# What happens with an object not originally rotating around its principal axis?

I'm wondering about this question stated in the title. Suppose we have a rigid body undergoing some sort of rotation. Let the rotational vector point in a different direction than the angular momentum vector. Obviously, if we want to keep the rotational vector constant, we'll have to add an external torque since the angular momentum will rotate around the rotational vector.

First question: in what way does the net torque affect the angular momentum? Is it added so that the angular momentum becomes parallel to the rotational vector?

Also, if the rotational vector actually does point in the direction of the angular momentum vector, it's rotating around one of it's principal axis, meaning that we won't have to give it a net torque in order for it to rotate (if it's already in rotation from the beginning so say).

Second question: So let's suppose we don't rotate it around one of its' principal axes, will the body eventually come at rest to rotate around it's principal axes? If so, a net torque must have been added, and where will it come from?

I'm new to this concept in mechanics. So I'd be glad if you could give me a detailed explanation of the questions above.

For the first question, you're talking about a constraint, an axle in this case. If the axle isn't parallel to a principal axis, it imposes oscillating torques on the body about the axes perpendicular to the axle. The angular momentum of the body thus oscillates.

For the second, you get torque-free precession. The angular momentum remains constant, but the rotation vector doesn't. So, the rotating body wobbles. In the absence of friction and external torques, this persists indefinitely. However, if the body has internal moving parts subject to friction they may damp the wobble. The result is a body rotating around its axis of maximum inertia. This is the principle behind "nutation dampers" on spacecraft. This effect was a surprise to early spacecraft engineers.

• Thank you, very clear! Apr 27, 2022 at 11:57

First question.

If a rigid body is forced to rotate about a fixed axis that is not a principal axis, the angular momentum vector does not lie along the axis of rotation (the angular velocity vector is along the axis of rotaton), but varies in direction describing a cone around the axis of rotation. Since the angular momentum is not constant, there is an external torque exerted on the body. The torque is at right angles to the axis of rotation. In this case, the body is not dynamically balanced. If the rotation is about a principal axis, the angular momentum vector is aligned with the axis of rotation and there is no torque. In this case, the body is dynamically balanced..

Second question.

For torque free rotation, the angular momentum is constant even if the rotation is not around a principal axis. The kinetic energy is constant. Also, $$\vec \omega \cdot \vec J$$ is constant where $$\vec \omega$$ is the angular velocity and $$\vec J$$ is the angular momentum; that is, the projection of $$\vec \omega$$ onto $$\vec J$$ is constant. For initial rotation about a non-principal axis, if a torque is applied the subsequent motion depends on the inertia tensor of the body and the torque, and will not necessarily result in rotation about a principal axis.

The general motion of a rigid body is described using Euler's equations and the Eulerian angles; also, the Lagrangian formulation is frequently used. See an intermediate/advanced physics mechanics textbook, such as Mechanics by Symon, or Classical Mechanics by Goldstein. The general motion is very complicated; for example, entire books have been written describing the motion of a spinning moving top, subject to gravity and friction.

• Thank you! We'll be checking in on general motion using the Lagrangian formulation soon, so I'll certainly check the textbook out. Apr 27, 2022 at 11:58
• You are welcome. This is not simple! Apr 27, 2022 at 12:52