Doubts on rotation

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  • #26
stevendaryl
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[disclaimer: The following is my dim recollection of how it works; you should really look it up in a text book to make sure you have it right.]

I want to know how the angles are determined, i.e. what orientation are they describing the body with respect to? And how do we get the formula for kinetic energy in terms of them (which I wrote on the last page.)?

Pick 3 orthogonal axes fixed in space to be [itex]\hat{x}, \hat{y}, \hat{z}[/itex]. Take your object, and identify the three principle axes [itex]\hat{A_1}, \hat{A_2}, \hat{A_3}[/itex]. To start with, align the principle axes with [itex]\hat{x}, \hat{y}, \hat{z}[/itex]. These axes are fixed to the object, rather than fixed in space.

First, rotate the object in the x-z plane through an angle [itex]\theta[/itex].
Next, rotate the object in the x-y plane through an angle [itex]\phi[/itex].
Finally, turn the object on its 3rd principle axis (the one that was initially aligned with the z-axis) through an angle [itex]\psi[/itex]. That determines the final orientation.

Now, if the object is rotating, all three of the angles are changing simultaneously, instead of one at a time, and so there is a corresponding rotational kinetic energy associated with the rotation.

For changes of [itex]\theta[/itex], the object is rotating about principle axis [itex]\hat{A_2}[/itex].

For changes of [itex]\phi[/itex], the object is rotating about an axis [itex]\hat{A}(\theta)[/itex] that is a combination of principle axes: [itex]\hat{A}(\theta) = \hat{A_3} cos(\theta) + \hat{A_1} sin(\theta)[/itex].

For changes of [itex]\psi[/itex], the object is rotating about axis [itex]\hat{A_3}[/itex]

So if all three angles are changing simultaneously, then there will be an angular velocity about [itex]A_1[/itex] of [itex]\omega_1 = \dot{\phi} sin(\theta)[/itex]. There will be an angular velocity about [itex]A_2[/itex] of [itex]\omega_2 = \dot{\theta}[/itex]. There will be an angular velocity about [itex]A_3[/itex] of [itex]\omega_3 = \dot{\phi} cos(theta) + \dot{\psi}[/itex].

Associated with the rotation is a kinetic energy: [itex]\dfrac{1}{2} I_1 (\omega_1)^2 + \dfrac{1}{2} I_2 (\omega_2)^2 + \dfrac{1}{2} I_3 (\omega_3)^2[/itex] where [itex]I_1, I_2, I_3[/itex] are the moments of inertia associated with the three axes [itex]\hat{A_1}, \hat{A_2}, \hat{A_3}[/itex].
 
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robphy
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1. Why are the Coriolis and Centrifugal forces "fictitious"? I think I might be getting confused by the terminology, but do they represent physical forces? My lecture notes say they only appear due to choice of coordinate systems. But I want to develop physical understanding rather than mathematical.

The classic "Frames of Reference" video by Hume and Ivey (1960) may be helpful:
http://archive.org/details/frames_of_reference
Non-inertial reference frames are discussed near the 16min mark.
Rotating frames are near the 17min mark.
 
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stevendaryl, it is starting to make more sense now. Sometimes I get questions which ask for solutions of the equations of motion for the system (using the Lagrangian and the Euler-Lagrange equation) in which one of the Euler angles is constant. Does this correspond to anything special physically?

A second thing I was wondering was how to recognize the Euler angles for a system. Any leads on doing that?

robphy, I'll have a look at the video when I get time, thanks.
 

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