Physlib.ClassicalMechanics.RigidBody.SolidSphere

For a given dimension $d \in \mathbb{N}$, mass $m \in \mathbb{R}_{\geq 0}$, and radius $R \in \mathbb{R}_{\geq 0}$, this definition represents a solid sphere as a rigid body in the $d$-dimensional Euclidean space $\text{Space}(d)$. The mass distribution $\rho$ is uniform over the closed ball $\bar{B}_R(0)$ of radius $R$ centered at the origin. Formally, for a function $f$, the distribution is defined by the linear functional: $$\rho(f) = \frac{m}{\text{Vol}(\bar{B}_R(0))} \int_{\bar{B}_R(0)} f(x) \, dx$$ where $\text{Vol}(\bar{B}_R(0))$ denotes the volume of the closed ball.

For any natural number $d$, and any non-negative real numbers $m$ and $R$ where $R \neq 0$, the total mass of a solid sphere rigid body in $(d+1)$-dimensional space with radius $R$ and mass parameter $m$ is equal to $m$.

For any natural number $d$, mass $m \in \mathbb{R}_{\geq 0}$, and radius $R \in \mathbb{R}_{\geq 0}$, the center of mass of a solid sphere in $(d+1)$-dimensional space with mass $m$ and radius $R$ centered at the origin is the zero vector $0$.

For a solid sphere in 3D space with mass $m \in \mathbb{R}_{\geq 0}$ and radius $R > 0$, the moment of inertia tensor through its center of mass is equal to $\frac{2}{5} m R^2 \mathbf{I}$, where $\mathbf{I}$ denotes the $3 \times 3$ identity matrix.