Urmi Roy said:
Of course the directions of the principal axes are those where the products of inertia vanish as the arbitrary rotational axis varies.[/I]
Except those axes that we choose as the x, y, and z axes are anything but arbitrary. They were chosen to be what they are because they have special meaning.
Consider an automobile. It has a steering wheel on one side. If you pop the hood you will see other asymmetries. That single large battery is always off to one side. The engine block isn't quite symmetric, either. The same is true for lots of other machines. There are various reasons why they just can't made as symmetric as one would like.
1. how does the product moment of inertia make the object rotate in a cone? Has it got to do with a torque due to the weight of bodies? What if gravity is switched off?
It isn't the products of inertia that do this. After all, you can always choose a set of axes that make the inertia tensor diagonal. What causes this apparent torque is an angular velocity that is not directed along a principal axis.
Angular velocity is inevitably expressed with respect to the rotating body. (Compare to translational velocity, which is typically expressed with respect to some external frame such as north-east-down.) There are many reasons for doing this. One reason is that this is how rotational sensors work. Another reason is that expressing angular velocity with respect to inertial would mean using τ
ext=d\dt(Iω), which is a nightmare. The inertia tensor as expressed in inertial coordinates is a time-varying 3x3 beast.
Just as you can use Newton's laws in a rotating frame with the introduction of inertial forces such as the centrifugal and coriolis forces, you can use the rotational equivalent in a rotating frame -- but you need to introduce an inertial torque. Specifically,
I\frac{d\omega}{dt} = \tau_{\text{ext}} - \omega\times(I\omega)
The \omega\times(I\omega) term vanishes if the rotation is along a principal axis. It doesn't vanish if the rotation is not along a principal axis.
2. So what we define as our principal axis is up to us...and we can just as easily align out rotational axis with the principal axis (by choosing the principal axis in that way)...?
In theory, yes. In practice, hardly ever. Those axes that we decided to label as x, y, and z have some real meaning to us. A plane, for example, typically has the x-axis directed along the body of the plane, y out one of the wings. That the plane happens to have a non-zero Ixy component to inertia tensor does not mean we should designate the x-axis as mostly forward but slightly out the left wing. What happens the next flight when the cargo is loaded slightly differently?
3. Again, in the example you stated, there was nothing such as xy axis, so what exactly do you mean by "cross contributions"..? Why are we able to calculate them by simply making a product between the distances from the x-axis and y axis...?
Because that is how it is defined. Why is it defined that way? Because those definitions are very useful.
