mod08lec72
mod08lec72


In the previous 2 or 3 lectures what we have
learnt is that the dynamics of a rigid body is governed by change in its angular momentum
by the externally applied force However we dealt with simple problems where the axis
of rotation was fixed in space and the body was rotating about it In this case the entire
relationship became very simple and we showed that L equals I omega where I is the moment
of inertia and omega is the angular speed In the fixed axis rotation where the axis
was either fixed in space or it could move parallel to itself all we could do in rotation
was change the magnitude of omega and therefore change the magnitude of L We also applied
the principle of conservation of angular momentum where we in one demonstration changed the
value of I by pulling my hands in and out All we did in these problems was only changed
the momentum the magnitude of angular momentum and angular velocity However when we started L as a vector quantity
which is equal to for a given distribution of masses MIRI cross VI So in principle if
there is a vector L I should also be able to cause a change in it not just by changing
its magnitude but changing its direction so that if this was L1 and this was L1 plus Delta
L this would be the change Delta L in its magnitude In that case the axis of rotation may also
change And these kind of changes in Delta L give rise to a little more complicated dynamics
The body may change orientation in many many different ways This I show by a few demonstrations
after this In this demonstration I have these 4 cylinders
Let me show these shapes clearly to you One is plain cylinder the other one is cut like
this the 3rd one is like this and the 4th one is shaped curve like this The question
we ask now is which of these cylinders is going to go along this curve path when rolling
from this side to that side Let us see what happens Let us 1st take this
plain cylinder and roll it And you see it goes straight but falls out of the track after
time Let us look at this cylinder and let this roll and let us see what it does It grows
and you see it curves along the curves properly Let us do it once more You see this one goes through clearly How
about this cylinder This goes over to one side And how about this cylinder This also
falls over to one side In this demonstration I have this bicycle
wheel which can spin on its axis Let us see what happens when I put this end here If I
pivot it here it falls down If I leave it it will fall down Let us see what happens
if I give it a spin If I give it a spin and leave it here you see it rotates If I give
it a spin the other way you see it will rotate the other way What we observe is that when a rotation is
given instead of falling down this starts going around like this and this is known as
recession If I make it go faster it goes around slower If I let it go slow then it goes around
very fast The slower it gets faster it goes So these are the things that we should be
able to explain using rigid body dynamics In this demonstration what I want to show
you is if this wheel is not rotating I can lift it up like this You see when I apply
a torque like this it goes like this Now let me spin it and try to take it up You see it
does not go up It is going sideways If I try to push it down it goes sideways this way
Whereas when it is not spinning I can take up and down by applying a torque like this The moment I give it a spin I give a torque
up it goes that way I give a torque down it goes this way This is again a manifestation
of rigid body dynamics and how torques and angular momentum interplay makes the dynamics
very interesting Here I have a device known as the gyro compass
In this there are 2 frames that can rotate about 2 perpendicular axis independently This
frame can rotate like this about this axis this frame can rotate like this about this
axis and there is a spinning wheel which can rotate about axis perpendicular to both of
these like this So in a way X axis Y why axis and Z axis There are 3 independent axis about which rotation
can take place Now I will give the inner wheel a spin and you see what happens when I rotate
this If I rotate this you see the spinning wheel alliance with this rotation Let me show
it to you again I give the inner wheel a spin When I rotate this the moment I rotate the
outer frame the spinning wheel the spin axis aligns with the outer frame rotation axis
See it again I give it a spin rotate it The moment I rotate it it aligns with this This
can be used as a compass Another interesting aspect in rigid body dynamics
is I have a box of sweets here with 3 sides unequal Observe when I give it a rotation
like this and let it drop it drops observe it carefully it drops pretty much rotating
about the same axis On the other hand if I give it a rotation about this this axis it
again rotates in a very stable manner and drops Observe now when I do it about this axis what
happens By the time it comes down it has started rotating about all axis Observe it carefully
It has started tumbling We should be able to explain this using rigid body dynamics
equations of motion Whatever I have shown in these demonstrations
can be easily explained by using the equation that rate of change of angular momentum is
tao external Not only that you see I was also while doing these demonstrations showing you
how angular speed or angular velocity of the body was changing So after having solved these
equations I should also be able to tell you how omega of a rigid body
changes And therefore with time how the body changes
its orientation To develop the theory of this we need to be very very specific about what
does omega represent Is it a vector Is it a scalar How about the orientation being described
by vector theta Can we do that And how are L and omega related so that once I calculate
how L is changing I could relate it to change in omega and from omega I can find how the
angle of the body with respect to different axis is changing These are questions for which the answers
come when we develop relationship between L and omega We see how when L changes how
omega changes We see how theta is related to omega and so on So we do that now

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