🎯⭐ INTERACTIVE LESSON

Rotational Dynamics and Angular Momentum

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Rotational Dynamics and Angular Momentum - Complete Interactive Lesson

Part 1: Angular Momentum (L = Iω)

🌀 Angular Momentum

Part 1 of 7 — $L = I\omega$

Just as linear momentum $p = mv$ describes the "quantity of motion" in a straight line, angular momentum $L = I\omega$ describes the "quantity of rotational motion."

Defining Angular Momentum

For a rigid body rotating about a fixed axis:

$L = I\omega$

Where:

$L$ = angular momentum (kg·m²/s)
$I$ = rotational inertia (kg·m²)
$\omega$ = angular velocity (rad/s)

For a Point Mass

A particle of mass moving in a circle of radius :

Linear-Rotational Analogies

Linear	Rotational
Mass $m$	Rotational inertia $I$
Velocity $v$	Angular velocity $\omega$
Momentum $p = mv$

Angular Momentum Quiz 🎯

Angular Momentum Calculations 🧮

A solid disk ( $M = 4$ kg, $R = 0.5$ m) spins at $\omega = 10$ rad/s. What is its angular momentum? (in kg·m²/s)
A particle of mass 0.5 kg moves at 8 m/s in a circle of radius 2 m. What is its angular momentum? (in kg·m²/s)
A flywheel has kg·m²/s and kg·m². What is its angular velocity? (in rad/s)

Angular Momentum Concepts 🔍

Exit Quiz — Angular Momentum Basics ✅

Part 2: Conservation of Angular Momentum

🔄 Newton's Second Law for Rotation

Part 2 of 7 — $\tau_{\text{net}} = I\alpha$

Newton's Second Law $F = ma$ has a rotational analogue: net torque equals rotational inertia times angular acceleration.

The Rotational Second Law

Part 3: Ice Skater & Spinning Examples

⚡ Rotational Kinetic Energy

Part 3 of 7 — $KE_{\text{rot}} = \frac{1}{2}I\omega^2$

Part 4: Angular Impulse

🔒 Conservation of Angular Momentum

Part 4 of 7 — No External Torque → $L$ is Conserved

Just as linear momentum is conserved when there is no external force, angular momentum is conserved when there is no external torque.

The Conservation Law

If $\tau_{\text{net, ext}} = 0$ , then:

$L_i = L_f$

Part 5: Rotational Kinetic Energy

⭐ Figure Skater & Collapsing Star Examples

Part 5 of 7 — Conservation in Action

The conservation of angular momentum produces some of nature's most dramatic phenomena — from figure skaters spinning faster to neutron stars rotating hundreds of times per second.

The Figure Skater

A figure skater begins a spin with arms extended:

$I_i = 4.0$ kg·m², $\omega_i = 3$ rad/s

Part 6: Problem-Solving Workshop

🛠️ Problem-Solving Workshop

Part 6 of 7 — Angular Momentum Practice

Time to work through challenging problems involving angular momentum, rotational dynamics, and energy.

Problem-Solving Strategy

Identify the system and check for external torques
If no external torque → use conservation of $L$ : $I_i\omega_i = I_f\omega_f$

Part 7: Synthesis & AP Review

🎓 Synthesis & AP Review

Part 7 of 7 — Angular Momentum

Let's synthesize everything about angular momentum and tackle AP-level questions.

Complete Summary

Angular Momentum

$L = I\omega \quad \text{(rigid body)} \qquad L = mvr \quad \text{(point mass)}$

Newton's Second Law (Rotational)

$τ_{net} = I α$

(since $I = mr^2$ and $\omega = v/r$ , so $L = mr^2 \cdot v/r = mvr$ )

Like torque, angular momentum follows a sign convention:

CCW rotation → $L > 0$ (positive)
CW rotation → $L < 0$ (negative)

$[L] = \text{kg·m}^2\text{/s}$

Angular momentum is large when:

The object has a large rotational inertia (lots of mass far from axis)
The object spins fast (large $\omega$ )

A massive, slowly spinning flywheel can have the same $L$ as a tiny, rapidly spinning top.

$\tau_{\text{net}} = I\alpha$

This is the most important equation in rotational dynamics. It tells us:

A net torque causes angular acceleration
More rotational inertia means less angular acceleration for the same torque
The angular acceleration is in the same direction as the net torque

$\tau_{\text{net}} = \frac{\Delta L}{\Delta t}$

Net torque equals the rate of change of angular momentum — the direct analogue of $F = dp/dt$ .

When $\tau_{\text{net}} = 0$ :

$\frac{\Delta L}{\Delta t} = 0 \Rightarrow L = \text{constant}$

No net torque → angular momentum is conserved!

Applying $\tau = I\alpha$

Example 1: Spinning a Wheel

A solid disk ( $M = 5$ kg, $R = 0.4$ m) has a tangential force of $20$ N applied at its rim.

$I = \frac{1}{2}MR^2 = \frac{1}{2}(5)(0.16) = 0.4$ kg·m²

Example 2: Pulley Problem

A mass $m$ hangs from a string wrapped around a pulley (mass $M$ , radius $R$ , solid disk). The tension in the string provides the torque:

$\tau = TR = I\alpha = \frac{1}{2}MR^2 \cdot \frac{a}{R}$

Combined with $mg - T = ma$ , you can solve for both $a$ and $T$ .

Rotational Newton's Second Law Quiz 🎯

Rotational Dynamics Calculations 🧮

A solid cylinder ( $M = 8$ kg, $R = 0.25$ m) has a net torque of 5 N·m applied. What is $\alpha$ ? (in rad/s²)
A wheel ( $I = 2$ kg·m²) starts from rest and a constant torque of 6 N·m is applied for 4 seconds. What is the final angular velocity? (in rad/s)
A disk ( $I = 0.5$ kg·m²) decelerates from 40 rad/s to rest in 8 seconds. What is the magnitude of the braking torque? (in N·m)

Round all answers to 3 significant figures.

Rotational Dynamics Review 🔍

Exit Quiz — Rotational Second Law ✅

A spinning object has kinetic energy due to its rotation — even if its center of mass isn't moving. This rotational kinetic energy follows the same pattern as translational KE.

Rotational Kinetic Energy

$KE_{\text{rot}} = \frac{1}{2}I\omega^2$

Compare with translational: $KE_{\text{trans}} = \frac{1}{2}mv^2$

Linear	Rotational
$\frac{1}{2}mv^2$	$\frac{1}{2}I\omega^2$

Units

$KE_{\text{rot}}$ is measured in joules (J), just like any other form of energy.

Total Kinetic Energy for Rolling Objects

An object that both translates and rotates has:

$KE_{\text{total}} = \frac{1}{2}mv_{\text{cm}}^2 + \frac{1}{2}I\omega^2$

For rolling without slipping ( $v = R\omega$ ):

Shape	$KE_{\text{total}}$
Hoop	$\frac{1}{2}mv^2 + \frac{1}{2}mv^2 = mv^2$

Work-Energy Theorem for Rotation

The work done by a torque:

$W = \tau \cdot \theta$

The work-energy theorem:

$W_{\text{net}} = \Delta KE_{\text{rot}} = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2$

Power

The rotational power (rate of doing work):

$P = \tau\omega$

This is analogous to $P = Fv$ in linear motion.

Rotational KE Quiz 🎯

Rotational KE Calculations 🧮

A wheel ( $I = 4$ kg·m²) spins at 10 rad/s. What is its rotational KE? (in J)
A solid sphere (mass 3 kg, radius 0.1 m) spins at 20 rad/s (not translating). What is its rotational KE? (in J, round to 3 significant figures)
A torque of 8 N·m acts through an angle of 25 rad on a wheel. How much work is done? (in J)

Energy Concepts 🔍

Exit Quiz — Rotational KE ✅

I_i \omega_i = I_f \omega_f

What Counts as "No External Torque"?

External torque is zero when:

No external forces act on the system
External forces act at the axis of rotation ( $r = 0$ )
External forces are parallel to the axis

If $I$ decreases → $\omega$ must increase (and vice versa) to keep $L$ constant.

$\omega_f = \frac{I_i}{I_f} \omega_i$

Important Distinction

Angular momentum is conserved, but rotational kinetic energy is generally NOT conserved when $I$ changes:

$KE_f = \frac{1}{2}I_f\omega_f^2 = \frac{I_i}{I_f} \times \frac{1}{2}I_i\omega_i^2 = \frac{I_i}{I_f} KE_i$

If $I$ decreases, $KE$ increases — the energy comes from internal work (muscles, etc.).

Rotational Collisions

When two rotating objects interact (e.g., a disk drops onto a turntable), angular momentum is conserved:

$I_1\omega_1 + I_2\omega_2 = (I_1 + I_2)\omega_f$

Example

A disk ( $I_1 = 2$ kg·m², $\omega_1 = 10$ rad/s) has a ring ( $_{}$ kg·m², initially at rest) dropped on top:

$2(10) + 3(0) = (2 + 3)\omega_f$ $ω_{f} = 20 / 5 =$

Note: KE is NOT conserved (this is an inelastic rotational "collision").

Conservation Quiz 🎯

Conservation Calculations 🧮

A turntable ( $I = 0.5$ kg·m²) spins at 8 rad/s. A 2 kg block of clay ( $I = mr^2$ , $r = 0.3$ m) is dropped on it. What is the final $\omega$ ? (in rad/s, round to 3 significant figures)
A skater with $I = 4$ kg·m² and $\omega = 6$ rad/s pulls in her arms to $I = 1.5$ kg·m². What is her new $\omega$ ? (in rad/s)
In problem 2, by what factor does her KE increase? (round to 3 significant figures)

Conservation Concepts 🔍

Exit Quiz — Conservation of Angular Momentum ✅

L = I_i\omega_i = 12

kg·m²/s

She pulls her arms in:

$I_f = 1.2$ kg·m²
$\omega_f = L/I_f = 12/1.2 = 10$ rad/s

Speed increase: $\omega_f/\omega_i = 10/3 \approx 3.3\times$ faster!

$KE_i = \frac{1}{2}(4.0)(9) = 18$ J
$KE_f = \frac{1}{2}(1.2)(100) = 60$ J
Energy increase: $60 - 18 = 42$ J

Where does the extra 42 J come from? Internal work by the skater's muscles pulling her arms inward against the centripetal acceleration.

The Collapsing Star

When a massive star runs out of fuel, its core collapses from roughly the size of the Sun ( $R \sim 7 \times 10^8$ m) to a neutron star ( $R \sim 10^4$ m).

Before collapse

$R_i = 7 \times 10^8$ m, rotation period $T_i \approx 30$ days

After collapse

$R_f = 10^4$ m
$I \propto MR^2$ , so

By conservation: $\omega_f = (I_i/I_f)\omega_i$

$\omega_f \approx 5 \times 10^9 \times \omega_i$

The period goes from ~30 days to milliseconds! This explains why pulsars (rotating neutron stars) spin incredibly fast.

Other Examples

Helicopter tail rotor: prevents the body from spinning (reaction to main rotor torque)
Cat righting reflex: cats change their body shape mid-air to reorient
Diver's tuck: pulling into a tuck position reduces $I$ , increasing spin rate

Real-World Angular Momentum Quiz 🎯

Application Calculations 🧮

A diver ( $I = 14$ kg·m² extended) rotates at 2 rad/s. She tucks to $I = 3.5$ kg·m². What is her angular velocity while tucked? (in rad/s)
What is the ratio of her tucked KE to her extended KE?
A merry-go-round ( $I = 800$ kg·m², $\omega = 2$ rad/s) has a 40 kg child ( $r = 2$ m from center) jump off tangentially. What is the new $\omega$ ? (in rad/s, round to 3 significant figures)

Real-World Review 🔍

Exit Quiz — Conservation Examples ✅

If external torque exists → use

\tau = I\alpha

\tau = \Delta L/\Delta t

For energy questions → compute

KE = \frac{1}{2}I\omega^2

before and after

For rolling problems → remember

v = R\omega

and total

KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2

Object drops onto rotating platform (inelastic collision)
Person walks on turntable
Object changes shape while spinning
Atwood machine with massive pulley

Workshop Problems — Set 1 🎯

Workshop Calculations 🧮

A turntable ( $I = 1.2$ kg·m², $\omega = 6$ rad/s) has a ring ( $I = 0.8$ kg·m²) dropped on it from rest. Find the final $\omega$ . (in rad/s, round to 3 significant figures)
How much kinetic energy is lost in the collision above? (in J, round to 3 significant figures)
A 60 kg person stands on the edge of a 200 kg, 3 m radius turntable (uniform disk) initially at rest. The person begins walking at 1.5 m/s tangentially (relative to the ground). What is the turntable's angular velocity? (in rad/s, round to 3 significant figures)

Strategy Check 🔍

Exit Quiz — Workshop ✅

τ_{net} = I α = \frac{Δ L}{Δ t}

Rotational Kinetic Energy

$KE_{\text{rot}} = \frac{1}{2}I\omega^2$

Conservation of Angular Momentum

$\text{If } \tau_{\text{net, ext}} = 0: \quad I_i\omega_i = I_f\omega_f$

$L$ conserved ↔ no external torque
When $I$ decreases → $\omega$ increases → $KE$ increases (internal work done)
Rotational "collisions": $I_1\omega_1 + I_2\omega_2 = (I_1 + I_2)\omega_f$

AP-Style Questions — Set 1 🎯

AP Calculation Practice 🧮

A solid cylinder ( $M = 10$ kg, $R = 0.2$ m) starts from rest and a constant torque of 4 N·m is applied. What is its angular momentum after 5 seconds? (in kg·m²/s)
A hoop (mass 2 kg, radius 0.5 m) rolls without slipping at 3 m/s. What is its total kinetic energy? (in J)
A child ( $m = 30$ kg) runs at 4 m/s tangent to the edge of a stationary merry-go-round (uniform disk, $M = 100$ kg, $R = 2$ m) and jumps on. What is the final angular velocity? (in rad/s, round to 3 significant figures)

Comprehensive Review 🔍

Final AP Review ✅

\tau = FR = (20)(0.4) = 8

N·m

\alpha = \tau/I = 8/0.4 = 20

rad/s²

ω_{f} = 20/5 = 4 rad/s

I_f/I_i = (R_f/R_i)^2 = (10^4/7 \times 10^8)^2 \approx 2 \times 10^{-10}

Rotational Dynamics and Angular Momentum

Direction and Sign

Units

Key Insight

Equivalent Form

When $\tau_{\text{net}} = 0$ :

Applying $\tau = I\alpha$

Example 1: Spinning a Wheel

Example 2: Pulley Problem

Rotational Kinetic Energy

Units

Total Kinetic Energy for Rolling Objects

Work-Energy Theorem for Rotation

Power

What Counts as "No External Torque"?

Key Consequence

Important Distinction

Rotational Collisions

Example

Speed increase: $\omega_f/\omega_i = 10/3 \approx 3.3\times$ faster!

Energy Analysis

The Collapsing Star

Before collapse

After collapse

Other Examples

Common AP Scenarios

Rotational Kinetic Energy

Conservation of Angular Momentum

Key Relationships

Rotational Dynamics and Angular Momentum

Rotational Dynamics and Angular Momentum - Complete Interactive Lesson

Part 1: Angular Momentum (L = Iω)

🌀 Angular Momentum

Defining Angular Momentum

For a Point Mass

Linear-Rotational Analogies

Part 2: Conservation of Angular Momentum

🔄 Newton's Second Law for Rotation

The Rotational Second Law

Part 3: Ice Skater & Spinning Examples

⚡ Rotational Kinetic Energy

Part 4: Angular Impulse

🔒 Conservation of Angular Momentum

The Conservation Law

Part 5: Rotational Kinetic Energy

⭐ Figure Skater & Collapsing Star Examples

The Figure Skater

Part 6: Problem-Solving Workshop

🛠️ Problem-Solving Workshop

Problem-Solving Strategy

Part 7: Synthesis & AP Review

🎓 Synthesis & AP Review

Complete Summary

Angular Momentum

Newton's Second Law (Rotational)

Direction and Sign

Units

Key Insight

Equivalent Form

When τnet=0\tau_{\text{net}} = 0τnet​=0:

Applying τ=Iα\tau = I\alphaτ=Iα

Example 1: Spinning a Wheel

Example 2: Pulley Problem

Rotational Kinetic Energy

Units

Total Kinetic Energy for Rolling Objects

Work-Energy Theorem for Rotation

Power

What Counts as "No External Torque"?

Key Consequence

Important Distinction

Rotational Collisions

Example

Speed increase: ωf/ωi=10/3≈3.3×\omega_f/\omega_i = 10/3 \approx 3.3\timesωf​/ωi​=10/3≈3.3× faster!

Energy Analysis

The Collapsing Star

Before collapse

After collapse

Other Examples

Common AP Scenarios

Rotational Kinetic Energy

Conservation of Angular Momentum

Key Relationships

When $\tau_{\text{net}} = 0$ :

Applying $\tau = I\alpha$

Speed increase: $\omega_f/\omega_i = 10/3 \approx 3.3\times$ faster!