Cartesian Form of Discrete Moment of Inertia

Theorem

Let $B$ be a rigid body which is rotating in space about some axis $\LL$.

Let $B$ be embedded in Cartesian $3$-space whose origin lies on $\LL$.

Let $I_{xx}$, $I_{yy}$, and $I_{zz}$ be the moments of inertia of $B$ about the coordinate axes $\mathbf O x$, $\mathbf O y$ and $\mathbf O z$ respectively.

Let $B$ be composed of a countable number of particles $P_1, P_2, \ldots$ such that:

each $P_i$ has mass $m_i$

each $P_i$ has perpendicular distance $r_i$ from $\LL$.

each $P_i$ has coordinates $\tuple {x_i, y_i, z_i}$.

Then:

$\ds I_{xx}$

$=$

$\ds \sum m_i \paren { {y_i}^2 + {z_i}^2}$

$\ds I_{yy}$

$=$

$\ds \sum m_i \paren { {z_i}^2 + {x_i}^2}$

$\ds I_{zz}$

$=$

$\ds \sum m_i \paren { {x_i}^2 + {y_i}^2}$

Proof

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Sources

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): moment of inertia
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): moment of inertia