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