The derivation of the angular momentum of a rigid body in terms of the moment of inertia considers only the z component. Why are not the other components considered ? Derivation of angular momentum of Rigid Body

  • $\begingroup$ This is from Kleppner - pg. 253 $\endgroup$ Oct 18 '16 at 15:16
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    $\begingroup$ Don't use $$ $$ in title. $\endgroup$
    – user36790
    Oct 18 '16 at 15:24
  • $\begingroup$ Isn't the rotation about the z-axis? So the distance along the rotation axis aren't important. $\endgroup$ Oct 18 '16 at 15:29
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    $\begingroup$ Possible duplicate of Cancelling internal forces/moments term when deriving inertial matrix $\endgroup$ Oct 18 '16 at 15:33
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    $\begingroup$ I think the author is afraid to use cross products which is what you need when considering all the 3D components. $\endgroup$ Oct 18 '16 at 15:35

The $z$ axis is of course not special. However, the angular momentum is a vector quantity, and it is often easiest to compute a vector quantity by calculating each component separately. On the other hand, precisely because the $z$ axis is not special, the same calculation will apply to the $x$ and $y$ components, as long as you change what needs to be changed. This last part means that a rigid body really has three moments of inertia -- one for each axis. The formal way to do it is with an object called the inertia tensor. 1

You can choose the $z$-axis to be the axis of rotation, and then the $x$ and $y$ components will be zero, so that you only need the $z$ component. This amounts to demanding that $L = L_z\hat z$ where $\hat z$ is a unit vector, but then if $\tau$ is the torque $$\tau = \frac{d}{dt} L = \frac{d}{dt} L_z \hat{z}$$ so $\frac{d}{dt} \hat z$ has to be non-zero in general. Thus the cost is that you have to find how $\hat z$ changes with time. That can be done however, and any mechanics textbook should treat it.

  1. The inertia tensor has six independent components because it's symmetric, but three only encode the transformation between the chosen coordinate system and the principal axes. It's really only the principal moments that are properties of the rigid body itself. Three principal moments + three Euler angles = six components.

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