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Physics

Conservation of Angular Momentum

In an isolated system, the total angular momentum remains constant.

I_{1} ω_{1} = I_{2} ω_{2}

Study tools

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Source & review

Difficulty

Intermediate

Read time

6 min

Prerequisites

Rotational 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

Imagine a spinning ice skater. When they pull their arms in, they spin faster. This happens because their 'spinning inertia' decreases, and to keep the total 'spin quantity' constant, their spinning speed must increase. The conservation of angular momentum states that without an external torque, this total 'spin quantity' (angular momentum) remains unchanged.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$I_{1}$	Initial moment of inertia	$kg \cdot m^{2}$
$ω_{1}$	Initial angular velocity	$rad/s$
$I_{2}$	Final moment of inertia	$kg \cdot m^{2}$
$ω_{2}$	Final angular velocity	$rad/s$
$ω$	Angular velocity symbol used in angular momentum relationships	$kg \cdot m^{2} / s$

Real-World Applications

Where this formula is used in practice

Figure Skating and Diving

Explains why athletes spin faster or slower by changing their body shape (moment of inertia) to control rotation speed.

Planetary Orbits

Helps explain why planets move faster when they are closer to the sun, as their moment of inertia changes.

Gyroscopes and Stabilizers

Used in gyroscopes for navigation systems and to stabilize spacecraft or bicycles, leveraging their resistance to change in angular momentum.

Helicopter Rotors

Designing helicopter rotors involves considering the conservation of angular momentum to control the aircraft's stability and maneuverability.

Worked Example

Step-by-step calculation with real numbers

Problem

A figure skater has an initial moment of inertia of 4.0 kg·m² and spins at an angular velocity of 1.5 rad/s. She pulls her arms in, reducing her moment of inertia to 1.2 kg·m². What is her new angular velocity?

Given

I_{1} = 4.0 kg \cdot m^{2}

ω_{1} = 1.5 rad/s

I_{2} = 1.2 kg \cdot m^{2}

Solution

I_{1} ω_{1} = I_{2} ω_{2}

ω_{2} = \frac{I _{1} ω _{1}}{I _{2}}

ω_{2} = \frac{( 4.0 kg \cdot m ^{2} ) ( 1.5 rad/s )}{1.2 kg \cdot m ^{2}}

ω_{2} = \frac{6.0 kg \cdot m ^{2} / s}{1.2 kg \cdot m ^{2}}

Final Answer

ω_{2} = 5.0 rad/s

Formula Guide

Info

DifficultyIntermediate

Read Time6 min

PrerequisitesRotational 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.