Lectures 22 & 23
Rotational dynamics V: Kinetic energy, angular momentum and torque in 3-dimensions

 

You learnt in the previous lecture is that the angular velocity is a vector quantity pointing in the direction of the axis of rotation. Any vector that is rotating about also changes direction. Thus the vector changes even if its magnitude is constant. If the vector is then its rate of change purely on the basis of rotation is

Thus the velocity of a rotating particle at position from the origin is

I also derived the general expression for the angular momentum, which is given as

Here are the moments of inertia about the x, y and the z axes, respectively. The off diagonal elements like I_xy are the products of inertia. A simplification in the expression above arises by employing the principal axes for which the products of inertia vanish. For convenience in writing, the principal axes are usually denoted by (1,2,3 ) instead of (x,y,z). Using this notation the angular momentum vector can be written in a simple form as

where ω₁ , ω₂ and ω₃ are the components of the angular velocity along the principal axes. I now derive the expression for kinetic energy for a rigid body rotating with one point fixed.

Kinetic energy of a rotating rigid body: I consider a rigid body rotating with angular velocity . Its kinetic energy T is calculated as follows

Substituting   for of the velocities above and making use of some identities of vector products we get

In the principal axes therefore

This is the expression for the kinetic energy in terms of the principal moments of inertia and the components of angular velocity along the principal set of axes. Having obtained the general expressions for the angular momentum and kinetic energy of a rigid body, we now study the dynamics of a rigid body through the angular-momentum torque equation. Along the way I will explain the three observations that I had started my previous lecture with.

Dynamics of a rigid body: Dynamics of a rigid body is governed by the equation

and it is this equation that governs everything about the rigid-body rotation. What makes the motion of a rigid-body interesting is that there is a fantastic interplay between the angular momentum, angular velocity of a rigid body with or without an applied torque. For example if the angular velocity and the angular momentum of a rigid body are not parallel, the vector would rotate about and that would make change. However, if there is no torque applied on the body, angular momentum cannot change. Therefore to compensate the change in arising from its rotation, the angular velocity itself must change. Changing would make body rotate in a different way and this goes on. It is thus this interplay between and that makes a rigid body move in seemingly counterintuitive ways.

As a body rotates, its angular momentum changes on two counts: first because in general and are not parallel and therefore rotates about . With

and

the rate of change of only due to its rotation about is given as

If the components ω₁ , ω₂ and ω₃ were also changing, I would have to add an additional term on the right-hand side of the expression above to take care of that. This is the second reason for the change in angular momentum of the body. For the time being I focus on cases where the components of along the principal axis remain unchanged. This in turn implies that the magnitude of the angular momentum remains constant during the rotational motion of the body. This happens when the applied torque is always perpendicular to the angular momentum. Substituting for L₁ , L₂ and L₃ in the equation above, I get

So at any instant the components of   are

For a geometric interpretation of these equations I urge you to go back to the previous lecture and see how we obtained the changes in the coordinates of the end of a rod rotating infinitesimally. This gives the components of the torque required to be

To apply these equations I start with calculation of torque for the example that we solved at the end of the previous lecture.