🔄 Angular Quantities

Part 1 of 7 — $\theta$, $\omega$, $\alpha$ and Their Linear Analogues

Everything you learned about linear kinematics has a rotational counterpart. In this part, we introduce the angular quantities that describe rotational motion.

Angular Position $\theta$

The angular position $\theta$ describes how far an object has rotated from a reference direction.

Radians

One complete revolution = $2\pi$ radians = $360°$

$1 \text{ rad} = \frac{360°}{2\pi} \approx 57.3°$

Converting Degrees to Radians

$\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180°}$

Arc Length Connection

The arc length $s$ swept by a point at distance $r$ from the axis:

$s = r\theta$

This only works when $\theta$ is in radians!

Angular Velocity $\omega$

Angular velocity $\omega$ ("omega") is the rate of change of angular position:

$\omega = \frac{\Delta\theta}{\Delta t}$

Sign Convention

• CCW rotation → $\omega > 0$ (positive)
• CW rotation → $\omega < 0$ (negative)

RPM to rad/s

$\omega = \text{RPM} \times \frac{2\pi}{60}$

Period and Frequency

$\omega = 2\pi f = \frac{2\pi}{T}$

where $f$ is frequency (Hz) and $T$ is period (seconds).

Angular Acceleration $\alpha$

Angular acceleration $\alpha$ ("alpha") is the rate of change of angular velocity:

$\alpha = \frac{\Delta\omega}{\Delta t}$

Key Analogies

Speeding Up vs. Slowing Down

• If $\omega$ and $\alpha$ have the same sign → object speeds up (angular speed increases)
• If $\omega$ and $\alpha$ have opposite signs → object slows down (angular speed decreases)

Angular Quantities Quiz 🎯

Angular Quantities Calculations 🧮

1. Convert 90° to radians. Express as a decimal rounded to 2 places.

2. A merry-go-round completes one revolution in 8 seconds. What is its angular velocity? (in rad/s, round to 3 significant figures)

3. A fan blade accelerates from rest to 600 RPM in 10 seconds. What is the angular acceleration? (in rad/s², round to 3 significant figures, use $\pi \approx 3.14$)

Angular Analogues Review 🔍

Exit Quiz — Angular Quantities ✅

📐 Rotational Kinematic Equations

Part 2 of 7 — The Big Four, Rotational Edition

The four kinematic equations you mastered for linear motion have exact rotational counterparts. If you know one set, you know both!

The Rotational Kinematic Equations

For constant angular acceleration $\alpha$:

Same Structure, Different Variables!

Just replace:

• $x \to \theta$
• $v \to \omega$
• $a \to \alpha$

Key Requirement

These equations are valid only when $\alpha$ is constant (uniform angular acceleration).

Choosing the Right Equation

Example

A grinding wheel starts from rest and reaches $50$ rad/s in $10$ s.

• $\omega_0 = 0$, $\omega = 50$ rad/s, $t = 10$ s
• $\alpha = \frac{\omega - \omega_0}{t} = \frac{50}{10} = 5$ rad/s²
• $\theta = \omega_0 t + \frac{1}{2}\alpha t^2 = 0 + \frac{1}{2}(5)(100) = 250$ rad
• In revolutions: $250/(2\pi) \approx 39.8$ revolutions

Rotational Kinematics Quiz 🎯

Rotational Kinematics Calculations 🧮

1. A disk starts from rest and accelerates at $3$ rad/s². How many radians does it rotate in $4$ seconds?

2. A wheel spinning at $40$ rad/s decelerates at $-5$ rad/s². How long (in seconds) until it stops?

3. A turbine accelerates from $10$ rad/s to $50$ rad/s over $8$ seconds. How many revolutions does it make? (Round to 3 significant figures)

Equation Selection 🔍

Exit Quiz — Rotational Kinematics ✅

🔗 Connecting Linear and Angular

Part 3 of 7 — $v = r\omega$ and $a_t = r\alpha$

Points on a rotating object move in circles. Their linear (tangential) quantities are directly connected to the angular quantities through the radius.

Tangential Velocity

The tangential velocity of a point at distance $r$ from the rotation axis:

$v = r\omega$

Key Insights

• All points on a rigid body have the same $\omega$
• Points farther from the axis move faster (larger $v$)
• The direction of $v$ is always tangent to the circular path

Example

A merry-go-round rotates at $\omega = 2$ rad/s. A child sits $3$ m from the center and another at $1.5$ m.

• Child at 3 m: $v = (3)(2) = 6$ m/s
• Child at 1.5 m: $v = (1.5)(2) = 3$ m/s

Both have the same $\omega$ but different tangential speeds!

Tangential Acceleration

The tangential acceleration (rate of change of speed along the circular path):

$a_t = r\alpha$

Centripetal Acceleration

Don't forget — circular motion also has centripetal acceleration directed toward the center:

$a_c = \frac{v^2}{r} = r\omega^2$

Total Acceleration

The total acceleration is the vector sum of tangential and centripetal:

$a_{\text{total}} = \sqrt{a_t^2 + a_c^2}$

These components are always perpendicular, so we use the Pythagorean theorem.

Linear-Angular Connection Quiz 🎯

Linear-Angular Calculations 🧮

1. A wheel of radius 0.3 m rotates at 20 rad/s. What is the tangential speed of a point on the rim? (in m/s)

2. A disk accelerates at $\alpha = 4$ rad/s². What is the tangential acceleration of a point 0.5 m from the center? (in m/s²)

3. A point on a spinning wheel has tangential velocity 12 m/s and is 0.6 m from the center. What is the centripetal acceleration? (in m/s²)

Connection Review 🔍

Exit Quiz — Linear-Angular Relations ✅

🛞 Rolling Without Slipping

Part 4 of 7 — When Rotation Meets Translation

A ball rolling across the floor, a tire on a road, a bowling ball down a lane — these objects both rotate AND translate. When there is no slipping at the contact point, a special condition connects the two motions.

The Rolling Condition

For an object that rolls without slipping:

$v_{\text{cm}} = R\omega$

Where:

• $v_{\text{cm}}$ = velocity of the center of mass
• $R$ = radius of the rolling object
• $\omega$ = angular velocity

What Does "No Slipping" Mean?

The contact point between the rolling object and the surface is instantaneously at rest relative to the surface. Think of a tire on dry pavement — the rubber at the bottom isn't sliding.

Differentiating the Rolling Condition

$a_{\text{cm}} = R\alpha$

This connects the linear acceleration of the center of mass to the angular acceleration.

Distance Traveled

$d = R\theta$

The distance the center moves equals the arc length "unrolled."

Velocity at Different Points

For a rolling object (combining translation + rotation):

Why?

• At the center: pure translational velocity = $v_{\text{cm}}$
• At the top: translation + rotation = $v_{\text{cm}} + R\omega = 2v_{\text{cm}}$
• At the bottom: translation − rotation = $v_{\text{cm}} - R\omega = 0$

This is why the contact point has zero velocity — it's the condition for no slipping!

Rolling Without Slipping Quiz 🎯

Rolling Calculations 🧮

1. A tire of radius 0.4 m rolls without slipping. If the car travels at 20 m/s, what is the tire's angular velocity? (in rad/s)

2. A ball of radius 0.1 m rolls without slipping through 5 complete revolutions. How far does its center travel? (in m, round to 3 significant figures)

3. A cylinder rolls without slipping with $\omega = 15$ rad/s and $R = 0.2$ m. What is the speed of the top of the cylinder? (in m/s)

Rolling Concepts 🔍

Exit Quiz — Rolling Without Slipping ✅

⚙️ Rotational Inertia

Part 5 of 7 — $I = \sum mr^2$ and Common Shapes

In linear motion, mass resists changes in velocity ($F = ma$). In rotational motion, rotational inertia (moment of inertia) resists changes in angular velocity ($\tau = I\alpha$).

Defining Rotational Inertia

For a collection of point masses:

$I = \sum m_i r_i^2$

Where $r_i$ is the distance of each mass from the axis of rotation.

Key Features

• Units: $\text{kg·m}^2$
• $I$ depends on mass AND how that mass is distributed relative to the axis
• Moving mass farther from the axis increases $I$
• $I$ depends on the choice of axis

Example: Two Point Masses

Two 3 kg masses sit on a light rod, one at 0.5 m and one at 1.0 m from the axis.

$I = (3)(0.5)^2 + (3)(1.0)^2 = 0.75 + 3.0 = 3.75 \text{ kg·m}^2$

Rotational Inertia of Common Shapes

Pattern

The more mass is concentrated far from the axis, the larger the rotational inertia.

• Hoop ($MR^2$) > Disk ($\frac{1}{2}MR^2$) > Sphere ($\frac{2}{5}MR^2$)

All have mass $M$ and radius $R$, but the hoop has all mass at the rim.

The Parallel Axis Theorem

$I = I_{\text{cm}} + Md^2$

This lets you find $I$ about any axis that is parallel to one through the center of mass, displaced by distance $d$.

Rotational Inertia Quiz 🎯

Rotational Inertia Calculations 🧮

1. Three masses (2 kg each) are arranged on a light rod at distances 0.1 m, 0.3 m, and 0.5 m from the rotation axis. What is the total rotational inertia? (in kg·m², round to 3 significant figures)

2. A solid disk has mass 4 kg and radius 0.3 m. What is its rotational inertia about its central axis? (in kg·m²)

3. A solid sphere has $I = 0.8$ kg·m² and radius 0.2 m. What is its mass? (in kg)

Rotational Inertia Concepts 🔍

Exit Quiz — Rotational Inertia ✅

🛠️ Problem-Solving Workshop

Part 6 of 7 — Rotational Kinematics Practice

Let's apply all the rotational kinematics tools to solve challenging problems systematically.

Problem-Solving Strategy

1. Identify whether the problem involves pure rotation, pure translation, or both (rolling)
2. List knowns using angular variables ($\theta$, $\omega_0$, $\omega$, $\alpha$, $t$)
3. Choose the right kinematic equation
4. Connect linear and angular if needed: $v = r\omega$, $a_t = r\alpha$
5. Check units and signs

Common Pitfalls

• Mixing degrees and radians
• Forgetting the rolling condition $v = R\omega$
• Confusing $a_t$ (tangential) with $a_c$ (centripetal)
• Using the wrong rotational inertia formula

Practice Problems — Set 1 🎯

Comprehensive Calculations 🧮

1. A CD player accelerates a disc from rest to $50$ rad/s in $2$ seconds. How many revolutions does it make? (Round to 3 significant figures)

2. A solid cylinder (mass 5 kg, radius 0.2 m) rolls without slipping at 4 m/s. What is its angular velocity? (in rad/s)

3. A fan blade decelerates from $\omega_0 = 80$ rad/s to $\omega = 20$ rad/s while making 50 revolutions. What is the angular acceleration? (in rad/s², round to 3 significant figures. Include the negative sign)

Strategy Check 🔍

Challenge Problems 🏆

Exit Quiz — Workshop ✅

🎓 Synthesis & AP Review

Part 7 of 7 — Rotational Kinematics

Let's consolidate everything and practice AP-level questions covering all aspects of rotational kinematics.

Complete Summary

Angular Quantities

$\theta \leftrightarrow x, \quad \omega \leftrightarrow v, \quad \alpha \leftrightarrow a$

Kinematic Equations (constant $\alpha$)

$\omega = \omega_0 + \alpha t$ $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha\theta$

Linear-Angular Connection

$v = r\omega, \quad a_t = r\alpha, \quad a_c = r\omega^2$

Rolling Without Slipping

$v_{\text{cm}} = R\omega, \quad a_{\text{cm}} = R\alpha$

Rotational Inertia

$I = \sum m_i r_i^2$

• Hoop: $MR^2$ | Disk: $\frac{1}{2}MR^2$ | Solid sphere: $\frac{2}{5}MR^2$

AP-Style Questions — Set 1 🎯

AP Calculation Practice 🧮

1. A car accelerates from rest. Its tires (radius 0.3 m) reach $\omega = 80$ rad/s in 12 seconds. What is the car's speed at that time? (in m/s)

2. How far has the car traveled in those 12 seconds? (in m)

3. A solid sphere ($M = 2$ kg, $R = 0.1$ m) rolls without slipping at $v = 5$ m/s. What is its total kinetic energy (translational + rotational)? (in J)

Comprehensive Review 🔍

Final AP Review ✅