Spherical Shell

22 Nov 2024

Sometimes called a hollow sphere.

Parameterizing the Surface

We can do this with just standard spherical coordinates.

\[\b{r}(\theta, \phi) = R \sin \theta \cos \phi \; \u{x} + R \sin \theta \sin \phi \; \u{y} + R \cos \theta \u{z} = R \; \u{r} \\[10pt] A = \{ \b{r}(\theta, \phi) \ : \ 0 \leq \theta < \pi \quad 0 \leq \phi < 2\pi \}\]

Therefore

\[\frac{\partial \b{r}}{\partial \theta} = R \cos \theta \cos \phi \; \u{x} + R \sin \cos \sin \phi \; \u{y} - R \sin \theta \u{z} = R \; \u{\theta} \\[10pt] \frac{\partial \b{r}}{\partial \phi} = - R \sin \theta \sin \phi \; \u{x} + R \sin \theta \cos \phi \; \u{y} = R \sin \theta \; \u{\phi}\] \[d \b{A} = \left ( \frac{\partial \b{r}}{\partial \theta} d\theta \right ) \times \left ( \frac{\partial \b{r}}{\partial \phi} d\phi \right ) = R^2 \sin \theta \; d\theta \; d\phi \; \u{r}\] \[dA = \abs{ d \b{A} } = R^2 \sin \theta \; d\theta \; d\phi\]

Mass

\[\begin{align} M &= \int \; dm \\[10pt] &= \sigma \int \; dA \\[10pt] &= \sigma \int_{0}^{\pi} \int_{0}^{2 \pi} R^2 \sin \theta \; d\theta \; d\phi \\[10pt] &= \sigma R^2 \left ( \int_{0}^{\pi} \sin \theta \; d\theta \right ) \left ( \int_{0}^{2 \pi} d\phi \right ) \\[10pt] &= \sigma R^2 \left ( 2 \right ) \left ( 2 \pi \right ) \\[10pt] &= \sigma \cdot 4 \pi R^2 \end{align}\]

Moment of Inertia About Any Diameter

\[\begin{align} I &= \int r_{axis}^2 \; dm \\[10pt] &= \sigma \int r_{axis}^2 \; dA \\[10pt] &= \sigma \int_{0}^{\pi} \int_{0}^{2 \pi} r_{axis}^2 R^2 \sin \theta \; d\theta \; d\phi \\[10pt] &= \sigma \int_{0}^{\pi} \int_{0}^{2 \pi} (R \sin \theta)^2 R^2 \sin \theta \; d\theta \; d\phi \\[10pt] &= \sigma R^4 \left ( \int_{0}^{\pi} \sin^3 \theta \; d\theta \right ) \left ( \int_{0}^{2\pi} d \phi \right ) \\[10pt] &= \sigma R^4 \left ( \frac{4}{3} \right ) \left ( 2 \pi \right ) \\[10pt] &= \sigma \cdot \tfrac{8}{3} \pi R^4 \\[10pt] &= \tfrac{2}{3} M R^2 \end{align}\]

---

Inertia Tensor of a Spherical Shell

\[I = \begin{bmatrix} \frac{2}{3} M R^2 & 0 & 0 \\ 0 & \frac{2}{3} M R^2 & 0 \\ 0 & 0 & \frac{2}{3} M R^2 \end{bmatrix} = \tfrac{2}{3} M R^2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\]