Parallel Axis Theorem

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.

What is the Parallel Axis Theorem?

According to the Parallel Axis Theorem, the moment of inertia of a body about any axis is equal to the sum of two components, the product of the body’s mass and the square of the distance between the axis and the body’s center of mass, plus the moment of inertia of the body about a parallel axis passing through its center of mass.

History of the Parallel Axis Theorem

The concept of the moment of inertia was first introduced by the polymath Christiaan Huygens while studying a compound pendulum. Later, Jakob Steiner made significant contributions to this theorem. In recognition of their work, the parallel axis theorem is often referred to as the Huygens-Steiner theorem. This theorem provides a method for calculating the moment of inertia of a rigid body about an axis parallel to an axis passing through its center of mass.

Parallel Axis Theorem Definition

The Parallel Axis Theorem states that the moment of inertia of a rigid body about any axis parallel to a principal axis passing through its center of mass is equal to the sum of the moment of inertia about the principal axis and the product of the body’s mass and the square of the distance between the two axes.

What is the Formula of the Theorem of Parallel Axis?

The formula for the Parallel Axis Theorem is:

I = I_cm​ + md²

Where:

• I is the moment of inertia about a parallel axis,
• I_cm​ is the moment of inertia about the center of mass (also known as the centroid),
• m is the mass of the object, and
• d is the perpendicular distance between the two axes.

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.

Limitations of Parallel Axis Theorem

Following are the limitations of parallel axis theorem:

• The theorem relies on the assumption that the mass distribution within the object remains constant. If this assumption is violated, the theorem’s accuracy diminishes.
• It is valid only for rigid bodies, where the distance between the parallel axes remains fixed. For objects experiencing significant deformations, its applicability may be limited.
• While it can be extended to three-dimensional objects, calculations become more complex, potentially reducing accuracy.
• External forces acting on the object are not considered in the theorem, which may lead to inaccuracies in dynamic scenarios.
• It primarily addresses linear motion and may not adequately describe rotational motion in all cases, especially non-linear or chaotic motion.

Parallel Axis Theorem and Perpendicular Axis Theorem

The difference between parallel axis theorem and perpendicular axis theorem from the table given below:

Aspect	Parallel Axis Theorem	Perpendicular Axis Theorem
Definition	Relates the moment of inertia about a parallel axis to the moment of inertia about the centroidal axis and the distance between them.	Relates the moment of inertia about an axis perpendicular to the plane of a two-dimensional object to the sum of its moments of inertia about two perpendicular axes lying in the plane.
Applicability	Applicable to calculating the moment of inertia of a rigid body about any parallel axis, given the moment of inertia about its centroidal axis.	Applicable to calculating the moment of inertia of a two-dimensional object about an axis perpendicular to its plane, given its moments of inertia about two perpendicular axes lying in the plane.
Formula	Iparallel​ = Icentroidal​ + md2	Iz ​= Ix ​+ Iy
Example	Calculating the moment of inertia of a thin rod about an axis parallel to one end, given its moment of inertia about the centroidal axis.	Calculating the moment of inertia of a flat plate about an axis perpendicular to its plane, given its moments of inertia about two perpendicular axes lying in the plane.
Dimensionality	Applicable to three-dimensional rigid bodies.	Applicable to two-dimensional objects.
Ease of Application	Requires knowledge of the centroidal moment of inertia and the distance between the centroidal and parallel axes.	Requires knowledge of the moments of inertia about two perpendicular axes lying in the plane of the object.
Widely Used in	Widely used in engineering and physics for analyzing the rotational motion of rigid bodies.	Less commonly used but still important for calculating moments of inertia in two-dimensional systems.

Application of Parallel Axis Theorem

Following are the applications of parallel axis theorem:

• Rotational Dynamics: The theorem is widely used in rotational dynamics to calculate the moment of inertia of objects about axes parallel to their centroidal axis. This facilitates the analysis of rotational motion in various engineering and physics applications.
• Engineering Design: Engineers use he theorem in designing machinery, vehicles, and structures, where understanding rotational motion is critical. It aids in optimizing designs for stability, efficiency, and performance.
• Aerospace Industry: In aerospace engineering, the theorem helps in designing aircraft components, such as wings, propellers, and control surfaces. Accurate moment of inertia calculations are crucial for flight dynamics and stability.
• Robotics: Parallel axis theorem is employed in robotics for designing robotic arms, joints, and end-effectors. It enables engineers to determine the required torque and motor specifications for precise and efficient movement.
• Materials Science: In materials science, the theorem is applied to analyze the mechanical properties of materials, such as stiffness and strength. Understanding the moment of inertia aids in predicting the behavior of materials under various loading conditions.

FAQs on Parallel Axis Theorem

What is the theorem of parallel forces?

The theorem of parallel forces states that when two or more forces act in the same direction on an object, their combined effect can be calculated by adding their magnitudes.

What is the second parallel axis theorem?

The second parallel axis theorem, also known as the parallel axis theorem for moments of inertia, states that the moment of inertia of a rigid body about any axis parallel to an axis passing through its center of mass is equal to the sum of the moment of inertia about the center of mass and the product of the body’s mass and the square of the distance between the two axes.

What is the perpendicular axis theorem?

The perpendicular axis theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes lying in the plane of the object and intersecting at the point where the perpendicular axis intersects the object.

What is the parallel axis theorem of energy?

The parallel axis theorem of energy is a concept in mechanics that states the total kinetic energy of a system can be calculated as the sum of the kinetic energies of its individual components, considering both their translational and rotational motion.

How do you prove parallel axis theorem?

The parallel axis theorem is typically proved using mathematical derivations based on principles of mechanics and geometry, such as the application of the definition of moment of inertia and the parallel axis theorem itself. It involves analyzing the distribution of mass around different axes and understanding how the moments of inertia relate to each other.