Perpendicular & Parallel Axis Theorem: Statement, Formula

Since we know that moment of inertia plays an important role in studying rotational motion of an object. It's not always the case that the body will rotate about its geometric axis. Thus to find moment of inertia about other axes becomes difficult using the conventional integral method. Therefore theorems like perpendicular axis and parallel axis aids to find moment of inertia about different axes. 

Table of contents

• Perpendicular axis theorem
• Parallel axis theorem
• Practice problems
• FAQs

Perpendicular axis theorem

For any plane body, the moment of inertia about any of its axes that are perpendicular to the plane is equal to the sum of the moments of inertia about any two perpendicular axes in the plane of the body which intersect the first axis in the plane.

Let the and axes be chosen in the plane of the body and the z-axis be perpendicular to this

plane. The three axes are mutually perpendicular. The perpendicular axis theorem states the following:

This theorem is applicable only for two-dimensional objects like circular disc, ring, lamina, etc.

However

The point of intersection of three axes need not be at the centre of the body. It can be any point in the plane of the body, which lies on the body or even outside it.

Parallel Axis Theorem

The moment of inertia of a body about an axis parallel to a centroidal axis and separated by perpendicular distance 𝑑 is given by

Or

Where,

is the moment of inertia of the system about an axis AA’.  

is the moment of inertia of the system about an axis passing through the centre of massand parallel to AA’.

is the moment of inertia of body about an axis AA’.  

Parallel axis theorem is applicable for any type of body, while the perpendicular axis theorem is only applicable to planar or 2D bodies. To apply the parallel axis theorem, the axis has to be parallel not skew.

Practice problem

Q. Find the moment of inertia of a uniform ring of mass M and radius R along the tangent parallel to the diameter of the ring.

A.

Here, axis AA’ is not parallel to Icom. The following diagram shows the correct axis of rotation using which we can apply the parallel axis theorem correctly.

By applying the perpendicular axis theorem, we get the following:

Using the parallel axis theorem,

Q. Find the moment of inertia of the two uniform joint rods having mass M, each about an axis passing through point P.

A.

The moment of inertia about the same axis can be added or subtracted. Using this property, the moment of inertia about point P is given by,

Here

Q. A uniform disc of radius R has a round disc of radius  cut, as shown in the figure. The mass of the remaining (shaded) portion of the disc equals M. Find the moment of inertia of such a disc about the axis passing through the geometrical centre of the original disc and perpendicular to the plane of the disc.

A.

Mass per unit area, i.e., σ is given by,

Now, the moment of inertia of the system about the z-axis is the differnce of moment of inertia of the parent disc about the z-axis and the moment of inertia of the extracted disc about its z-axis (for this, we apply parallel axis theorem).

Q. Find the moment of inertia of a uniform square plate of side length and mass about an axis passing through its centre as shown in figure. .

A.

As we know,

Moment of inertia of a square plate about an axis perpendicular to its plane passing through its geometric centre O is

Let us draw an axis CD which is perpendicular to axis AB in the plane of the square plate.

Axis through O is to axis CD and AB and passing through their intersection, hence from perpendicular axis theorem,

where

refers to axis AB and CD.

Since due to symmetry of square plate

FAQs

Q. Necessary condition for the application of Perpendicular axes theorem i.e. , where are the axes of rotation of body is

1. Body must be two dimensional
2. axes must lie in the plane of the body and axis must be perpendicular to the plane of body
3. must be mutually perpendicular axes
4. All of these

A. (d)

Necessary condition for the application of perpendicular axes theorem i.e. 

where \(x,y\text{ and }z\) are the axes of rotation of body are:

1. Body must be two dimensional.

2. and axes must lie in the plane of the body and -axis must be perpendicular to the plane of the body.

3. must be mutually perpendicular axes.

Q. Can we use the perpendicular axis theorem to find the moment of inertia for 3D objects?

A. For objects like cylinders, perpendicular axis theorem could be a valuable tool in finding their moment of inertia. By breaking them up into planar disks and summing the moments of inertia of the composite disks, we can calculate the moment of inertia.

Q. Let be the moment of inertia about the centre of mass of a thick asymmetrical body. Let be the moment of inertia about an axis parallel to . The distance between the two axes is ‘a’ & the mass of the body is ‘m’. Find the relation between & .

a)

b)

c)

d) Parallel axis theorem can’t be used for a thick asymmetrical body

Q. Let be the moment of inertia about the centre of mass. Let be the moment of inertia about an axis parallel to . The distance between the two axes is ‘d’ & the mass of the body is ‘M’. Find the relation between & .

a)

b)

c)

d)

A. Using the parallel axis theorem,

Q. Why do we use the perpendicular and parallel axis theorem?

A. To study rotational motion, one has to be familiar with the concept of moment of inertia. The perpendicular and parallel axis theorem serves as a valuable tool to find this moment of inertia.