Perpendicular Axis Theorem

What is the Perpendicular Axis Theorem?

The parallel and perpendicular axis theorem deals with the moment of inertia. So, before studying the theorems, let's know about the moment of inertia. It is the property of a body by virtue of which a body resists angular acceleration. 

Angular acceleration is the sum of the product of masses of particles of the body with the square of the distance from the axis of rotation.

Moment of inertia I = ∑miri2

The equation is applicable only in-plane lamina. 

This theorem is used for symmetric objects, i.e. when two out of three axes are symmetric. The moment of inertia of the third axis can be calculated by this equation when the moment of inertia of the other two axes is known.


State and Prove Theorem of Perpendicular Axis

Perpendicular axis theorem statement - The perpendicular axis theorem states that the moment of inertia, for any axis which is perpendicular to the plane, is equal to the sum of any two perpendicular axes of the body which intersects with the first axis. 

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Let us consider a plane lamina made up of a large number of particles in the x-y plane as represented by the figure. Consider a particle having mass 'm' at point P. 

From P, draw PN and PN' perpendicular to the x and y-axis, respectively.

The moment of inertia about the x-axis = my².

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

Ix = ∑my²-----(1)

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

Iy = ∑mx²-------(2)

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

Iz = ∑mr²

But r² = x² + y²

Therefore,

Iz = ∑m (x² + y²)

From eq(1) and (2), we get:

i.e., Iz = ∑mx² + ∑my²

         (or)

Iz = Ix + Iy.

The perpendicular axis theorem helps calculate the moment of inertia of a body where it's difficult to access one vital axis of the body. 


Parallel Axis Theorem and Perpendicular Axis Theorem

The parallel and perpendicular axis theorem is mentioned below:

1. Parallel Axis Theorem

The parallel axis theorem states that the moment of inertia of a body about an axis which is parallel to the axis of the body and passing through its center is equal to the sum of moments of inertia of the body about the axis passing through the center of the product of the mass of the body and the square of the distance between two axes.

Statement

The parallel axis theorem is represented by the following equation:

I = Ic + Mh2

Where,

I = moment of inertia of the body.

Ic = moment of inertia about the center of the body.

M = mass of the body.

h2 = square of the distance between the two axes.

2. Perpendicular Axis Theorem

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Let’s see an example for this theorem:

Let us suppose that we want to calculate the moment of inertia of a uniform ring about its diameter.

Let the center of the ring be MR2/2,

Where,

M = mass of the ring

R = radius of the ring

According to perpendicular axis theorem,

I= Ix + Iy

Since it is a uniform ring; so all its diameter is equal

Therefore, 

I= Iy

∴ Iz = 2 Ix

I= MR²/2

Hence, the moment of inertia of a ring about its diameter is MR2/2.


Parallel Axis Theorem Derivation

Let Ic be the moment of inertia of an object with the axis passing through the center of mass of the object. Let, I be the moment of inertia about the axis A'B' (from figure AB) at a distance h.

Let us consider a particle having mass ‘m’ and located at a distance ‘r’ from the center of gravity of the body.

Distance from A’B’ = r + h

I = ∑m (r + h)2

I = ∑m (r2 + h2 + 2rh)

I = ∑mr2 + ∑mh2 + ∑2rh

I = Ic + h2∑m + 2h∑mr

I = Ic + Mh2 + 0

I = Ic + Mh2


Difference Between Parallel and Perpendicular Axis Theorem

Parallel Axis Theorem

Perpendicular Axis Theorem

The MOI around any axis is equal to the sum of moments of inertia about an axis parallel to the axis passing through the center of mass (COM) of the object and the product of the mass of the object with the square of the perpendicular distance from the axis in consideration and the COM axis parallel to it.

Moment of inertia of an object about an axis perpendicular to it is equal to the sum of moments of inertia of the object about two mutually perpendicular axes lying in the plane of the object.

Io = Ic + md2

Here,

Io = M.O.I of the object about the point O.

Ic = M.O.I of the object about the centroid C.

md2 = added M.O.I due to the distance between O and C.

Izz = Ixx + Iyy

IZZ = MOI of the object about the 3D plane along the z-axis

Ixx = MOI of the object about the 3D plane along the x-axis

Iyy = MOI of the object about the 3D plane along the y-axis

This theorem is applicable for any object, there is no such restriction.

This theorem applies to the planar bodies, which means two-dimensional bodies.

FAQs (Frequently Asked Questions)

Q1. Why is the Perpendicular Axis Theorem not applicable for 3-D objects?

Ans: Perpendicular axis theorem is not applicable for 3D objects in the case of a planar object in the x-y plane. 

However, the perpendicular axis theorem doesn't work for three-dimensional objects because the equation is derived by the assumption that the object is planar.

Q2. What are the applications of the Perpendicular Axis Theorem?

Ans: Some of the most important application of perpendicular axis theorem is outlined here below:

  • With the help of the perpendicular axis theorem, the moment of inertia about a third axis can be calculated.

  • By using the perpendicular axis theorem, the moment of inertia for a three-dimensional object can be calculated.

Q3. The MOI of a thin uniform rod with mass M and length L about an axis perpendicular to the rod, through its center, is I. What is the MOI of a rod on an axis perpendicular to the rod at its endpoint?

a. 4I

b. 2I

c. I/4

d. I/2

Ans: a. Icenter = ML²/12, and

Iendpoint = ML²/3 = 4I

Q4. When do we use a Parallel Axis Theorem?

Ans: It is used to calculate the moment of inertia of the area of a rigid object, whose axis is parallel to the axis of the object, and passes through the center of gravity of the object.