Moment of inertia and force needed to tilt/change axis of rotation

[birulami]

Consider a freely rotating body. Let the axis of rotation be the z-axis. For simplicity assume all the mass of the body is concentrated in the x-y-plane, i.e. the plane in which the body rotates.

I have read about the moment of inertia tensor on wikipedia, but I don't see how I would combine it with a torque to tilt the axis of rotation.

Suppose the above rotating body indeed has a solid axis, albeit of zero mass, sticking out at one end with length $\gt l$. At $z=l$ we apply a force perpendicular to the axis for a distance of $\Delta s$ in the direction of $-x$.

  |<- apply force 
  |
  |
=====  <- x-y plane of rotation

What will happen to the to the overall rotation.

a) Will the axis tilt only a certain amount or does the force applied induce a rotation that keeps going and combines with the previous rotation.

b) What is the formula to get the tilt angle or the angular speed? I assume it somehow combines the inertia tensor and the force F or torque $l\times F$?

Thanks,
Harald.

 

[birulami]

Found it myself. In general it is

$$\tau = I\cdot\dot{\vec{\omega}}$$

where $\tau$ is the torque, the equivalent of force for linear motion, $I$ is the moment of inertia tensor (i.e. 3x3 matrix) and $\dot{\vec{\omega}}$ is the three-vector of angular acceleration. The rest seems to be to put in the special case values. And I reckon that applying a torque that that is not just parallel to $\dot{\vec{\omega}}$ will result in an angular velocity component, not just in a tilt of the rotational axis.