# Rotation of a rod after a roll and a pitch

There's a rod in free space (there's no other torques that contributes on rotating it). A torque is exerted on an axis such that the rod rolls (rotating about the principle axis parallel to the rod). While it's rolling, another torque is briefly applied to it to make it rotate on the other principle axis. What's going to happen in the subsequent motion?

My intuitive guess is that it will continue to roll and rotate along the other principle axis, but it doesn't seem to be the case since it implies that the angular momentum changes (and it's supposed to be conserved because there's no other torques exerted).

• Is the torque that makes the rod roll removed? – Farcher Mar 22 '17 at 13:42
• Are you looking for the Euler equations? – Robin Mar 22 '17 at 14:08
• Farcher no the torque just isn't along the axis that the roll is involved in. – user148792 Mar 22 '17 at 15:06
• Robin ermmm yeah I think i can use Euler equations, but how do I find a qualitative description of it without involving Euler's equations? – user148792 Mar 22 '17 at 15:07
• delickcrow123, if you use the @Robin syntax in comments, you ensure that the commenter is notified of your new comment. Otherwise, the commenter may not notice your comment answer. – Robin Mar 22 '17 at 16:15

For rigid bodies the equivalent of mass in rotation is the moment of inertia, which is a tensor. This complicates stuff.

Have a look at the wikipedia article on precession and the nice animation there. Your rod will move similarly as long as no external torque is applied.

According to your comment, your rod is not torque-free. Therefore, the precession will change when additional torque is applied:

• Increasing the roll rate (with respect to the pitch rate) will reduce the precession angle, making it less noticeable.

• Reducing the roll rate will increase the precession angle.

In addition to the precession movement, the additional torque applied will of course also increase or decrease the roll rate of your rod.

So the answer to your question is: The movement becomes more complicated, and you cannot simply superpose the two movements.

Please note that this kind of precession movement occurs with rods and similar objects only. As soon as you have a more complicated object, rotation becomes more complicated. There are some nice animations in the German wikipedia article for Poinsot's ellipsoid.