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I know that,

$$L=I\omega$$

where $L$ is the angular momentum vector, $I$ is the inertial tensor, and $\omega$ is the angular velocity.

Now here are my doubts :-

  1. Before I was taught the moment of inertia tensor concept, we were taught that moment of inertia is always calculated about an axis. However this tensor matrix seems to calculate it about a point. Am I right?

  2. Assuming that I am correct in my previous doubt and moment of inertia is calculated about a point (the same point about which $L$ is to be found),
    is the formula

$$L_{0}=I_{0}\omega$$

correct even if $0$ is not the center of mass of the body?

Assuming that it is correct about any point, is it correct even if point $0$ is moving with any velocity and acceleration (say uniform circular motion)?

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  • $\begingroup$ Why do you think the tensor is calculated about a point? Each element is given by an integral in terms of distances from an axis $\endgroup$ – Triatticus Oct 24 '19 at 20:23
  • $\begingroup$ @Triatticus the components of the inertia tensor are given in a coordinate system $\endgroup$ – Eli Oct 25 '19 at 7:50
  • $\begingroup$ Yes I'm aware of the definition but in practical calculation of any element of the tensor a reference axis is used, that even follows immediately after they mention the position vector to the mass element. $\endgroup$ – Triatticus Oct 25 '19 at 18:40
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If rotation is restricted about a fixed axis, then there is a single mass moment of inertia component associated with this axis. It is defined as

$$\text{(angular momentum)} = \text{(mass moment of inertia)} \text{(rotation)}$$

$$ L_{\rm axis} = I_{\rm axis} \omega_{\rm axis} \tag{1}$$

The rotation vector is really just a direction and a magnitude. Any single MMOI component relates only the magnitude along a specified direction to the resulting angular momentum.

To generalize this problem, you can take all possible rotation directions and describe the resulting angular momentum vectors as a mass moment of inertia tensor. Mathematically this is a 3×3 matrix that transforms a 3×1 rotation vector into a 3×1 angular momentum vector

$$ \boldsymbol{L} = \mathbf{I}\, \boldsymbol{\omega} $$ $$ \pmatrix{L_x \\ L_y \\ L_z} = \begin{vmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & I_{yz} \\ I_{xz} & I_{yz} & I_{zz} \end{vmatrix} \pmatrix{\omega_x \\ \omega_y \\ \omega_z } \tag{2}$$

Now in the general sense, the rotation vector is not associated with a particular location. I mean the components of $\boldsymbol{\omega}$ do not change from point to point, like the components of linear velocity $\boldsymbol{v}$. So in the above equation rotation only defines direction and magnitude.

But angular momentum is defined at a point. Meaning, that if expressed at a different location the components change in a manner similar to velocities and torques:

$$ \begin{aligned} \boldsymbol{v}_A & = \boldsymbol{v}_B + \boldsymbol{r} \times \boldsymbol{\omega} & & \text{transformation of velocities} \\ \boldsymbol{L}_A & = \boldsymbol{L}_B + \boldsymbol{r} \times \boldsymbol{p} & & \text{transformation of angular momentum} \\ \boldsymbol{\tau}_A &= \boldsymbol{\tau}_B + \boldsymbol{r} \times \boldsymbol{F} & & \text{transformation of torque} \\ \end{aligned} \tag{3}$$

Here the vector $\boldsymbol{r}$ goes from $A \rightarrow B$, and $\boldsymbol{p}$ is linear momentum.

So the definition of mass moment of inertia needs to include locational information, making equation (2) incorrect unless the location is specified somehow.

$$ \boldsymbol{L}_A = \mathbf{I}_A \,\boldsymbol{\omega} \tag{4}$$

and transformation from one point to another is done using the parallel axis theorem

$$ \begin{aligned} \boldsymbol{L}_A & = \boldsymbol{L}_B + \boldsymbol{r} \times \boldsymbol{p} \\ & = \boldsymbol{L}_B + \boldsymbol{r} \times (m \boldsymbol{v}) \\ & = \mathbf{I}_B\, \boldsymbol{\omega} + \boldsymbol{r} \times (m \boldsymbol{\omega} \times \boldsymbol{r}) \\ \boldsymbol{L}_A &= \mathbf{I}_A\,\boldsymbol{\omega} \end{aligned} \tag{5}$$

where $$\mathbf{I}_A = \mathbf{I}_B - m [\boldsymbol{r}\times] [\boldsymbol{r} \times] $$

and $[\boldsymbol{r}\times] = \begin{vmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{vmatrix} $ is mathematical construct to represent cross products. It is called the cross product matrix operator.

In summary:

  • Rotation vectors are not defined at a point, but rotation components must be specified along a direction. Rotation vectors are a property of an entrire rigid body.
  • Velocity vector are defined at a point, although there might be multiple points (on a line) that do not change the components of the vector. These are defined as the axis of rotation of a body. When a component of velocity is specified, its location and direction need to be specified also.
  • Momentum vector is not defined at a point but a property of an entire rigid body as it is always defined as mass times the velocity vector at the center of mass.
  • Angular momentum vector is defined at a point just as velocity is, and there is a line in space where the components of the vector do not change. This is called this axis of percussion. Again when a component of angular momentum is specified, its direction, as well as its location, needs to be specified as well.
  • Mass moment of inertia tensor is defined at a location, and along with a specified coordinate system. This has to be the same coordinate system rotational velocity is defined at.
  • Force vector is not location specific, but shared with an entire body. As far as the motion of the center of mass of a body only the net force matters, not the location of where any load is applied.
  • Torque vector is location specific, and the axis in space where the components of torque do not change is called the line of action of the force.
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