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AP Physics 1AP Physics 192 views·Updated May 22, 2026·2 pages

Understanding Moment of Inertia for Common Shapes

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priscilla@cilla

Moment of inertia is a crucial concept in rotational motion,... Show more

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Moment of Inertia Newton's 2nd Law for rotation: ang. acceler.d= $\frac{T_{net}}{I}$ $\underset{inertia}{\leftarrow}$ common shapes obje

Moment of Inertia Basics

When objects rotate, Newton's Second Law takes a special form: angular acceleration = net torque ÷ moment of inertia α=τ/Iα = τ/I. The moment of inertia (I) represents an object's rotational inertia - how much it resists changes in rotational motion.

Different shapes have specific formulas for calculating moment of inertia. For example, a thin rod rotating about its center has I = (1/12)ML², while the same rod rotating about its end has I = (1/3)ML². Notice how the value increases when rotation occurs farther from the center of mass!

Other common shapes include a plane I=(1/12)Ma2aboutcenterI = (1/12)Ma² about center, a cylinder/disk I=(1/2)MR2aboutcenterI = (1/2)MR² about center, and a solid sphere I=(2/5)MR2aboutdiameterI = (2/5)MR² about diameter. The formula always includes both the object's mass and the square of a distance measurement.

Think About It: Why does moment of inertia depend on both mass AND the distribution of that mass? The further mass is from the rotation axis, the harder it is to start or stop rotation!

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of 2
Moment of Inertia Newton's 2nd Law for rotation: ang. acceler.d= $\frac{T_{net}}{I}$ $\underset{inertia}{\leftarrow}$ common shapes obje

Calculating Moment of Inertia

For systems with multiple objects, the total moment of inertia equals the sum of each individual part's moment of inertia. This can be written as I = m₁R₁² + m₂R₂² + ... for objects at different distances from the axis.

When all mass is at the same distance from the rotation axis (like in a cylindrical hoop), the formula simplifies to I = MR², where M is the total mass and R is the radius to all mass points.

Let's see how to apply these formulas with an example: For a solid cylinder with mass 4 kg, radius 0.5 m, and length 0.8 m rotating about its central axis, we use I = (1/2)mR². Plugging in the values: I = (1/2)(4 kg)(0.5 m)² = 0.5 kg·m². This tells us exactly how much the cylinder resists rotational acceleration.

Pro Tip: When solving moment of inertia problems, first identify the shape and rotation axis before selecting the correct formula. The axis location makes a huge difference in the final value!

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AP Physics 1AP Physics 192 views·Updated May 22, 2026·2 pages

Understanding Moment of Inertia for Common Shapes

user profile picture
priscilla@cilla

Moment of inertia is a crucial concept in rotational motion, representing an object's resistance to changes in its rotation. Think of it as the rotational equivalent of mass, determining how easily an object can be accelerated in a spinning motion.

1
of 2
Moment of Inertia Newton's 2nd Law for rotation: ang. acceler.d= $\frac{T_{net}}{I}$ $\underset{inertia}{\leftarrow}$ common shapes obje

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Moment of Inertia Basics

When objects rotate, Newton's Second Law takes a special form: angular acceleration = net torque ÷ moment of inertia α=τ/Iα = τ/I. The moment of inertia (I) represents an object's rotational inertia - how much it resists changes in rotational motion.

Different shapes have specific formulas for calculating moment of inertia. For example, a thin rod rotating about its center has I = (1/12)ML², while the same rod rotating about its end has I = (1/3)ML². Notice how the value increases when rotation occurs farther from the center of mass!

Other common shapes include a plane I=(1/12)Ma2aboutcenterI = (1/12)Ma² about center, a cylinder/disk I=(1/2)MR2aboutcenterI = (1/2)MR² about center, and a solid sphere I=(2/5)MR2aboutdiameterI = (2/5)MR² about diameter. The formula always includes both the object's mass and the square of a distance measurement.

Think About It: Why does moment of inertia depend on both mass AND the distribution of that mass? The further mass is from the rotation axis, the harder it is to start or stop rotation!

2
of 2
Moment of Inertia Newton's 2nd Law for rotation: ang. acceler.d= $\frac{T_{net}}{I}$ $\underset{inertia}{\leftarrow}$ common shapes obje

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

Calculating Moment of Inertia

For systems with multiple objects, the total moment of inertia equals the sum of each individual part's moment of inertia. This can be written as I = m₁R₁² + m₂R₂² + ... for objects at different distances from the axis.

When all mass is at the same distance from the rotation axis (like in a cylindrical hoop), the formula simplifies to I = MR², where M is the total mass and R is the radius to all mass points.

Let's see how to apply these formulas with an example: For a solid cylinder with mass 4 kg, radius 0.5 m, and length 0.8 m rotating about its central axis, we use I = (1/2)mR². Plugging in the values: I = (1/2)(4 kg)(0.5 m)² = 0.5 kg·m². This tells us exactly how much the cylinder resists rotational acceleration.

Pro Tip: When solving moment of inertia problems, first identify the shape and rotation axis before selecting the correct formula. The axis location makes a huge difference in the final value!

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Most popular content in AP Physics 1

9

Most popular content

9

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5App Store
4.7/5Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan SiOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha KlichAndroid user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

AnnaiOS user