Definition of angular momentum about an axis and conservation of the angular momentum about the axis

Question

I am reading this book Vector Mechanics for Engineers. On page 1332, I read about the Conservation of the angular momentum about the axis of precession and Conservation of the angular momentum about the axis of spin.

How am I supposed to apply these conservation theorems? I can't find the definition of angular momentum about an axis in the book. Throughout the book, only definition of angular momentum about a point O or center of mass G is stated. What were the authors thinking to put the readers in so much confusion? I have read this question but it does not help much.

Again: In a Newtonian reference frame, we have an axis (fixed or moving) and a moving rigid body. How do we defined the angular momentum of the rigid body about this axis at an instant when looking from this Newtonian reference frame?

Answer 1

Angular momentum is a vector, with a magnitude and a direction.

For a general 3D shape rotating in space with rotational velocity vector $\boldsymbol{\omega}$, it is calculated by

$$ \boldsymbol{L} = \mathbf{I}\, \boldsymbol{\omega} $$

$$ \pmatrix{L_x \\ L_y \\ L_z} = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & I_{yz} \\ I_{xz} & I_{yz} & I_{zz} \end{bmatrix} \pmatrix{ \omega_x \\ \omega_y \\ \omega_z } \tag{1}$$

where the 3×3 symmetric matrix $\mathbf{I}$ is the mass moment of inertia tensor expressed in the same orientation as $\boldsymbol{\omega}$.

Now to say that the (scalar) angular momentum is to be conserved along a particular direction, say $\boldsymbol{n}$ means that

$$ L_n = \boldsymbol{n} \cdot \boldsymbol{L} = n_x L_x + n_y L_y + n_z L_z = \text{(const)} \tag{2}$$

That is all there is to it. I assume the vector $\boldsymbol{n}$ is a unit vector above, in order to correctly project the momentum vector along the target direction and get its magnitude.

You can interpret the above as $L_n = \| \boldsymbol{L} \| \cos \theta$ where the angle $\theta$ is the angle between the $\boldsymbol{L}$ vector and the $\boldsymbol{n}$ vector.

The above value for angular momentum is only about the center of mass. To measure angular momentum about any other point you need a transformation law, just as you need a transformation law for velocities

$$ \boldsymbol{L}_A = \boldsymbol{L}_O + \boldsymbol{r}_{O/A} \times \boldsymbol{p} $$ where $\boldsymbol{p} = m \,\boldsymbol{v}_O$ is the linear momentum vector and $\boldsymbol{r}_{O/A}$ is the position of O relative to the reference point A. You interpret this as when you measure from any point away from the center of mass, angular momentum increases as the mass is further away.

Answer 2

The angular momentum of a point $M$ about a point $A$ is defined by: $$\vec L_{A}(M)=\vec{AM}\times m\vec v=\vec{AM}\times\vec p$$ If $\Delta$ is an axis through $A$, with a unit direction vector $\vec u$, then: $$\vec L_\Delta(M)=\vec L_{A}(M)\cdot\vec u$$ Also, note that for $A'\in\Delta$: $$\begin{align*} \vec L_{A}(M)\cdot\vec u &=\left[\left(\vec{AA'}+\vec{A'M}\right)\times\vec p\right]\cdot\vec u\\ &=\left[\vec{AA'}\times\vec p+\vec{A'M}\times\vec p\right]\cdot\vec u\\ &=\vec u\cdot\left(\vec{AA'}\times\vec p\right)+\left(\vec{A'M}\times\vec p\right)\cdot\vec u\\ &=\vec p\cdot\left(\vec u\times\vec{AA'}\right)+\left(\vec{A'M}\times\vec p\right)\cdot\vec u\\ &=\left(\vec{A'M}\times\vec p\right)\cdot\vec u\\ &=\vec L_{A'}(M)\cdot\vec u \end{align*}$$ Taking the derivative of the angular momentum about $A$: $$\begin{align*} \frac{d}{dt}\left(\vec{AM}\times\vec p\right) &=\left(\frac{d}{dt}\vec{AO}+\frac{d}{dt}\vec{OM}\right)\times\vec p+\vec{AM}\times\frac{d}{dt}\vec p\\ &=-\vec v_A\times\vec p+\vec{AM}\times\vec F\\ &=\vec p\times\vec v_A+\vec{AM}\times\vec F \end{align*}$$ If $A$ is fixed, then: $$\frac{d}{dt}\vec L_{A}(M)=\vec{AM}\times\vec F$$

Answer 3

The angular momentum of a rigid body rotating about a fixed axis is L = Iω , where I is the rotational inertia of the body (relative to the axis), and ω is the angular velocity of rotation. The L and ω are generally defined as vectors directed along the axis of rotation, using a right hand rule. (Wrap your right hand fingers around the axis in the direction of rotation, and our thumb gives the direction of the vectors.)