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Moment of Inertia of a Circle

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Last updated date: 17th Apr 2024
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The Basic Concept of Inertia

First of all, let us discuss the basic concept of moment of inertia, in simple terms. It can be inferred that inertia is related to the mass of a body. When a body starts to move in rotational motion about a constant axis, every element in the body travels in a loop with linear velocity, which signifies, every particle travels with angular acceleration. Yes, the proper definition of the moment of inertia is that a body tends to fight the angular acceleration. Mathematically, it is the sum of the product of the mass of each particle in the body with the square of its length from the axis of rotation.


The moment of Inertia formula can be coined as:

I = Moment of inertia = Σ miri2

Here, m = mass of the body 

r = radius of the circular path

  • The moment of a circle area or the moment of inertia of a circle is frequently governed by applying the given equation:


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The moment of inertia = I = πR4/4

  • Similarly, the moment of inertia of a circle about an axis tangent to the perimeter(circumference) is denoted as:


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The moment of inertia = I = 5πR4/4

  • In the case of a circle, the polar moment of inertia is given as:

The moment of inertia = I = 5πR4/2

  • In the case of a semi-circle the formula is expressed as:

The moment of inertia = I = πR4/8

  • In the case of a quarter circle the expression is given as:

The moment of inertia = I = πR4/16


Moment of Inertia of a Circle about its Diameter

If we consider the diameter of a circle D, then we must also take ‘r’ the radius as D/2. 

The axis passes through the center.

This expression for the moment of inertia of a circle about its diameter can be given as

I = π D4/64.


Moment of Inertia of a Circle Derivation

Do you know how to find the moment of inertia of a circle? To learn about this, we need to understand the derivation of the moment of inertia of a circle, as explained below. It will be easier to understand how to find the moment of inertia of a circle with this derivation.


The moment of inertia formula of a circle, as per the derivation, the circular cross-section will be calculated with the radius and an axis going exactly through the center. This explanation will follow certain steps, such as:

  1. Describe the coordinate system.

  2. Discover the differential area.

  3. Finally, Integrate it.


Step: 1

Let us just begin by remembering the equation for the second-moment area. That equation is expressed below:

A∫ z2 by DZ = Iy 

∫ z2 da = Iy 

It is the component of MOI in the y-axis. 

Presently, let us just describe the coordinates by applying the polar system.

After applying, we acquire;

r sin θ = z 

r cos θ = y


Step: 2

After the completion of step-1, we need to calculate the differential area, which can be achieved by declaring the area of the sector. 

From the image, the area of the sector can be expressed as;

A = ½r2

Now after the differentiation, the expression will be obtained as;

dA = rdrdθ


Step: 3

Now, let us just transcribe the integral for the moment of inertia of a circle. 

The expression will be;

oor(r sin θ)2rdrdθ = Iy

Let us just simplify this integration with proper steps:

oor r3 sin2θ dr dθ = Iy  

osin2θ [or r3dr ] dθ = Iy 

o sin2θ [r4/4]ordθ = Iy  

r4/4osin2θ = Iy 

As we know about the trigonometric identity i.e. 

sin2θ = 1 - cos2θ/2 -------eq(1)

Let us just put the value in the integral. The expression will be:

r4/4 o1 - cos 2θ/2 x dθ = Iy

r4/4 o½ – ½cos(2θ)dθ = Iy 

r4/4

½θ–½sin(2θ)

½θ–½sin(2θ)o = Iy 

r4/4

½(2π)–¼sin(4π)–½(0)+¼sin(0)

½(2π)–¼sin(4π)–½(0)+¼sin(0)o = Iy

r4/4 (π – 0 – 0 + 0) = Iy

πr4/4 = Iy 

In some cases, there is a possibility to calculate the M.O.I of the circle regarding its axis tangent to the perimeter, then we will practice the parallel axis theorem.

According to the theorem;

I + Ad2 = II

Well, we need to consider some of the factors such as,

d = r 

I = πR4/4

A = πR2

Let us just put the value in eq (1):

πR4/4 + πR2/4= II

5πR4/4 = II


Moment of Inertia of a Circle Formula

  1. I = πR4/4 - MOI of a circle

  2. I = πD4/64 - MOI of a circular section about an axis perpendicular to the section

  3. I = 5πR4/4 - MOI of a circle about an axis tangent to the perimeter(circumference) 

  4. I = 5πR4/2 - The polar moment of inertia

  5. I = πR4/8 - The case of a semi-circle

  6. I = πR4/16 - The case of a quarter circle


Law of Inertia-Application in Mechanics.

According to Newton's first  law of motion nobody by itself can change its state of rest or of uniform motion in a straight line. This inertness or   in ability of the body is called inertia. Obviously an external force is required to change the state of rest or of uniform motion in a  straight line of the body.Larger the force required to change the state of the body, greater  is its inertia.  therefore to produce or for destroy a given  even acceleration force is required  is directly proportional to the mass of the body .clearly heavier  the body greater is the force  required to change it state and hence greater is it's inertia.The reverse is also true.mass of the body is therefore a measure of its inertia in translatory motion.the inertia of body depends only upon its mass, the moment of inertia of a body about a given axis depends upon its mass , position and direction of the axis of rotation.It also depend on the shape of the body. It is due to  the reason that  two bodies having the same mass, but different shapes will have different moments of inertia about a given axis of rotation.


Table of Contents:

The following contents are present in the moment of inertia in Physics of Class 11

  1. What is the Law of Inertia? 

  2. Moment of inertia of a rigid body

  3. Moment  of inertia of a straight rod about an Axis through its center

  4. Moment of inertia of a square lamina about its diagonals


Types of Inertia

Following are the three types of inertia:

  1. Inertia of Rest

  2. Inertia of Direction

  3. Inertia of Motion

FAQs on Moment of Inertia of a Circle

1. What is Newton's first law? What is its formula?

Newton’s First Law of Motion

According to Newton's first  law of motion nobody by itself can change its state of rest or of uniform motion in a straight line. This inertness or   in ability of the body is called inertia.


Newton’s First Law Formula

Newton’s first law of motion equation is

F=dp/dt=d(mv)/dt.



Newton’s first law of motion equation

where

V is the velocity of the object,

t is the time,

And F is the force.

2. What would happen if there was no inertia on the earth?

Inertia = mass. There wouldn't be any  Earth in existence if there was no mass and things on  Earth would simply float apart.


According to the law of inertia, everybody continues to be in its state of rest or of uniform motion in a straight line unless it is compelled to change by external force. Inertia is the measure of mass and  If there is no inertia then there will be no mass and things would be stagnant or move apart. The particles will have infinite acceleration.

3. State a few more examples of inertia?

  • Sudden start of the lift.

  • Jerk experienced when the lift operates suddenly.

  • We lean back. It's due to the inertia of motion that the satellite keeps moving in a                  circular motion.

  • Swirl milk even after you have stopped stirring.

  • Falling a coin into the glass.

  • The Tomato sauce bottle keeps the bottom upside down and if you give it an upwards jerk the sauce1. Leaves get detached from the trees when shaken hard.

  • Dust particles come out when the mat is beaten.

  • If a car or any other vehicle starts

Some more examples of inertia can be found in the below link of Vedantu's website.

4. What is the perpendicular axis theorem in inertia?

This theorem is valid only for two-dimensional objects or laminar object which are rotating about an Axis passing through the center of mass .the sum of moments of inertia of a lamina object about two mutually perpendicular axis lying in the plane of lamina is equal to the moment of inertia about an Axis normal to the plane of the lamina and passing through the two perpendicular axis. This equation is known as the perpendicular axis theorem.

5. What points are studied under the moment of inertia of a rigid body?

  • A moment of inertia is not constant for a body.It depends on the axis of rotation.

  • The higher the mass the higher is the moment of inertia

  • Moment of inertia does not change if the mass is shifted parallel to the axis of the rotation

  • If it is rotated with a constant radius about the axis of rotation clockwise or anticlockwise that means the moment of inertia does not depend upon the sense of rotation.