00:24:37,530
understand how to write the equations of equilibrium.
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00:24:37,530 --> 00:24:43,230
Now I have a few challenges for you. So far
what we have done should be enough for you
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00:24:43,230 --> 00:24:50,230
to answer these questions. This is the rigid
body, let me call this is A, a single rigid
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00:24:50,360 --> 00:24:57,360
body which is supported on a straight ground
three rollers supports. Question B, this time
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00:25:01,549 --> 00:25:07,700
with two rollers supports but at some angle
to each other as you can see here. Third one
184
00:25:07,700 --> 00:25:14,700
is three roller supports, two rollers support
like this in the horizontal and one roller
185
00:25:15,750 --> 00:25:22,750
support for vertical direction. Here there
are 3 roller supports on something that is
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00:25:29,140 --> 00:25:36,140
curved. Fifth problem is I have a slider over
here but it is not a slider which is straight.
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00:25:39,370 --> 00:25:46,370
I need to find out degrees of freedom for
this system of bodies A and B. I am sure you
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00:25:47,940 --> 00:25:52,770
will be able to get the answers.
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00:25:52,770 --> 00:25:59,770
Let’s just look at whether we are correct
in this particular problem. We have this body
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00:26:02,179 --> 00:26:09,179
A and this body can move around or rotate
which means it has 3 degrees of freedom. What
191
00:26:09,380 --> 00:26:16,380
we have there is one pin over here like this,
there is one more pin that connects A and
192
00:26:20,820 --> 00:26:27,820
C. So you have A, B and C. Let me just connect
them. This is what we have. As you can see
193
00:26:31,720 --> 00:26:34,299
A, B and C.
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00:26:34,299 --> 00:26:39,870
We need to find out the degrees of freedom
of the system. We can see that you can move
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around like this. Now how do we find it out?
We take this body A, we know that it can rotate,
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it can translate and it can move which means
this body as such we can say as three degrees
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of freedom then I fixed this to the fixed
frame of reference. Just assume that my hand
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is a fixed frame of reference, left hand is
of reference. I am just holding is A completely
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and don’t let it move in any way possible.
If I look at only B, only possibility for
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B is to rotate about this particular joint
which means if I focus only on B, B has only
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one degree of freedom or in other words only
one quantity needed to describe the motion
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of that with respect to A. If I fix this particular
body B also with respect to the fixed frame
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of reference, body C has only one way in which
it can move or in other words just one single
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of degree of freedom which is rotation, can
describe what is the motion of this with respect
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to A and B.
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In other words 3 plus 1 plus 1, we have 5
degrees of freedom whereas the other problem
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is like this. I have body A, body B, body
C but all the three are connected through
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a single pin. Let me just put it properly
so that it is easy for you to understand.
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Let’s examine how many degrees of freedom
it will have, like before we will have 3 degrees
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of freedom for body A. Let me just fix it.
Now let’s look at B, B has one degree of
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freedom and body C has another degree of freedom,
if I fixed A and B which means it has 5 degrees
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of freedom.
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I have put a small slot over here through
which a pin can move. I can just show you
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like this or like this. It cannot move this
way, it can move this way along this particular
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slot. Now what I am going to do is I am going
to attach through this particular pin, the
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rigid body B and ask the question what is
the total degree of freedom that this particular
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system of bodies A and B will have. Very simple,
again I will do the same thing supposing I
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take only this body A. This has three degrees
of freedom, I will take this pin has another
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body, this has one degree of freedom.
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Now I am going to attach this B and like before
if I fix A and look at only this pin, the
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pin will have one degree of freedom. So 3
degrees of freedom plus 1 degree of freedom
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and
if I fix the pin as well as the rigid body
A, B has only one rotation possible. Therefore
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A has 3, the pin has 1 and B has 1, total
of 5 degrees of freedom.
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