Instability of free rigid body rotation about middle axis

• Jack Davies
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Jack Davies

Hi everyone, I was recently talking to someone with a non-maths background about rotational stability, in particular how rotation is stable around the largest and smallest principal moments but not the intermediate one. He asked me if there was any 'obvious' reason for this, but one didn't spring to mind.

Obviously to anyone with a mathematical background you would tell them to write down Euler's equations and linearise them for a small perturbation. But I was curious if someone knew of any physical reason which would enable explanation to someone without a maths background.

Jack Davies said:
Obviously to anyone with a mathematical background you would tell them to write down Euler's equations and linearise them for a small perturbation. But I was curious if someone knew of any physical reason which would enable explanation to someone without a maths background.
Can't one explain the small perturbation without math, using a simple body, like two masses connected with a rod.

That is why we use math.

vanhees71
A.T. said:
Can't one explain the small perturbation without math, using a simple body, like two masses connected with a rod.
I tried thinking along those lines, but I was getting the wrong intuitive answer. It is treacherous.

Here is an interesting approach that is more geometric (starting at ~ 1:15) than algebraic. But there is still math graphing involved.
NOTE: This youtube video is a little careless. Both surfaces are ellipsoids. The intersections of the surfaces retain the significant characteristics that are needed to make the main point.. You can visualize the axes being scaled so that one of the ellipsoids is a sphere.

Last edited:
Jack Davies
FactChecker said:
I tried thinking along those lines, but I was getting the wrong intuitive answer. It is treacherous.

Here is an interesting approach that is more geometric (starting at ~ 1:15) than algebraic. But there is still math graphing involved.

Here is more visualization of the geometric approach:

Jack Davies and FactChecker

1. What is the definition of free rigid body rotation about middle axis?

Free rigid body rotation about middle axis refers to the motion of a rigid body around its middle axis without any external forces acting on it. This type of rotation is characterized by a constant angular velocity and no translation.

2. What causes instability in free rigid body rotation about middle axis?

Instability in free rigid body rotation about middle axis is caused by an imbalance in the distribution of mass or the location of the axis of rotation. This imbalance leads to a torque being applied to the body, causing it to deviate from its original axis of rotation.

3. How does the moment of inertia affect the stability of free rigid body rotation about middle axis?

The moment of inertia, which is a measure of an object's resistance to rotational motion, plays a crucial role in the stability of free rigid body rotation about middle axis. A higher moment of inertia leads to a more stable rotation, while a lower moment of inertia increases the likelihood of instability.

4. Can external forces affect the stability of free rigid body rotation about middle axis?

External forces, such as friction or air resistance, can affect the stability of free rigid body rotation about middle axis. These forces can create torque and disrupt the balance of the body, leading to instability.

5. How can instability in free rigid body rotation about middle axis be prevented?

To prevent instability in free rigid body rotation about middle axis, the distribution of mass and the location of the axis of rotation must be carefully considered. Increasing the moment of inertia or minimizing external forces can also help to maintain stability. Additionally, using gyroscopic stabilization or actively adjusting the axis of rotation can prevent instability in certain situations.