Rotation of a rectangular prism.

In summary, it is impossible for a rectangular prism to have pure rotation along one of its axes due to air resistance and the instability of rotation about one of the three principal axes. This can be derived from perturbative solutions to Euler's equations, also known as the "tennis racket theorem". For more rigorous derivations, one can refer to Landau&Lifgarbagez, Arnold, or Goldstein.
  • #1
Fusiontron
108
2
For some reason it is impossible for a rectangular prism to have pure rotation along one of its axises. For example, if you throw a hardback book in the air and try to spin it about its width then there is some distortion. What is the explanation for this?
 
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  • #2
Fusiontron said:
For some reason it is impossible for a rectangular prism to have pure rotation along one of its axises. For example, if you throw a hardback book in the air and try to spin it about its width then there is some distortion. What is the explanation for this?

What is your evidence that this is so? I don't know one way or the other but I find it unlikely. I can certainly agree that it would be hard for a person to do it, but imagine a machine that spins the book about a line that give you the rotation that you want, and then it shoots the book straight up into the air. Why would that not work?
 
  • #3
The distortion is probably due to air resistance. If you had a really dense rectangular prism (and your arm was strong enough), then I'm sure you could make it work.
 
  • #4
Rotation about one of the three principal axes is unstable. The "unstable" axis is the one with intermediate moment of inertia. The axes with maximum and minimum moment of inertia are stable. It is a general property, not related to air drag.
Try to throw the book when spinning around a different axis.
 
  • #5
nasu is correct; I do this demonstration in class.

http://www.aerostudents.com/files/dynamicsAndStability/rigidBodies.pdf

(section 3.3)
 
  • #6
Andy Resnick said:
nasu is correct; I do this demonstration in class.

http://www.aerostudents.com/files/dynamicsAndStability/rigidBodies.pdf

(section 3.3)

Thank you. I'll look through this document.

EDIT: Is there a more rigorous derivation of this? I was told it involves complex variables.
 
  • #7
Interesting stuff. And thanks Andy for the pdf. Unfortunately, it doesn't give an explanation of why one of the principle axis is unstable, it simply says "It can be derived"
 
  • #8
Fusiontron said:
Thank you. I'll look through this document.

EDIT: Is there a more rigorous derivation of this? I was told it involves complex variables.

BruceW said:
Interesting stuff. And thanks Andy for the pdf. Unfortunately, it doesn't give an explanation of why one of the principle axis is unstable, it simply says "It can be derived"

I'll look around and see what I can find.
 
  • #9
http://forum.allaboutcircuits.com/showthread.php?t=33367&highlight=spinning

see post#14
 
  • #10
Fusiontron said:
Is there a more rigorous derivation of this? I was told it involves complex variables.

See http://farside.ph.utexas.edu/teachin...on/node71.html [Broken]
 
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  • #11
This is sometimes called the "tennis racket theorem". It can be derived from a perturbative solution to Euler's equations (torque free).

If you have rotation around any of the 3 principle axes, perturbations about 2 of the axes will lead to nice oscillations in the rotation (I forget if it's called nutation or w/e), but perturbations about the intermediate axis will lead to exponential growth in the perturbation (and therefore violating the validity of your perturbative solution).

Euler's equations (torque free):

[tex]I_1\frac{d}{dt}\omega_1=(I_2-I_3)\omega_2\omega_3[/tex]
[tex]I_2\frac{d}{dt}\omega_2=(I_3-I_1)\omega_3\omega_1[/tex]
[tex]I_3\frac{d}{dt}\omega_3=(I_1-I_2)\omega_1\omega_2[/tex]

So, if we take the rotation to be almost all in the 1-axis (e.g. omega2 and omega 3 are small), and work to first order

We find that omega1 is roughly constant because we neglect the second order in smallness for the omega2*omega3 term; however, for omega 2 term (for example):

[tex]\frac{d^2}{dt^2}\omega_2=\left[\frac{(I_3-I_1)(I_1-I_2)}{I_3 I_2}\omega_1^2\right]\omega_2[/tex] (hopefully I did the algebra right).

We see then that for the omega2 term to remain small (and oscillate), then I1 must either be the largest or the smallest moment of inertia (thereby making the coefficient on the right hand side negative). If I1 is an intermediate moment of inertia, the coefficient on the right hand side is positive, and that means the solution for omega2 is exponential growth and not oscillatory.

One should note, though, that if you can PERFECTLY make the rotation ONLY on the 1-axis (with omega2 and omega3 being identically 0), then even if the 1-axis is the unstable axis, you won't get any wobbling. But this only works if you can make the rotation ONLY around the 1-axis.
 
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  • #12
Thanks, Matterwave. I found derivations in Landau&Lifgarbagez (section 37) and Arnold's "Mathematical Methods of Classical Mechanics" (section 29), but they are both horribly ugly. The derivation in Goldstein (chapter 5.6) is close to what you posted.
 

1. What is the definition of rotation in terms of a rectangular prism?

Rotation of a rectangular prism refers to the movement or change in position of the prism around a fixed point or axis.

2. How is the rotation of a rectangular prism different from translation or reflection?

Translation involves moving an object in a straight line while reflection involves flipping the object across a line. Rotation, on the other hand, involves turning the object around a fixed point or axis.

3. What is the formula for calculating the volume of a rotated rectangular prism?

The formula for calculating the volume of a rotated rectangular prism is V = lwh, where l is the length, w is the width, and h is the height of the prism.

4. Can the rotation of a rectangular prism affect its surface area?

Yes, the rotation of a rectangular prism can affect its surface area. When a prism is rotated, its length and width may change, causing its surface area to be different from its original orientation.

5. How does the number of rotations affect the orientation of a rectangular prism?

The number of rotations affects the final orientation of a rectangular prism. For example, if a prism is rotated 180 degrees, it will end up in the opposite orientation from its original position. However, if it is rotated 90 degrees, it will end up in a different orientation but still have the same base.

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