Suppose we have an object with a different moment of inertia along each axis.
It turns out that for any object with a different moment of inertia along each axis, rotations around the axis with the intermediate moment of inertia is always unstable.
If we apply forces causing the object to rotate around the axis with the smallest moment of inertia, the object will happily rotate around this axis.
If we apply forces causing the object to rotate around the axis with the largest moment of inertia, the object will also happily rotate around this axis.
But, if we apply forces causing the object to rotate around the axis with the intermediate moment of inertia, the object will refuse to rotate around this axis, and instead also start rotating in other directions as well.
In this video, we will present an intuitive explanation of why this is the case.
To do this, let us replace the object with the one shown.
Rotation around the red axis has the smallest moment of inertia.
Rotation around the yellow axis has the largest moment of inertia.
The blue axis is the intermediate axis, and it is this rotation that is unstable.
To understand why this is the case, let us consider what happens when the object has a component of rotation around more than one axis at the same time.
For example, suppose the object has a component of rotation around both the red axis and the yellow axis.
Consider what happens when the object begins to rotate in both of these directions simultaneously.
This momentum wants to cause the object to rotate around the blue axis.
This momentum wants to cause the object to rotate around the blue axis in the opposite direction.
If the object had the same moment of inertia around the red axis and yellow axis, then these effects would cancel out, and no rotation would be caused around the blue axis.
However, in our case, these effects are unequal.
Therefore, we end up starting a rotation around the blue axis.
Let us now consider a new scenario.
Suppose the object has a component of rotation around both the blue axis and the yellow axis.
Consider what happens when the object begins to rotate in both of these directions simultaneously.
This momentum wants to cause the object to rotate around the red axis.
This momentum wants to cause the object to rotate around the red axis in the opposite direction.
If the object had the same moment of inertia around the blue axis and yellow axis, then these effects would cancel out, and no rotation would be caused around the red axis.
However, in our case, these effects are unequal.
Therefore, we end up starting a rotation around the red axis.
Let us now suppose that we have no rotation around the yellow axis, and we just have these rotations around the blue axis and around the red axis.
As before, consider what happens when the object begins to rotate in both of these directions simultaneously.
This momentum wants to cause the object to rotate around the yellow axis.
This momentum wants to cause the object to rotate around the yellow axis in the opposite direction.
If the object had the same moment of inertia around the red axis and blue axis, then these effects would cancel out, and no rotation would be caused around the yellow axis.
However, in our case, these effects are unequal.
Therefore, we end up starting a rotation around the yellow axis.
Note that this rotation around the yellow axis is in the same direction as in our previous example.
We can now address the question of why rotation around the intermediate axis is unstable.
The intermediate axis is the blue axis.
Suppose we start out with just a large rotation around the blue axis.
Suppose that just a single air molecule collides with the object, giving the object just the slightest imperceptible motion around the yellow axis.
The rotation around the yellow axis will interact with the rotation around the blue axis to start a rotation around the red axis,
which will then increase the rotation around the yellow axis even more,
which will then increase the rotation around the red axis even more, and so on.
In theory, the object could rotate just around the intermediate blue axis, if the rotations around the red axis and yellow axis are precisely zero.
However, if the object has even the slightest rotation around the red axis or yellow axis, even if it just from the collision of a single air molecule, then the rotations around both the red axis and the yellow axis will end up growing.
Hence, rotation around the intermediate axis is always unstable.
But then, why are the rotations stable around the red axis and around the yellow axis?
Suppose we start out with a large rotation around the yellow axis.
Suppose that an air molecule collides with the object, causing it to now also have some rotation around the blue axis.
As we have seen earlier, the combination of the rotations around the yellow axis and around the blue axis will try to produce a rotation around the red axis.
This new rotation around the red axis will interact with the large rotation around the yellow axis to try to produce a rotation around the blue axis, but it will be in the opposite direction of the existing blue rotation, thereby cancelling it out.
If the object develops a rotation around the blue axis in the opposite direction, then this will interact with the large yellow rotation to try to create a rotation around the red axis in the opposite direction of the existing red rotation, thereby cancelling it out.
If the object develops a rotation around the red axis in the opposite direction, then this will interact with the large yellow rotation to try to create a rotation around the blue axis in the opposite direction of the existing blue rotation, thereby cancelling it out.
The end result is that with a large rotation around the yellow axis, the system is stable.
Let’s now consider the case where we start out just with a large rotation around the red axis.
Suppose that an air molecule collides with the object, causing it to now also have some rotation around the blue axis.
As we have seen earlier, the combination of the rotations around the red axis and around the blue axis will try to produce a rotation around the yellow axis.
This new rotation around the yellow axis will interact with the large rotation around the red axis to try to produce a rotation around the blue axis, but it will be in the opposite direction of the existing blue rotation, thereby cancelling it out.
If the object develops a rotation around the blue axis in the opposite direction, then this will interact with the large red rotation to try to create a rotation around the yellow axis in the opposite direction of the existing yellow rotation, thereby cancelling it out.
If the object develops a rotation around the yellow axis in the opposite direction, then this will interact with the large red rotation to try to create a rotation around the blue axis in the opposite direction of the existing blue rotation, thereby cancelling it out.
The end result is that with a large rotation around the red axis, the system is stable.
But, if we apply forces causing the object to rotate around the axis with the intermediate moment of inertia, the object will refuse to rotate around this axis, and instead also start rotating in other directions as well.
More information about stability analysis for all different types of systems in general is available in the video
"Stability Analysis, State Space, 3D visualization."
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