Let me again take a rod lying along the x-axis with one end fixed at the origin and the other at (l,0,0). However, this time I consider infinitesimal rotations about the y and the z axes. I do so because I want both the rotations to cause change in the orientation of the rod; first rotation about the x-axis does not do that. Before I present the calculations, I would like you to recall from the first lecture how different components of a vector change when the frame is rotated. I would be making use of those relationships now with one change: rotating a vector by an angle Δθ about an axis is same as viewing it from a frame rotated by the angle -Δθ about the same axis. I perform a rotation of the rod about the y-axis by an angle Δθ_y and that about the z-axis by angle Δθ_z. Let me first consider the case of rotation about the y-axis that is followed by a rotation about the z-axis. Rotation of the rod about the y-axis gives the new coordinates of it free end as

Now rotate the rod about z-axis to get coordinates of its free end as

Let us now do it the other way. Rotation about the z-axis gives

Now give a rotation about the y-axis to get

When we compare the two boxed results above, we find that the coordinates of the end point of the rod come out to be the same. We conclude that two infinitesimal rotations will give the same final result irrespective of the order in which they are applied. Thus infinitesimal rotations can be treated as vectors . But what about the direction of rotation? To assign a direction, notice that the change in the position vector of the end coordinate of the rod considered above can be written as

where I have written the second line above to emphasize that the order in which infinitesimal rotations are performed does not affect the end result of these operations. The equations above suggest that an infinitesimal rotation about an axis be assigned a direction parallel to the axis following the right hand convention: If the thumb of the right hand points in the direction of the infinitesimal rotation, the movement of fingers gives the sense of rotation. With this definition, the change in the position vector of a point after it is rotated by an infinitesimal angle Δθ about an axis in the direction of unit vector (sense of rotation given by right hand convention) is given as

It is obvious that the vector . The corresponding derivative with respect time is called the angular velocity, usually denoted by . Thus

I now point out that although the above equation is written for a position vector, there is nothing in its derivation that limits it to position vectors only. It is in fact true for any vector as can be easily proved by replacing the (xyz) coordinates by the corresponding components of the vector in the derivation above. Thus if a vector is given an infinitesimal rotation , its will change by

This is shown pictorially in figure 6.

Let us now see how much does a vector change when we apply two infinitesimal rotations about two different axes. Let the vector be denoted by after the first rotation and by after the second one. Then we have

thereby showing that for several infinitesimal rotations the final effect can indeed be expressed by adding the effect of each one of them.

Next we consider the rate of change of a vector rotating with an angular velocity . It is obtained as follows:

This is the rate of change of a vector only due to its rotation. If it changes additionally due to some other causes, that has to be added to the above change separately. If we take the vector to be the position vector , we get the formula

for linear velocity of a particle due to pure rotation of its position vector.

You may ask this point why is it that we want to take as vector quantities. The answer is that we in doing our calculations, we should know whether a quantity is a scalar or a vector or something else so that mathematical operations on it can be appropriately defined. For example, now that we know that is a vector quantity, we can take its components and deal with them independently. Let me give you an example.

 

Example 1: A ball is given a spin at speed w and then put on a rough floor with making an angle q with the vertical. When the ball eventually rolls, what would be its rolling speed (see figure 7)?

In solving this problem, I make use of the vector nature of and split it into its two components. It is the horizontal component that is responsible for making the ball roll. The vertical component does not contribute to rolling, as you well know. Further, this component eventually goes to zero due to friction. So the question is: if a sphere rotating with angular speed is kept on a rough floor with axis of rotation horizontal, what is its find rolling speed. I will let you figure that out. The point that is emphasized here is that knowing that is a vector quantity helped us to solve the problem easily.

Now that we know is a vector, the next question we ask is: how does change when an external torque is applied on a body? So far we have learnt that external torques change angular momentum . So to know how changes, we should know the relationship between . We derive this relationship next.