Notes on Perpendicular Axis Theorem

In order to understand the perpendicular axis theorem, we will take an example. Let’s consider a ball or a ring that can rotate about its centre. You know what is the moment of inertia of an object regarding its centre. However, when you change the point of rotation of the ball or ring, you need to find the moment of inertia. In order to understand this, it is important to understand the perpendicular axis theorem.

The moment of inertia around a perpendicular axis to the plane of a planar object is the sum of the moments of inertia of the other two perpendicular axes passing through the same point in the plane of the object. This theorem’s importance extends beyond finding the moments of strictly planar objects. It is an important technique for calculating the moments of inertia of three-dimensional structures like cylinders by dividing them into planar discs and adding the moments of inertia of the composite discs.

Perpendicular Axis Theorem:

The moment of inertia is addressed in the parallel and perpendicular axis theorem. Thus, before we see the theorems, let us have some knowledge of the moment of inertia. It is a body attribute that allows it to withstand angular acceleration.

The total of the masses of the body’s particles multiplied by the square of the distance from the axis of rotation gives the moment of inertia.

Moment of inertia is given as

Ii = ∑miri2

This theorem is applied to symmetric objects, that is, things with two out of three symmetric axes. When the moment of inertia of the other two axes is known, this equation may be used to compute the moment of inertia of the third.

Example

Suppose we need to calculate the moment of inertia of a body in an industrial application, but the body is irregularly shaped. We may apply the parallel axis theorem to get the moment of inertia at any position as long as we know the body’s centre of gravity. This is a significant theorem in space physics. It calculates the moment of inertia of spacecraft and satellites, allowing us to reach the outer planets and even deep space. The perpendicular axis theorem is helpful when we do not have access to one axis of a body and need to determine the moment of inertia.

Proving Perpendicular Axis Theorem

According to the perpendicular axis theorem, the moment of inertia for any axis perpendicular to the plane is equal to the sum of any two perpendicular axes of the body that intersect with the first axis.

Consider the case of a particle with mass ‘m’ at position K.

Draw perpendicular to the x and y axes, respectively, from K.

my2 is the moment of inertia about the x-axis.

The moment of inertia about the x-axis of the whole lamina is given by

Ix = ∑my2−−−−−(i)

The moment of inertia about the y-axis of the whole lamina is given by

Iy = ∑mx2−−−−−(ii)

The moment of inertia about the z-axis of the whole lamina is given by

Iz = ∑mr2

But r2 = x2 + y2

Therefore,

Iz = ∑m(x2 + y2)

From equation (i) and (ii), we derive:

i.e., Iz = ∑mx2 + ∑my2

      (or)

Iz = Ix + Iy

The perpendicular axis theorem aids in the calculation of a body’s moment of inertia when one crucial axis is impossible to access.

Example

Let us look at an illustration of this theorem:

Let us say we wish to find the moment of inertia of a uniform ring as a function of its diameter.

The moment of inertia of the ring be MR2/2

Here, M is the mass of the ring and R is the radius of the ring.

According to the perpendicular axis theorem,

Iz = Ix + Iy

As it is a uniform ring, hence the diameter is equal.

Therefore,

Ix = Iy

Therefore, Iz = 2Ix

Iz = MR2/2

As a result, a ring’s moment of inertia around its diameter is MR2/2.

Conclusion:

This article explains the perpendicular axis theorem. According to the Perpendicular Axis Theorem, The moment of inertia of the planar object about an axis perpendicular to the plane is equal to the sum of the moments of inertia of two perpendicular axes passing through the same location in the plane. This theorem is applied to symmetric objects, that is, things with two out of three symmetric axes. When the moment of inertia of the other two axes is known, this theorem is used to compute the moment of inertia of the third.