Moment of Inertia

What is Moment of Inertia?

Now we are about to learn the Moment of inertia, which is a vital topic, and the questions mostly asked in physics from this topic cover the concept of mass in rotational motion. 

Moment of inertia is applied for the calculation of the angular momentum.

Let us start with the definition of the Moment of inertia. This can be defined as the amount of resistance to change possessed by the body while encountering the angular acceleration. This is the amount of the multiplication of the mass of each particle with its square of a length from the axis of rotation. 

Let us explain the moment of inertia in simple words. 

It can be defined as an amount that elects the quantity of torque required for a specific angular acceleration in a rotational axis. This is also recognized as the rotational inertia or the angular mass. 

The alteration in the moment of inertia depends on the selected axis. In general, the moment of inertia is identified for a preferred axis of rotation. 

Moment of Inertia formula can be transcribed as the amount of the resistance of a body to do some modification in its rotational motion. 

I = ∑mi ri2 is the Moment of Inertia equation.


Moment of Inertia of Different Shapes

The moment of inertia list is given below with their formulas.

  1. A System of Particles’ Moment of Inertia 

I = ∑ mi ri2 

This is the primary equation of the moment of inertia.

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  1. Rigid Bodies’ Moment of Inertia

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I = ∫ r2 dm 

Here, dm = mass of the element

  1. An Even Rod’s Moment of Inertia about its Perpendicular Bisector 

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Here, the moment of inertia will be

I = ML2/12.

Where L is the length of the rod.

  1. A Round Ring’s Moment of Inertia about its Axis 

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A round ring’s moment of inertia about its axis 

I = MR2.

  1. Moment of Inertia of a Four-sided Dish about a Line Parallel to an Edge and Passing through the Centre

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Here, the moment of inertia can be written as:

I = Ml2/12.

But, there is a condition such as; if the mass of the element is selected parallel to the length of the plate, then the moment of inertia would be, 

I = Mb2/12.

  1. Moment of Inertia of an Unchanging Rounded Dish about its Axis

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Here, the moment of inertia can be written as

I = MR2/2.

  1. Moment of Inertia of thin Sphere-shaped Case or Identical Hollow Globe

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Here, the moment of inertia can be written as

I = 2MR2/5

  1. An unchanging solid sphere’s Moment of Inertia  

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Here, the moment of inertia can be written as

I = 2MR2/5.


Moment of Inertia of Different Objects

Let us take a closer look at the moment of inertia of different bodies as mentioned in the moment of inertia table (moment of inertia chart), which is given below with their respective formulas:

These pictures describe the moment of inertia formulas for different shapes.

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Moment of Inertia Questions

1. If there are two spheres linked with a bar as shown in the image below. Then what will be the moment of inertia of the system? Let us not consider the rod’s mass.

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I = mX rX2 + mY rY2

I = (0.4) × (0)2 + (0.6) × (0.4)2

I = 0 + 0.096

I = 0.096 kg-m2

Moment of inertia of the system is 0.096 kg-m2

2. As given in the figure, 300 grams is the mass of each sphere. These spheres are linked via cable. Here, 60 cm is the length of the cable and 30 cm is the width of the cable. Can you calculate the moment of inertia of the spheres about the axis of rotation by ignoring the cable’s mass?

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Given,

m1 = m2 = m= m= Mass of ball = 300 gram = 0.3 kg

r1 = Distance between ball 1 and the axis of rotation = 30 cm = 0.3 m

r2 = Distance between ball 2 and the axis of rotation = 30 cm = 0.3 m

r3 = Distance between ball 3 and the axis of rotation = 30 cm = 0.3 m

r4 = Distance between ball 4 and the axis of rotation = 30 cm = 0.3 m

MOI (I) = m1 r12 + m2 r2+ m3 r3+ mr42

I = (0.3) × (0.3)2 + (0.3) × (0.3)+ (0.3) × (0.3)2 + (0.3) × (0.3)2

I = 0.027+ 0.027 + 0.027 + 0.027

I = 0.108 kg-m2

Moment of inertia of the spheres about the axis 0.108 kg m2.


Application of Moment of Inertia

  • When you are riding any kind of vehicle, if it unexpectedly moves forward, a backward jerk can be felt by you.

  • The equipment available in your house will move from one position to another when they move, or they will stay in the same spaces. This is due to the effect of resistance to resist change by undergoing motion or remaining in the same state.

FAQ (Frequently Asked Questions)

Q1. Let us solve this MCQ. Select which of the following is necessary for the Moment of Inertia.

a) The body’s shape and size 

b) The body’s mass

c) Circulation of mass about the axis of revolution 

d) All of them are necessary

Ans: The answer is option d: all of them are necessary. All of these aspects govern a body’s moment of inertia.

Q2. Give your opinion about the mass Moment of Inertia.

Ans: The opposition given by the element to revolve because of its mass is recognized as the mass moment of inertia. 

Here are some examples of the mass moment of inertia, such as a ceiling fan, flywheel, and impeller.

Q3. What is the importance of the Moment of Inertia?

Ans: The moment of inertia is significant in nearly all concepts of physics questions that contain mass in rotational motion. The application of the moment of inertia is to evaluate the angular momentum. It also gives us a chance to clarify exactly how the rotational motion alters when the circulation of mass alters through the conservation of angular momentum.

Q4. What is the moment of inertia of the system about AB, if the moment of two spheres are linked by a pole, as displayed in the picture below, the mass of sphere (X) is 800 grams, and the mass of sphere (Y) is 600 grams?

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Ans: 

mX rX2 + mY rY2 = I 

I = (0.8) × (0.2)2 + (0.6) × (0.5)2

I = 0.032 + 0.15

I = 0.182 kgm2

Moment of inertia of the system is 0.182 kgm2.