Moment of Inertia

Moment

When one begin classical mechanics you’ve likely taken for granted that the moment is the moment of inertia. The moment is defined as

\[\begin{split} I = \sum{m_\alpha r_\alpha} \\ I = \int{r \d m} \end{split}\]

• \(r_\alpha\) : distance away from the origin (often the origin of rotation).

Product of Inertia

Inertia Tensor and Inertia Matrix

The moment of inertia can be described as a rank/degree 2 tensor (a \(3 \times 3\) matrix) :

\[ \mathbf{I} = \sum_{i=1}^3{\sum_{j=1}^3{I_{ij}~\mathbf e_1 \otimes \mathbf e_2}} \]

If you’re not familiar with tensors you can simply transform the line above to the following matrix,

\[\begin{split} \mathbf{I} = \begin{pmatrix} I_{xx} & I_{xy} & \ldots \\ I_{yx} & \ddots & \vdots \\ \vdots & \ldots & \ddots \end{pmatrix} \end{split}\]

\[\begin{split} I_{ij} = \sum{m_\alpha(r_\alpha^2\delta_{ij} - i_\alpha j_\alpha)} = \begin{cases} \sum{m_\alpha (r_\alpha^2 - j^2)} & i = j \\ \sum{m_\alpha ij} & i = j \end{cases} \end{split}\]

Properties :

• \(\mathbf I\) : This matrix is symmetric \(\mathbf I = \mathbf I^T\)

Useful volume differentials :

Shape	Volume	Differential Volume	Relations
Rectangular	\(V = a_xa_ya_z\)	\(\d V = \d x\d y\d z\)
Spherical	\(V = \frac{4}{3}\pi R^3\)	\(\d V = 4\pi r^2 \d r\)
Conic	\(V = \frac{1}{3}\pi R^2h\)	\(\d V = 2\pi r h \d r\)	\(r = z \cdot \tan\theta \\ r = z \cdot \frac{R}{h}\)

Surface Area of a Cone

Naive

\[\begin{split} r(z) = \frac{R}{h}z\\ \boxed{\int_0^h{2\pi r \d z} }\end{split}\]

Correct

\[\begin{split}\int_0^h{2\pi r \d s}\\ ds^2 = dr^2 + dz^2 \implies ds = \sqrt{1 + \frac{dr^2}{dz^2}} \d z = \sqrt{1 + \frac{R^2}{h^2}} \d z\\ \boxed{\int_0^h{2\pi r \sqrt{1 + \frac{R^2}{h^2}} \d z}} \end{split}\]

• \(ds\): Differential path along the inclined plane of the cone