Parallel Axis Theorem Proof

To prove the Parallel Axis Theorem, we start with the definition of moment of inertia about a parallel axis:

[Tex]I = \sum m_i r_i^2[/Tex]

We know that the center of mass of the object is located at O. Therefore, the perpendicular distance between the center of mass and any particle of mass m_i​ is r_i​ = d + r_i′​, where r_i′​ is the perpendicular distance between the particle and the parallel axis O′

Substituting r_i ​= d + r_i′​ into the expression for moment of inertia, we get:

[Tex]I = \sum m_i (d + r’_i)^2 = \sum m_i (d^2 + 2dr’_i + r’^2_i) [/Tex]

Expanding the square and rearranging terms, we get:

[Tex]I = md^2 + 2d \sum m_i r’_i + \sum m_i r’^2_i [/Tex]

The second term in the above expression represents the moment of inertia about the center of mass, I_cm​. The third term is the sum of the moments of inertia of each particle about the parallel axis.

[Tex]I = I_{\text{cm}} + 2dm \cdot \frac{\sum m_i r’_i}{m} + \sum m_i r’^2_i [/Tex]

[Tex]I = I_{\text{cm}} + 2dm \cdot d + \sum m_i r’^2_i[/Tex]

Since the center of mass of the object is located at O, the term ∑m_i​r_i′​ = 0. Therefore, the second term becomes zero.

[Tex] I = I_{\text{cm}} + md^2 + \sum m_i r’^2_i [/Tex]

[Tex]I = I_{\text{cm}} + md^2 [/Tex]

It shows that the moment of inertia about a parallel axis is equal to the moment of inertia about the center of mass plus the product of the mass of the object and the square of the perpendicular distance between the two axes.

Parallel axis theorem states that the moment of inertia of a rigid body about any axis parallel to its centroidal axis is equal to the sum of the body’s moment of inertia about its centroidal axis and the product of its mass and the square of the distance between the two axes.

In this article, we will understand the meaning of the parallel axis theorem, its history, proof, limitations and applications of parallel axis theorem.