Parallel Axis Theorem

The Parallel Axis Theorem is another principle in classical mechanics that relates the moments of inertia of an object about two parallel axes. This theorem allows the calculation of the moment of inertia about an axis parallel to an axis passing through the center of mass.

Statement of the Parallel Axis Theorem:
If I_cm is the moment of inertia of an object about an axis passing through its center of mass, and m is the total mass of the object, and d is the perpendicular distance between the two parallel axes, then the moment of inertia I about a parallel axis is given by:

I_z = I_x+I_y

Proof:

Let’s assume an object with mass elements distributed in space, and I_cm is the moment of inertia about an axis passing through its center of mass.

The moment of inertia I’ about an axis parallel to this and displaced by a distance d is given by:

I′ = ∑m_i r_i^‘2

Where r_i‘ is the perpendicular distance from the i-th mass element to the parallel axis.

Now, express r_i‘ in terms of r_i‘ (the distance from the i-th mass element to the axis passing through the center of mass) and d:

r_i′​=r_ii + d

Substitute this into the expression for I’:

I′=∑m_i(r_i​+d)²

Expand and simplify:

I′=∑m_i²​(r_i² ​+ 2r_id + d²)

Now, use the fact that ∑m_i²​r_i² ​ is the moment of inertia about the center of mass (I_cm):

I′ = I_cm ​+ 2d∑m_i r_i + md²

Since ∑m_iir_i is zero (center of mass), the term reduces to zero:

I’ = I_cm + md²

This is the Parallel Axis Theorem, providing a simple way to calculate the moment of inertia about a parallel axis given the moment of inertia about an axis passing through the center of mass.