Parallel and perpendicular axis theorems
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Perpendicular Axis Theorem
We will state the perpendicular axis theorem, illustrate it, and then finally prove it. It is best understood through illustration. Perpendicular Axis Theorem Let Ox and Oy be two perpendicular axes. Let the moment of inertia of a plane body about Ox be Ix and the moment of inertia of the same plane body about Oy be Iy. Let Oz be the axis that is perpendicular to both Ix and Iy. Then the moment of inertia Iz of the body about Oz is [Equation goes here - download the original to see it.] Example (1) A plane rectangular lamina of mass M has length 6m and breadth 4m. Find its moment of inertia about an axis of rotation passing through its centre of mass and perpendicular to the plane. [Diagram goes here - download the original to see it.] Standard results give us [Equation goes here - download the original to see it. Hence [Equation goes here - download the original to see it.] Example (2) Find the moment of inertia of a thin uniform disc of radius z and mass M about a diameter. The standard result gives us that a disc of radius z and mass M has moment of inertia [Equation goes here - download the original to see it.] through an axis z passing through its centre and perpendicular to its plane: [Diagram goes here - download the original to see it.] Let x and y be any two distinct diameters of this disc perpendicular to each other. Since they are diameters they are each other. Since they are diameters they are each perpendicular to z. Then by the perpendicular axis theorem [Equation goes here - download the original to see it.] But by symmetry, [Equation goes here - download the original to see it.] hence [Equation goes here - download the original to see it.]
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Contents of
Parallel and perpendicular axis theorems
1 Parallel Axis Theorem
2 Proof of the parallel axis theorem
3 Perpendicular Axis Theorem
4 Proof of the Perpendicular Axis Theorem
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