# Perpetual Rotation of Rigid Body

Assuming no external torque or forces acting can a rigid body be set in perpetual rotation motion about an axis which is not its principal axis?

If no then does the earth continuously change its axis of rotation? as the principal axes of the earth changes changes continuously because of tectonic plate movements, animals humans moving etc.

• Possible duplicate of Object Pushed by Multiple Forces – BioPhysicist Mar 30 '19 at 15:35
• The Earth has external forces and torques acting on it – BioPhysicist Mar 30 '19 at 15:36
• @AaronStevens The first question has nothing to do with the possible duplicate and the answer is not contained therein. – GiorgioP Mar 30 '19 at 16:11
• @GiorgioP This is why the word "possible" is there. This is also why it takes 5 votes to mark as such. If you disagree then just don't vote to close. If it stays open that's fine. I recognize what I think isn't the only thing that matters – BioPhysicist Mar 30 '19 at 16:19
• – anna v Mar 30 '19 at 16:38

The answer to your first question is no, any rotation about a non-principal axis will result in the axis of rotation changing. To see why, lets go into the body frame of the object, where with no torques the angular momentum follows the equation

$$\mathbf{\dot{L}}+\boldsymbol{ \omega} \times \mathbf{L=0}$$

Where $$\mathbf{L}$$ is the angular momentum of the system and $$\boldsymbol{ \omega}$$ is the axis of rotation. Now, $$\mathbf{L}=I\boldsymbol{ \omega}$$, where $$I$$ is the moment of inertia tensor. By definition, a principal axis is an eigenvector of the moment of intertia, so if $$\boldsymbol{ \omega}$$ is not along a principal axis, $$\mathbf{L}$$ can't be pointing in the same direction as $$\boldsymbol{ \omega}$$. In our body frame of reference, $$\dot{I}=0$$. Differentiating our previous expression for $$\mathbf{L}$$ and combining with our first equation, we have

$$I\boldsymbol{\dot{\omega}}+\boldsymbol{ \omega} \times \mathbf{L=0}$$

Now, a stationary axis of roation requires that within the body frame of reference, $$\boldsymbol{\dot{\omega}=0}$$. But from our equation above, this leaves us with the requirement that $$\boldsymbol{ \omega} \times \mathbf{L=0}$$. The cross product of two non-zero vectors is only zero if they are parallel, which we already ruled out by virtue of choosing a non-principal axis of rotation. Thus, it is impossible to have steady rotation about a non-principal axis.

As to your second question, the Earth's axis does indeed change over time, although this is mainly because of torques exerted on the Earth due to gravitational interactions. The Earth is close enough to being spherical that every axis is very nearly a principal one, so the precession of the axis caused by obliquity of the Earth is rather negligible.

Assuming no external torque or forces acting can a rigid body be set in perpetual rotation motion about an axis which is not its principal axis?

In a nutshell? No.

If a body rotates about a principal axis (an axis through its centre of mass) and no external torques act on it, then it will forever keep rotating.

Now look at the case where the rotation in NOT about a principal axis: In order to keep the rotation going, a force has to act on it, called the centripetal force, $$F_c$$. It's a vector pointing from the centre of mass of the object to the axis of rotation.

In vector notation we can write:

$$\mathrm{F_c}=-m\omega^2\mathbf{r}$$

where $$m$$ is the object's mass, $$r$$ the radius of the orbit and $$\omega$$ its angular velocity.

If this force ceases to act then the object will leave its orbital trajectory.

If no then does the earth continuously change its axis of rotation? as the principal axes of the earth changes changes continuously because of tectonic plate movements, animals humans moving etc.

It doesn't really. The effects you list are minimal, compared to the inertial moment/inertial momentum of the Earth.

• I like your answer to the first question. I think that the best answer to the second question is that tectonic plate movement and human activity are examples of how the the earth is NOT a rigid body. Thus is would be folly to apply a law for rigid bodies to analyze those effects. – Duncan Harris Mar 30 '19 at 16:36
• I might be misinterpreting your answer, so please correct me if I am, but you seem to be saying that any axis that goes through the center of mass is a principal axis. If you are saying this, I'm fairly sure it's incorrect-- there are plenty of axes that go through the center of mass that aren't principal as long as the object isn't spherically symmetric. – el duderino Mar 30 '19 at 17:25