Parallel and Perpendicular Axis Theorem:

Depending on the shape of the item and the distribution of mass around a specific rotating axis, the mass MOI takes several forms. The MOI is the product of the mass times the distance from the axis squared in the case of a single-point mass. This MOI works with ring forms that are uniformly weighted and have a rotational axis that is perpendicular to the ring. By using the point mass for each distribution, the point mass MOI can also assist in calculating the value for an object having scattered masses. This measurement will not be adequate because most things do not have a single, uniformly distributed mass across the edge. MOI must be calculated differently in objects having scattered mass. The MOI for scattered masses is the total of each point mass. That’s it for today’s article. Hopefully, you find it useful and enjoy it.

Introduction: Parallel and Perpendicular Axes Theorem

Moment of Inertia for a body about a given axis is defined as the sum of the product of masses of the constituent particles and squares of their respective perpendicular distance from the axis of rotation.

I=m r ²

The mass moment of inertia measures how resistant an object is to changes in its rotational rate around an axis. The mass MOI comes from Newton’s first law of motion, which deals with inertia. Without external forces acting on them, objects at rest will oppose being propelled into motion, and objects in motion will resist stopping. The force required to move something in linear motion is equal to the product of its mass and acceleration (F=ma). To convert their motion from standing still to moving, higher masses demand more force.

We are well aware of the fact that what is the moment of inertia about a given axis, but when it comes to finding the moment of inertia of a body about a different or another axis, then we have two theorems that come quite handy. In this article, we will discuss the Perpendicular axis theorem and the parallel axis theorem and also cover the Statement of the parallel and perpendicular axis theorem their formula, and such relatable definitions and examples

Perpendicular axis theorem or Theorem of Perpendicular axis:

Statement of Perpendicular axis theorem: The sum of the moments of inertia around two mutually perpendicular axes in the plane of the lamina such that the three mutually perpendicular axes share a common point of intersection is equal to the moment of inertia of a planar laminar body about an axis perpendicular to the plane.

The perpendicular axis theorem is only applicable to a lamina (2D body).

Perpendicular axis theorem derivation

In the below-given picture, OX and OY are two mutually perpendicular axes lying in the plane of the lamina. OZ is an axis that is perpendicular to the plane of the lamina and passes through point O.

Now let us consider that the lamina is constitute of hue number of particles

• m ₁, m ₂…..= mass of these particles
• r ₁, r ₂….= distances of these particles from the axis OZ
• x ₁, x ₂,….=Perpendicular distance of these particles from the axis OY
• y ₁, y ₂,….=Perpendicular distance of these particles from the axis OX

MOI of the lamina about the axis OX can be written as

I_x=m ₁ y ₁ ²+m ₂ y ₂ ²+……=m y ²

Similarly,

MOI of lamina about OY axis can be written as

I _y=m ₁ x ₁ ²+m ₂ x ₂ ²+……=m x ²

And MOI about the OZ axis can be written as

I _z=m ₁ z ₁ ²+m ₂ z ₂ ²+……=m z ²

Therefore, I_x+ I _y= m y ²+m x ²

m(x ²+y ²)=m r ²= I _z (as r ²=x ²+y ²)

Hence, I _z=I_x+I _y

Parallel Axis Theorem or Theorem of Parallel Axis:

Statement of parallel axis theorem: The sum of a body’s moment of inertia along a parallel axis through its center of gravity and the product of the mass of the body and the square of the distance between the two axes equals the moment of inertia about any axis.

The parallel axis theorem is applicable to the lamina as well as the 3 D body.

Parallel axis theorem Derivation:

As mentioned in the figure given below, AB is an axis of rotation that is parallel to an axis CD passing through the center of mass O of the body.

Image source: Self-made using the book as a reference

Let us take M, as the mass of the body, and h, as the distance between the axes AB and CD.

m ₁, m ₂…..= mass of various particles constituting the body

r ₁, r ₂….= distances of these particles from the axis OZ

I _cm= Moment of inertia of the body about CD

I= Moment of inertia of the body about any axis AB

Therefore,

⇒I=∑m ₁(r ₁+h ₂)+m ₂(r ₂+h ₂)+…….

⇒∑m(r+h)²=∑m(r ²+h ²+2 r h)

⇒∑m r ²+∑m h ²+∑m ×2 r h

⇒∑m r ²+h ² ∑m+2 h ∑m r

However, ∑m r ²=I _cm and ∑ m=M

Furthermore, because the body is balanced at its center of mass O, the algebraic total of all the particles’ weight moments around the axis passing through O is zero.

Hence we conclude that,

I=I _cm+M h ²

NCERT Solutions for Class 11 Physics Chapter 7

Parallel Axis Theorem of Rod:

Image source: Self-made using book as a reference

Assuming the whole mass of the rod to be uniformly distributed along its length.

The mass of a rod per unit length =M ⁄ l

Mass of element under consideration = (M ⁄ l)d x=d m (assume)

MOI of the element about YOY’

=(d m) x ²=(M ⁄ l)x ² d x

To find the Moment of Inertia in the entire rod using the parallel axis theorem, we need to integrate the above expression:

I= ^l/2 ∫ _-l/2 (M/l)x ² d x=(M/l) ⌈ x ³ ⁄ 3 ⌉ ^l/2_-l/2

I _cm=M l ² ⁄12

Let’s say I _AB is the MOI of the rod about the axis AB, passing through one of its perpendicular ends to its length which means parallel to the axis YOY’

I _AB=I _cm+ M (l/2)²

Now applying the parallel axis theorem

⇒M l ² ⁄12+M l ² ⁄4

⇒I _AB=M l ² ⁄3

What Is the Importance of Moment of Inertia?

One of the many mass qualities that explain an object’s stability and the forces required to change its motion is the moment of inertia (MOI). Stability is critical in the design and manufacture of air and spacecraft in aerospace engineering. Knowing the MOI of different axes is crucial for determining how well a device can withstand external and internal stresses. Accurate measurement of this value will result in better-produced craft or equipment to suit the unique requirements of numerous projects in the demanding aviation and space sector.

Also read: Moment of Inertia

FAQs: Parallel and Perpendicular Axis Theorem

Why does mass Moment of Inertia matter?

An object's MOI affects how much torque is required to achieve a given angular acceleration. The mass MOI must be known for calculating torque or rotational force. When you multiply the MOI by the required angular acceleration, you'll get the torque value needed to produce that angular acceleration. Greater torque is required to produce a given acceleration with higher MOI values. The designer's ability to precisely estimate these values allows them to size their components correctly while meeting the high-performance requirements of the aerospace and military industries. To meet the mission objectives, the designer can optimize the trade-off between size, weight, and power. In other circumstances, MOI can be used to check that manufacturing and assembly tolerances and process goals are within acceptable limits. In a third scenario, improved performance can be attained by reducing MOI while maintaining strength.

Is the rotational chapter important for JEE exams?

One of the most important topics in mechanics is rotation, which demands a solid understanding of previous topics such as kinematics, NLM, and so on. As a result, make sure you understand the previously studied concepts of work, power, and energy, as well as the ideas of the center of mass and momentum conservation.

Do rotation questions repeat in NEET?

Yes, but there's a catch! Questions repeated in NEET may not be identical to prior questions, but they may be similar in terms of values or data, with the concept of solving the problem being the same. NEET is a nationwide medical entrance exam administered by the National Testing Agency.