M5L1 Rotational Kinematics

"The universe is under no obligation to make sense to you — but it is, remarkably, under an obligation to be consistent."
— Physics principle on symmetry and analogy

Here is the most important idea in this entire module: you already know rotational kinematics. Every equation, every concept, every problem-solving strategy you learned in Module 1 has an exact rotational twin. Angular position mirrors linear position. Angular velocity mirrors linear velocity. Angular acceleration mirrors linear acceleration. The kinematic equations are identical in form — only the symbols change. This lesson is not about learning new physics; it is about recognizing the same physics wearing different clothes. Master the analogy scaffold presented here, and every rotational problem becomes a problem you have already solved.

The Analogy Scaffold: Linear ↔ Rotational

Every rotational quantity is the angular twin of a linear quantity you already know. Return to this table whenever you are stuck.

Linear (from M1)	Symbol	Units
Position	x	m
Velocity	v	m/s
Acceleration	a	m/s²
v = v₀ + at	—	kinematics eq. 1
x = x₀ + v₀t + ½at²	—	kinematics eq. 2
v² = v₀² + 2aΔx	—	kinematics eq. 3

Rotational (this lesson)	Symbol	Units
Angular position	θ	rad
Angular velocity	ω	rad/s
Angular acceleration	α	rad/s²
ω = ω₀ + αt	—	kinematics eq. 1
θ = θ₀ + ω₀t + ½αt²	—	kinematics eq. 2
ω² = ω₀² + 2αΔθ	—	kinematics eq. 3

Bridge equations connecting both worlds: s = rθ • v = rω • a_t = rα

Course Competencies and Learning Objectives

A ★ indicates that this page contains content related to that LO.

CC5.1 Analyze rotational motion using angular kinematic quantities

★ LO5.1.1 Define angular position, angular velocity, and angular acceleration and state their SI units

★ LO5.1.2 Convert between degrees, revolutions, and radians

★ LO5.1.3 Apply the rotational kinematic equations to constant angular acceleration problems

★ LO5.1.4 Use the bridge equations (s = rθ, v = rω, a_t = rα) to connect rotational and linear quantities

★ LO5.1.5 Identify the analogy between linear and rotational kinematic quantities and equations

Required Reading

Click the blue buttons to go to the OpenStax reading assignments. Read in order — each section builds directly on the previous one.

10.1 Rotational Variables 10.2 Constant Angular Acceleration 10.3 Relating Angular and Translational Quantities

Optional Reading

Explore More

Want to go deeper into rotational kinematics? These resources offer different explanations and visualizations that may click differently than the textbook.

Physics Classroom: Rotational Kinematics Khan Academy: Rotational Kinematics PhET: Rotation Simulation

Media

Watch these videos in order. Video 1 builds the conceptual foundation; Video 2 shows the equations in use; Video 3 demonstrates the bridge to linear quantities with real-world examples.

Video 1: Angular Position, Velocity, and Acceleration

What Are Angular Quantities?

This video introduces the three core rotational variables and explains why radians are the natural unit for angular measurement in physics.

What angular position θ means geometrically
Why the radian is defined as arc length / radius (s/r)
Angular velocity ω as the rate of change of θ
Angular acceleration α as the rate of change of ω
Direction conventions: counterclockwise positive by default

Time: 8:00

Video 2: Rotational Kinematic Equations

The Same Equations, New Symbols

Side-by-side comparison of linear and rotational kinematic equations, followed by worked examples using constant angular acceleration.

Deriving ω = ω₀ + αt by direct analogy
Using θ = θ₀ + ω₀t + ½αt² to find angular displacement
Applying ω² = ω₀² + 2αΔθ to find angular velocity
Worked example: a wheel spinning up from rest
Worked example: a grinding wheel slowing to a stop

Time: 10:30

Video 3: Connecting Rotational and Linear Quantities

Bridge Equations: s = rθ, v = rω, a_t = rα

Any point on a rotating rigid body also has linear velocity and acceleration. This video shows how to move between the two descriptions.

Why points at different radii on the same disk have different linear speeds
Calculating tangential velocity v = rω
Tangential acceleration a_t = rα vs centripetal acceleration a_c = rω²
Real-world application: CD/DVD drive — why the disc spins faster when reading the inner track
Gear and pulley systems as practical bridge equation applications

Time: 9:15

Deeper Look: Why Radians and the Right-Hand Rule

Why Radians Are Not Optional

The Hidden Reason Radians Matter

You can compute angles in degrees for everyday geometry, but in physics, radians are the only unit that makes the bridge equations work without a conversion factor.

Consider the arc length equation: s = rθ

In radians: s = r × θ ✔ (no extra factor needed)
In degrees: s = r × (π/180) × θ ✘ (requires correction every time)

The same issue propagates into every bridge equation. Angular velocity in rad/s gives tangential velocity in m/s directly via v = rω. In degrees/s it would not. Radians are not a preference — they are the unit in which the physics is written.

Calculus Insight

This is also why d(sinθ)/dθ = cosθ only when θ is in radians. In degrees, a factor of π/180 appears. Radians are the natural unit because they make calculus clean.

The Right-Hand Rule for Direction

Angular Velocity Is a Vector

Angular velocity ω is not just a scalar magnitude — it is a vector pointing along the axis of rotation. The right-hand rule tells you which direction:

Curl the fingers of your right hand in the direction of rotation.
Your extended thumb points in the direction of the angular velocity vector ω.

Examples:

A wheel rotating counterclockwise (viewed from the front): ω points toward you (out of the page).
A wheel rotating clockwise (viewed from the front): ω points away from you (into the page).
Earth rotating west-to-east: ω points toward the North Pole.

Why It Matters Later

You will need this in M5L3 when angular momentum L = Iω is a vector, and in advanced courses when torque and angular momentum are related by τ = dL/dt. The right-hand rule is not optional at the university level.

Unit Conversions: Degrees, Revolutions, Radians

A Common Source of Errors

All three unit systems appear in real problems. Memorize one conversion chain and derive the rest:

1 revolution = 360° = 2π radians

Derived from that single fact:

1 rad = 360°/(2π) ≈ 57.3°
1 rpm (rev/min) = 2π/60 rad/s ≈ 0.1047 rad/s
1 Hz (rev/s) = 2π rad/s

Angular frequency ω vs frequency f vs period T:

ω = 2πf = 2π/T

These relationships connect rotational kinematics to wave physics and circular motion — you will see the same ω appear in oscillations (M7) and waves (M8).

Practice and Apply — Conceptual

Problem-Solving Steps

Order the Steps for Solving a Rotational Kinematics Problem

Arrange these steps in the correct order. Notice how identical this procedure is to solving linear kinematics problems in Module 1.

Identify the known quantities (θ, ω, α, t) and the unknown
Convert all angles to radians and all speeds to rad/s if needed
Select the rotational kinematic equation that contains your unknown and your knowns
Substitute values with correct signs (CCW positive)
Solve for the unknown algebraically, then substitute numbers
Check units and verify the answer is physically reasonable
If a linear quantity is needed, apply the appropriate bridge equation (s = rθ, v = rω, or a_t = rα)

Linear or Rotational?

Sort These Quantities Into Their Categories

Drag each quantity into the correct column. Some quantities are purely rotational, some are purely linear, and some are bridge quantities that connect both.

Quantities to Sort

Angular velocity ω (rad/s)
Linear speed v (m/s)
Arc length s (m)
Angular acceleration α (rad/s²)
Displacement x (m)
Tangential acceleration a_t = rα
Period T (s)
Angular position θ (rad)
Radius r (m)
Linear acceleration a (m/s²)

Purely Rotational

Purely Linear

Bridge / Both Worlds

Conceptual Checkpoints

Test the Analogy: Can You Predict the Rotational Answer?

Each card states a linear kinematics fact. Before flipping, try to state the rotational equivalent yourself — then check.

LINEAR: An object at rest has v = 0 but can have a ≠ 0. What is the rotational equivalent?

Rotational Equivalent

A non-spinning object has ω = 0 but can have α ≠ 0. A wheel that has just stopped spinning still has angular acceleration if a net torque acts on it — it will start spinning again.

LINEAR: All points on a rigid rod moving linearly have the same velocity. Is the rotational equivalent true?

Important Difference!

No — this is where the analogy breaks down. All points on a rigid rotating body share the same angular velocity ω, but their linear speeds differ: v = rω. Points farther from the axis move faster. This is why the tip of a fan blade moves much faster than the hub.

A car wheel makes 500 revolutions coming to a stop. Is that 500 rad of angular displacement?

Unit Conversion Answer

No. Δθ = 500 rev × 2π rad/rev = 3,142 rad. This is one of the most common errors in rotational kinematics. Always convert revolutions to radians before using the kinematic equations.

A CD spins at 500 rpm. A point 3 cm from the center and a point 6 cm from the center: which has greater angular velocity?

Key Insight

They have the same angular velocity — both 500 rpm = 52.4 rad/s. But their linear speeds differ: v₃ = (0.03)(52.4) = 1.57 m/s, v₆ = (0.06)(52.4) = 3.14 m/s. Same ω, very different v. This is the essence of v = rω.

Practice and Apply — Computational

Quick Reference: Rotational Kinematic Equations

                        Eq. 1: ω = ω₀ + αt

                        (no Δθ)

                        Eq. 2: Δθ = ω₀t + ½αt²

                        (no ω)

                        Eq. 3: ω² = ω₀² + 2αΔθ

                        (no t)

                        Bridge: s = rθ   v = rω   at = rα

                        (θ must be in radians!)

Problem 1 — Unit Conversion Warm-Up

A motor shaft rotates at 1800 rpm. Convert this to (a) rad/s and (b) Hz.

(a) Convert to rad/s:

1800 rev/min × (1 min / 60 s) × (2π rad / 1 rev)

= 1800 × 2π / 60 = 60π ≈ 188.5 rad/s

(b) Convert to Hz:

1800 rev/min × (1 min / 60 s) = 30 Hz

Engineering note: 1800 rpm motors are common in 60 Hz AC electrical systems because 60 Hz × 60 s/min × 1/2 pole-pairs = 1800 rpm. The analogy to wave physics (ω = 2πf) is already at work.

Problem 2 — Constant Angular Acceleration (Spin-Up)

A flywheel starts from rest and reaches an angular velocity of 240 rpm in 8.0 seconds with constant angular acceleration. Find: (a) α in rad/s², (b) the angular displacement in radians, (c) the number of revolutions completed.

Given: ω₀ = 0, ω = 240 rpm = 8π rad/s, t = 8.0 s, α = constant

(a) Angular acceleration:

Using ω = ω₀ + αt:

8π = 0 + α(8.0)

α = π ≈ 3.14 rad/s²

(b) Angular displacement:

Using Δθ = ω₀t + ½αt²:

Δθ = 0 + ½(π)(8.0)² = 32π ≈ 100.5 rad

(c) Number of revolutions:

32π rad × (1 rev / 2π rad) = 16 revolutions

Problem 3 — Spin-Down (Braking)

A grinder wheel rotating at 3600 rpm is switched off. It comes to rest in 40.0 seconds with constant deceleration. Find: (a) α, (b) total angular displacement, (c) angle traveled in the last 10 seconds.

Given: ω₀ = 3600 rpm = 120π rad/s, ω = 0, t_total = 40.0 s

(a) Angular acceleration:

α = (ω − ω₀)/t = (0 − 120π)/40.0 = −3π ≈ −9.42 rad/s²

(b) Total angular displacement:

Δθ = ω₀t + ½αt² = (120π)(40) + ½(−3π)(40)² = 4800π − 2400π = 2400π ≈ 7540 rad = 1200 rev

(c) Angle in last 10 seconds (t = 30 s to t = 40 s):

ω at t = 30 s: ω = 120π + (−3π)(30) = 30π rad/s

Δθ_last = (30π)(10) + ½(−3π)(10)² = 300π − 150π = 150π ≈ 471 rad = 75 rev

Notice the wheel covers far fewer revolutions in the last 10 seconds than in the first 10 — exactly as a decelerating car covers less distance per second near the stop.

Problem 4 — Bridge Equations: From Rotation to Linear

A bicycle wheel of radius 0.35 m accelerates uniformly from rest to 2.5 rev/s in 4.0 seconds. For a point on the rim, find: (a) the final tangential speed, (b) the tangential acceleration, (c) the total arc length traveled.

Given: r = 0.35 m, ω₀ = 0, ω = 2.5 rev/s = 5π rad/s, t = 4.0 s

Step 1 — Find α:

α = (ω − ω₀)/t = 5π/4.0 = 5π/4 rad/s²

(a) Final tangential speed:

v = rω = (0.35)(5π) = 1.75π ≈ 5.50 m/s

(b) Tangential acceleration:

a_t = rα = (0.35)(5π/4) = 7π/16 ≈ 1.37 m/s²

(c) Arc length traveled:

Δθ = ω₀t + ½αt² = 0 + ½(5π/4)(16) = 10π rad

s = rΔθ = (0.35)(10π) = 3.5π ≈ 11.0 m

Problem 5 — Engineering Application: CD/DVD Drive

A CD-ROM drive reads data at a constant linear velocity (CLV) of 1.2 m/s. The disc has an inner track radius of 25 mm and an outer track radius of 58 mm. Find: (a) the angular velocity when reading the inner track, (b) the angular velocity when reading the outer track, (c) why the disc must change speed as the laser moves outward.

(a) ω at inner track (r = 0.025 m):

v = rω ⇒ ω = v/r = 1.2/0.025 = 48 rad/s ≈ 459 rpm

(b) ω at outer track (r = 0.058 m):

ω = v/r = 1.2/0.058 = 20.7 rad/s ≈ 198 rpm

(c) Why the disc slows down as the laser moves out:

Data is stored at constant linear density (bits per meter of track). To read at a constant bit rate, the linear velocity v must stay constant. Since v = rω, a larger radius r requires a smaller ω. The CD drive motor must continuously decrease its angular velocity as playback progresses from the inner to the outer track.

This is a real engineering constraint in optical storage design — and a perfect demonstration that the same angular velocity does not mean the same linear speed.

Ready to Move On?

Self-Check Before M5L2

Before moving to torque and rotational dynamics, make sure you can answer all of these from memory or by reasoning:

State the three rotational kinematic equations from memory. Which linear equation does each correspond to?
A wheel starts at 100 rpm and decelerates at 2 rad/s². How long before it stops? How many revolutions does it complete?
A point is 0.5 m from the axis of a rotating disk. The disk has α = 4 rad/s². What is the tangential acceleration of the point?
Why do all points on a rigid rotating body share the same α but not the same linear acceleration?
Using the right-hand rule: if a wheel rotates clockwise when viewed from the front, which direction does ω point?

If any of these give you trouble, revisit the Analogy Scaffold table at the top of this lesson and the computational problems above. M5L2 (Torque and Rotational Dynamics) builds directly on every concept here.