Hoop (Cylindrical Shell)

21 Sep 2023

Some also call this a “zero-thickness torus”.

Parametarizing the Surface

We can do this with just standard cylindrical coordinates. Recall that in the circle post we showed that $\frac{\partial}{\partial \phi} \u{s} = \u{\phi}$

\[\b{r}(\phi, z) = R \; \u{s} + z \; \u{z} \\[10pt] A \{ \b{r}(\phi, z) \ : \ 0 \leq \phi < 2\pi \quad -L/2 \leq z \leq L/2 \}\]

Therefore

\[\frac{\partial \b{r}}{\partial \phi} = R \; \u{\phi} \qquad \frac{\partial \b{r}}{\partial z} = \u{z}\] \[d \b{A} = \left ( \frac{\partial \b{r}}{\partial \phi} d\phi \right ) \times \left ( \frac{\partial \b{r}}{\partial z} dz \right ) = R \; d\phi \; dz \; \u{s}\] \[dA = \abs{ d \b{A} } = R \; d\phi \; dz\]

Mass

\[\begin{align} M &= \int dm \\[10pt] &= \sigma \int dA \\[10pt] &= \sigma \int_{0}^{2 \pi} \int_{-L/2}^{L/2} \; R \; d\phi \; dz \\[10pt] &= \sigma R \left ( \int_{0}^{2 \pi} d \phi \right ) \left ( \int_{-L/2}^{L/2} dz \right ) \\[10pt] &= \sigma R \left ( 2 \pi \right ) \left ( L \right ) \\[10pt] &= \sigma \cdot 2 \pi R L \end{align}\]

Moment of Inertia About Central Axis

\[\begin{align} I &= \int r_{axis}^2 \;dm \\[10pt] &= \int R^2 \;dm \\[10pt] &= R^2 \int dm \\[10pt] &= \sigma \cdot 2 \pi R^3 w \\[10pt] &= M R^2 \end{align}\]

Moment of Inertia About Central Diameter

\[\begin{align} I &= \int r_{axis}^2 \;dm \\[10pt] &= \sigma \int r_{axis}^2 \;dA \\[10pt] &= \sigma \int_{0}^{2\pi} \int_{-L/2}^{L/2} r_{axis}^2 \;s \; d\phi \; dz \\[10pt] &= \sigma \int_{0}^{2\pi} \int_{-L/2}^{L/2} (R \ \sin \phi)^2 \;R \; d\phi \; dz \\[10pt] &= \sigma R^3 \left ( \int_{0}^{2\pi} \sin^2 \phi \; d\phi \right ) \left ( \int_{-L/2}^{L/2} dz \right ) \\[10pt] &= \sigma R^3 \left ( \pi \right ) \left ( L \right ) \\[10pt] &= \sigma \cdot \pi R^3 L \\[10pt] &= \tfrac{1}{2} M R^2 \end{align}\]

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Inertia Tensor of a Hoop

\[I = \begin{bmatrix} \frac{1}{2} M R^2 & 0 & 0 \\ 0 & \frac{1}{2} M R^2 & 0 \\ 0 & 0 & M R^2 \end{bmatrix} = \tfrac{1}{2} M R^2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}\]