State and prove perpendicular axis theorem

The Perpendicular Axis Theorem is a principle in classical mechanics that relates the moments of inertia of a planar object about three perpendicular axes. The theorem is applicable to planar objects that lie in the xy-plane.

Statement of the Perpendicular Axis Theorem:
If a planar object lies in the xy-plane and the z-axis is perpendicular to this plane, then the sum of the moments of inertia Ix and Iy about any two perpendicular axes in the plane (x and y) is equal to the moment of inertia Iz about an axis perpendicular to the object’s plane (z). Mathematically, this can be expressed as:

[Iz = Ix + Iy]

Proof:

Consider a planar object with mass elements distributed along the xy-plane. Let’s assume the object is continuous and its mass distribution is described by a mass density function ( \rho(x, y) ). The moments of inertia about the x, y, and z-axes are given by:

[Ix = \int \int \rho(x, y) \cdot y^2 \, dx \, dy]

[Iy = \int \int \rho(x, y) \cdot x^2 \, dx \, dy]

[Iz = \int \int \rho(x, y) \cdot (x^2 + y^2) \, dx \, dy]

Now, we want to prove that (Iz = Ix + Iy).

Let’s consider the expression (Ix + Iy):

[Ix + Iy = \int \int \rho(x, y) \cdot y^2 \, dx \, dy + \int \int \rho(x, y) \cdot x^2 \, dx \, dy]

Combine the terms:

[Ix + Iy = \int \int \rho(x, y) \cdot (x^2 + y^2) \, dx \, dy]

But, ((x^2 + y^2)) is the distance squared from the z-axis in the xy-plane, which is the definition of the moment of inertia Iz.

Therefore, (Ix + Iy = Iz), and we have proved the Perpendicular Axis Theorem:

[Iz = Ix + Iy]