Perpendicular Axis Theorem

To understand what is perpendicular axis theorem, let’s take an object like a ball or a rotating disc which can rotate about its centre. You already know the moment of inertia of an object about its centre. But, after changing the point of rotation of this object, how do you find the moment of inertia? To understand this, we should know the perpendicular axis theorem.

Now, before discussing the perpendicular axis theorem, first, we will see what moment of inertia is.

Moment of Inertia

The property to resist angular acceleration is defined as the moment of inertia of an object. It is written as a summation of the products of the mass of each particle within the object, with the square of its distance from the axis of rotation.

• Let’s consider an object which is having mass m.
• It consists of small particles which have masses m₁, m₂, m₃……., respectively.
• The perpendicular distance of each particle from the centre of mass is r₁, r₂, r₃…… (as shown in the figure)

According to the definition of the moment of inertia, the mass moment of Inertia for the whole object is:

I = m₁r₁² + m₁r₂² + m₁r₃² + …..

As we consider the mass of a body (m) to be concentrated at a point, this point is its centre of mass. If this mass m is situated at a perpendicular distance of r from the centre of mass, then the moment of inertia of the whole object is,

I = ∑ mr²

So, to calculate the moment of inertia, we use two important theorems. The first one is the parallel axis theorem, and the second one is the perpendicular axis theorem. In this article, we will emphasise only on perpendicular axis theorem. Let us understand what this concept is all about.

What Is Perpendicular Axis Theorem?

The MOI of a 2-dimensional object about an axis passing perpendicularly from it is equal to the sum of the MOI of the object about 2 mutually perpendicular axes lying in the plane of the object.

According to the above definition, the perpendicular axis theorem can be written as,

I_ZZ = I_XX + I_YY

Also Read: How to Calculate Moment of Inertia

Proof of Perpendicular Axis Theorem

Let us assume there are three mutually perpendicular axes, namely X, Y and Z, and they meet at origin O.

Now, consider the object lies in the XY plane having a small area dA. It is having y distance from X-axis and x distance from Y-axis. Its distance from the origin is r.

Let I_Z, I_X and I_Y be moments of inertia about the X, Y and Z axis, respectively.

Moment of inertia on the Z-axis,

i.e.,

I_Z = ∫ r².dA …………. (i)

Here, r² = x² + y²

Put this value in the above equation

I_ZZ = ∫ (x² + y²) . dA

I_ZZ = ∫ x².dA + y².dA

I_ZZ = I_XX + I_YY

Hence proved.

Difference between Perpendicular Axis Theorem and Parallel Axis Theorem

Application of Perpendicular Axis Theorem

• This theorem is most useful when considering a body which is symmetrical about two out of the three axes. If the value of the moment of inertia about these two axes is known, then we can easily calculate the moment of inertia about the third axis by applying this theorem.
• To calculate the MOI of 3-dimensional objects such as cylinders, we can use this theorem. For this, break the cylinder into planner disks and then sum up all the MOI of the composite disks.

There is a table representing the moment of inertia of different bodies.