Perpendicular Axis Theorem – Definition & Application

Table of Contents

Introduction

The perpendicular axis theorem is a fundamental concept in physics and engineering, which helps us to calculate the moment of inertia of a rigid body about an axis that is perpendicular to the plane of the body. It is one of the most important theorems in rotational mechanics and is widely used in various engineering and scientific applications.

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 the moments of inertia about two perpendicular axes in the plane passing through the same point on the perpendicular axis”.

Definition:

The perpendicular axis theorem states that the moment of inertia of a planar body about an axis perpendicular to the plane of the body is equal to the sum of the moments of inertia of the body about two mutually perpendicular axes lying in the plane of the body and intersecting at the point where the perpendicular axis passes through the plane of the body. Mathematically, it can be written as:

Iz = Ix + Iy

where Iz is the moment of inertia of the body about the perpendicular axis, and Ix and Iy are the moments of inertia of the body about two mutually perpendicular axes lying in the plane of the body.

Derivation:

The perpendicular axis theorem can be derived using basic calculus and geometry. Let us consider a planar body of mass M and shape S, which is rotating about a perpendicular axis passing through the origin O. Let us also consider two mutually perpendicular axes X and Y lying in the plane of the body and intersecting at the point P, where the perpendicular axis passes through the plane of the body. Let us assume that the body is rotating with an angular velocity ω.

Now, let us consider an element of mass dm of the body located at a distance r from the origin O. The moment of inertia of this element of mass about the perpendicular axis passing through the origin O is given by:

dIz = r² dm

Similarly, the moment of inertia of this element of mass about the X and Y axes is given by:

dIx = (y² + z²) dm

dIy = (x² + z²) dm

where x, y, and z are the distances of the element of mass from the X, Y, and perpendicular axes, respectively.

Now, let us integrate these equations over the entire body to obtain the moments of inertia of the body about the perpendicular axis and the X and Y axes, respectively:

Iz = ∫S r² dm

Ix = ∫S (y² + z²) dm

Iy = ∫S (x² + z²) dm

By using the Pythagorean theorem, we can simplify these equations as follows:

Ix = Iy = ∫S (x² + y²) dm

Substituting this result into the perpendicular axis theorem equation, we obtain:

Iz = Ix + Iy = 2 ∫S (x² + y²) dm

Using the perpendicular axis theorem, we have reduced the calculation of the moment of inertia about a perpendicular axis to the calculation of the moment of inertia about two mutually perpendicular axes lying in the plane of the body.

Applications:

The perpendicular axis theorem has numerous applications in physics and engineering. Some of its important applications are:

1. Calculating the moment of inertia of a planar body: The perpendicular axis theorem is commonly used to calculate the moment of inertia of a planar body about a perpendicular axis passing through the origin O. By using the theorem, we can reduce the calculation of the moment of inertia to the calculation of the moment of inertia about two mutually perpendicular axes lying in the plane of the body.

2. Understanding the rotation of rigid bodies: The perpendicular axis theorem is fundamental to our understanding of the rotation of rigid bodies. It helps us to determine the moment of inertia of a rigid body with different axes, which is necessary.

Perpendicular Axis Theorem Example

Let’s consider a square plate with a mass of 4 kg and sides of length 2 meters. We want to find the moment of inertia of the square plate about the z-axis passing through its center. To apply the Perpendicular Axis Theorem, we need to find the moments of inertia of the square plate about two perpendicular axes (x and y) in the plane passing through the center of the square.

The moment of inertia of a square plate about its center of mass and perpendicular to its plane (z-axis) is given by:

Iz = (1/6) x m x a²

where m is the mass of the square plate and a is the length of the side of the square plate.

Iz = (1/6) x 4 x (2)²

Iz = 8/3 kg.m²

To find the moments of inertia about the x and y axes, we need to use the parallel axis theorem. According to the parallel axis theorem, the moment of inertia of a planar object about an axis parallel to its plane is equal to the moment of inertia about its center of mass plus the product of its mass and the square of the distance between the two axes.

Ix = Iz + md²

where d is the distance between the center of the square and the x-axis.

From the geometry of the square, we can determine that d = a/2 = 1 meter.

Ix = Iz + md²

Ix = 8/3 + 4 x (1)²

Ix = 20/3 kg.m²

Similarly, the moment of inertia about the y-axis passing through the center of the square can be found using the same formula:

Iy = Iz + md²

where d is the distance between the center of the square and the y-axis, which is also equal to a/2.

Iy = Iz + md²

Iy = 8/3 + 4 x (1)²

Iy = 20/3 kg.m²