📂Classical Mechanics

Vertical Axis Theorem

Vertical Axis Theorem

The moment of inertia about an axis perpendicular to a plane is equal to the sum of the moments of inertia about any two perpendicular axes lying in the plane and passing through the perpendicular axis.

$$\color{red}{I_{z}}=\color{blue}{I_{x}+I_{y}}$$

Proof

$$I_{z}=\sum\limits_{i} m_{i}{r_{i}}^{2}$$

By Pythagoras’ theorem, since ${r_{i}}^{2}={x_{i}}^{2}+{y_{i}}^{2}$, substituting this into the above equation gives:

$$I_{z}=\sum\limits_{i} m_{i}({x_{i}}^{2}+{y_{i}}^{2})=\sum\limits_{i} m_{i}{x_{i}}^{2}+\sum\limits_{i} m_{i}{y_{i}}^{2}$$

$x$ is the distance from the $y$-axis, and $y$ is the distance from the $x$-axis, so it follows that:

$$\sum\limits_{i} m_{i}{x_{i}}^{2}=I_{y}, \quad \sum\limits_{i} m_{i}{y_{i}}^{2}=I_{x}$$

Therefore,

$$I_{z}=I_{x}+I_{y}$$

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See Also

• Parallel Axis Theorem