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Physics

Moment of Inertia

The resistance of an object to rotational motion about an axis.

I = i = 1 \sum n m_{i} r_{i}^{2}

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Source & review

Difficulty

Intermediate

Read time

7 min

Prerequisites

Basic Physics

Review statusNeeds review

Source

FormuLab initial formula library

Initial content draft pending verification against authoritative course or textbook sources.

Definition

Understanding the core concept

Moment of inertia is like rotational mass. Just as mass resists linear acceleration, moment of inertia resists rotational acceleration. Objects with a larger moment of inertia are harder to spin up or slow down.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$I$	Moment of Inertia	$k g \cdot m^{2}$
$m_{i}$	Mass of the i-th particle	$k g$
$r_{i}$	Distance of the i-th particle from the axis of rotation	$m$
$n$	Total number of particles	$-$

Real-World Applications

Where this formula is used in practice

Flywheels in Engines

Flywheels store rotational energy in engines to smooth out power delivery.

Figure Skating

Skaters change their moment of inertia by bringing their arms closer to their body to spin faster.

Balancing a Bicycle

The rotating wheels of a bicycle provide stability due to their moment of inertia.

Design of Rotating Machinery

Engineers consider the moment of inertia when designing rotating parts to manage stress and vibration.

Worked Example

Step-by-step calculation with real numbers

Problem

A dumbbell consists of two weights, each with a mass of 5 kg, attached to the ends of a rigid rod 1 meter long. Calculate the moment of inertia when the dumbbell is rotated about its center.

Given

m_{1} = m_{2} = 5 kg

r_{1} = r_{2} = 0.5 m

Solution

The moment of inertia is the sum of the moments of inertia of each weight.

I = m_{1} r_{1}^{2} + m_{2} r_{2}^{2}

I = (5 kg) (0.5 m)^{2} + (5 kg) (0.5 m)^{2}

I = 5 \cdot 0.25 + 5 \cdot 0.25 kg \cdot m^{2}

I = 1.25 + 1.25 kg \cdot m^{2}

I = 2.5 kg \cdot m^{2}

Final Answer

I = 2.5 kg \cdot m^{2}

Formula Guide

Info

DifficultyIntermediate

Read Time7 min

PrerequisitesBasic Physics

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

FormuLab initial formula library

Initial content draft pending verification against authoritative course or textbook sources.

Assumptions

• Use SI units unless the formula or course context states otherwise.
• Check notation, sign conventions, and applicability against your course material.
• Confirm variables and assumptions before using the formula in graded or professional work.