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Perpendicular axis theorem
Intermediate
For a flat lamina, I_z equals I_x plus I_y.
I_z = I_x + I_yFor a flat lamina in the xy-plane, I_z = I_x + I_y. Useful for discs and rings: I_x = I_z/2.
Ring I_z = mR², so I about a diameter = ½mR².
Only for flat (planar) bodies.
In more detail
For a thin planar body, the moment of inertia about an axis perpendicular to the plane equals the sum of the moments about two perpendicular axes in the plane, I_z = I_x + I_y.
Real-world uses
- Flat plates and discs are analysed with this theorem to simplify moment calculations.
- Engineers compute moments of inertia of flat components in machinery.
- For a thin flat plate, the moment of inertia about an axis perpendicular to the plate equals the sum of the moments about two in-plane axes.
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Frequently asked questions
Is this for flat bodies only?
Yes, it applies to thin planar bodies, which are flat in the plane of the axes.
Can I use it for a square plate?
Yes. The moments about in-plane axes are equal, so you can find the perpendicular one quickly.
Does it work for three-dimensional bodies?
No. The perpendicular axis theorem applies only to flat laminae lying in a plane.
Related topics in Rotational Motion
Moment of inertia of a solid sphere · Moment of inertia of a hollow sphere · Parallel axis theorem · Radius of gyration · Newton's second law for rotation · Rotational kinetic energy