Why does the spinning book have an unstable axis?

[jpswensen]

I searched through the forums and couldn't find a topic or answer, so I will pose it (possibly again). I can work through deriving the equations of motion (either through Euler-Lagrange methods or Hamilton methods) for the spinning book problem and it is obvious that there *is* an unstable axis, but I am wondering if there is a more fundamental explanation as to *why* this occurs. I guess I can see that it happens in the equations and experiments, but don't have any intuition or understanding as to why this occurs.

I have searched the internet and various mechanics books that show derivations of this problem, but haven't seen an explanation of why. If someone could point me to a good book or article that tries to explain it, I would appreciate it.

 

[K^2]

Because moment of inertia is a tensor, so angular momentum and angular velocity are not co-linear in general. Angular momentum is conserved, which forces angular velocity vector to precess.

 

[Bob S]

There is a comprehensive discussion in Goldstein's book on Classical Mechanics. If an object has three unequal principal moments of inertia, spinning about the intermediate axis is unstable. Look up polhode in Google. There are some Youtube videos that demonstrate the rolling of the polhode on the invariable plane. See

http://www.google.com/url?sa=t&sour...g5ywCw&usg=AFQjCNFWuUf34mlgPymTSWQ5gdCwf552bw

http://www.google.com/url?sa=t&sour...pbCwCw&usg=AFQjCNF3ti1bv9375jQT-cFre4XXoNOyyg

Bob S

 

[Studiot]

Will this do?

(after Acheson)

 

[PJC01]

The reason for the unstable axis in the spinning book problem is due to the conservation of angular momentum. When the book is spinning, it has a certain amount of angular momentum, which is the product of its moment of inertia and angular velocity. This angular momentum is conserved, meaning it cannot change unless acted upon by an external torque.

The book has two axes of rotation, one along its length and one along its width. The axis along its length is stable because it is the axis with the largest moment of inertia. This means that it takes more torque to change the rotation around this axis compared to the rotation around the width axis.

However, the axis along the width has a smaller moment of inertia, meaning it takes less torque to change the rotation around this axis. This makes it unstable because any slight disturbance or external torque can easily cause the book to rotate around this axis, leading to its instability.

Furthermore, the distribution of mass in the book also plays a role in its unstable axis. If the mass is evenly distributed, the book will have a stable rotation around both axes. But if the mass is concentrated towards one end, it will have a lower moment of inertia around the width axis, making it more susceptible to instability.

In summary, the unstable axis in the spinning book problem is a result of the conservation of angular momentum, the difference in moments of inertia between the two axes, and the distribution of mass in the book. These factors all contribute to making the rotation around the width axis unstable.