In analytical mechanics, we always need to solve the eigenvalue problem of moment of inertia tensor to find out the principal axes and principal moment of inertia. Well, I know that if we know the principal moment of inertia, the total rotating energy of the system could be written as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\frac{1}{2}I_x\omega_x^2 + \frac{1}{2}I_y\omega_y^2 + \frac{1}{2}I_z\omega_z^2

[/tex]

where [tex]I_x, I_y, I_z[/tex] are principal moment of inertia.

Except for that, I don't quite understand how to use principal axes and principal moment of inertia in analysis physical problem.

For understanding more about this topic, I read an example in a textbook. But still feel confusing about the relative concepts between axes. Here it is the example: A dumbbell (with two point mass, each is m, attached to two ends of a weightless bar, length 2L) is rotating around z axis at constant angular momentum [tex]\omega[/tex], the dumbbell make an angle [tex]\theta[/tex] to z axis, at t=0, the dumbbell located on the x-z plane (so no y component at t=0). Find the torque of the system.

Here is how it solve the problem:

1) Setting up a coordinate system (call body coordinate system) attaching to the rotating object with z-axis along the rotating axis. Note that the coordinate system rotates exactly the same way as the object is rotating so x and y are relatively unchanged during the rotation.

[tex]

x = L\sin\theta, y = 0, z = L\cos\theta

[/tex]

2) With x, y and z, according to the definition, we can write the tenor of moment of inertia

[tex]

I = \left(

\begin{matrix}

2mL^2\cos^2\theta & 0 & -mL^2\sin2\theta \\

0 & 2mL^2 & 0 \\

-mA^2\sin 2\theta & 0 & 2mL^2\sin^2\theta

\end{matrix}

\right)

[/tex]

3) We can find the principal axes and principal moment of inertia by solving the corresponding eigenvalue problem.

4) From the problem, the body is rotating about z-axis with angular velocity [tex]\vec{\omega}[/tex]. Now, project this angular velocity to the principal axes to get three components [tex]\omega'_1, \omega'_2, \omega'_3[/tex]

5) Since we already found principal moment of inertia and components of angular velocity along principal axes. We could plug all of these into Euler equation to find the torque.

The procedure seems quite forward but I don't understand why we could do that. My question is:

1) when we obtain the principal axes, whether this axes change with respect to time? Does principal axes defined with respect to a space frame or a frame defined and attached to an rotating object?

2) if we use the following equation to find the torque

[tex]\vec{L} = I\vec{\omega}, \vec{N} = \frac{d\vec{L}}{dt}[/tex]

where [tex]\vec{L}[/tex] is the angular momentum, [tex]\vec{N}[/tex] is the torque, both of them are with respect to a origin of a fixed coordinate (or space coordinate). In this way,

[tex]\vec{N} = mL^2\omega^2\sin2\theta (\sin\omega t \vec{\hat{x'}} - \cos\omega t\vec{\hat{y'}})[/tex]

and [tex]\vec{\hat{x'}}, \vec{\hat{y'}}[/tex] is the unit vector of fixed (space) coordinate system.

However, if we apply Euler equation to find the torque, since everything of the equation is defined in principal coordinate system, I wonder what's physical significance of the torque we obtained in this way? Obviously, from Euler equation, we will obtain an time-independent solution of torque, how can I construct a time-dependent torque with that solution?

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# What does the principal axes help in rotation probelm?

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