--- title: "Unit 5.1 | Rotational Kinematics" description: "Learn rotational kinematics for AP Physics 1: angular velocity and acceleration, rpm conversions, v = rω, rolling, and ω vs. t graphs, with practice." featured_image: "https://nerd-notes.com/wp-content/uploads/2026/10/133280.jpg" url: "https://nerd-notes.com/unit-5-1-rotational-kinematics/" date_modified: "2026-10-31T09:35:19+00:00" --- # Unit 5.1 | Rotational Kinematics ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-5-1-rotational-kinematics/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-5-1-rotational-kinematics/ Every Unit 1 equation has a rotational twin. This lesson shows how to describe spin, link it to the speed of a point, and solve the four rotational kinematics problem types. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Unit 5 Breakdown You are on Lesson 1 of 4 1. [Unit 5.1 | Rotational Kinematics](https://nerd-notes.com/unit-5-1-rotational-kinematics/) 2. [Unit 5.2 | Torque](https://nerd-notes.com/unit-5-2-torque/) 3. [Unit 5.3 | Rotational Equilibrium](https://nerd-notes.com/unit-5-3-rotational-equilibrium/) 4. [Unit 5.4 | Rotational Inertia and Newton’s Second Law](https://nerd-notes.com/unit-5-4-rotational-inertia-and-newtons-second-law/) #### In this lesson:  - Describe rotation with angle, angular velocity and angular acceleration - Use the rotational kinematic equations, the same ones you already know from Unit 1 - Connect rotation to the motion of a point: [katex] s = r\theta [/katex], [katex] v = r\omega [/katex] and [katex] a_t = r\alpha [/katex] - The 4 problem types: spin-ups and slow-downs, points at different radii, rolling and unwinding, and angular motion graphs [Remove Ads](https://nerd-notes.com/pricing/?for=students) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-5-1-rotational-kinematics&utm_content=top) ### Introduction This lesson covers **rotational kinematics**: how to describe spin before asking what causes it. It starts Unit 5, torque and rotation, and ends with the four rotational kinematics problem types on the AP exam. Good news: you already know the math. Every rotational quantity has a linear twin, and every [kinematic equation](https://nerd-notes.com/unit-1-3-mastering-motion-applying-kinematic-equations-in-1-dimension/) from Unit 1 has a rotational version. Swap the letters and the problems solve the same way. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Angle, Angular Velocity and Angular Acceleration \(\theta\) \(r\) \(s = r\theta\) \(v = r\omega\) \(\omega\) \(\theta \text{ in radians}\) \(1 \text{ rev} = 2\pi \text{ rad}\) \(s = r\theta\) \(v = r\omega\) \(a_t = r\alpha\) *A point on a turning wheel. The whole wheel shares one angle \(\theta\) and one \(\omega\). The point at radius \(r\) travels an arc \(s = r\theta\).* - **Angular position** [katex] \theta [/katex]: how far the object has turned, in radians. One full turn is [katex] 2\pi [/katex] rad. - **Angular velocity** [katex] \omega = \frac{\Delta\theta}{\Delta t} [/katex]: how fast it turns, in rad/s. - **Angular acceleration** [katex] \alpha = \frac{\Delta\omega}{\Delta t} [/katex]: how fast [katex] \omega [/katex] changes, in rad/s². [Remove Ads](https://nerd-notes.com/pricing/?for=students) Every point on a rigid object turns through the same angle in the same time. So the whole object has one [katex] \theta [/katex], one [katex] \omega [/katex] and one [katex] \alpha [/katex]. That is what makes rotation simpler than it looks. #### Rotation has a direction Like velocity, angular velocity has a sign. Counterclockwise is positive and clockwise is negative, unless the problem says otherwise. Speeding up means [katex] \omega [/katex] and [katex] \alpha [/katex] have the same sign. Slowing down means opposite signs. Convert first. [katex] 1 \text{ rev} = 2\pi \text{ rad} [/katex], and rpm means revolutions per minute, so [katex] 1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/s} [/katex]. Many wrong answers in this unit come from skipping this step. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### The Rotational Kinematic Equations When the angular acceleration is constant, the Unit 1 equations carry straight over. Replace [katex] x [/katex] with [katex] \theta [/katex], [katex] v [/katex] with [katex] \omega [/katex] and [katex] a [/katex] with [katex] \alpha [/katex]: | **Linear** | **Rotational** | | --- | --- | | [katex] v = v_0 + at [/katex] | [katex] \omega = \omega_0 + \alpha t [/katex] | | [katex] \Delta x = v_0 t + \tfrac{1}{2}at^2 [/katex] | [katex] \Delta\theta = \omega_0 t + \tfrac{1}{2}\alpha t^2 [/katex] | | [katex] v^2 = v_0^2 + 2a\Delta x [/katex] | [katex] \omega^2 = \omega_0^2 + 2\alpha\Delta\theta [/katex] | | [katex] \Delta x = \tfrac{v_0 + v}{2}\,t [/katex] | [katex] \Delta\theta = \tfrac{\omega_0 + \omega}{2}\,t [/katex] | [Remove Ads](https://nerd-notes.com/pricing/?for=students) The strategy is the same as Unit 1: list what you know, then pick the one equation that holds your unknown and skips the variable you do not have. ### Linking Rotation to a Point on the Object The whole object shares [katex] \omega [/katex], but its points do not share a speed. A point far from the axis travels a bigger circle in the same time, so it moves faster. Here is where the link comes from: 1. Arc length is radius times angle, with the angle in radians: [katex] s = r\theta [/katex]. 2. Divide both sides by time: [katex] \frac{\Delta s}{\Delta t} = r\frac{\Delta\theta}{\Delta t} [/katex], so [katex] v = r\omega [/katex]. 3. Do the same with the change in speed: [katex] \frac{\Delta v}{\Delta t} = r\frac{\Delta\omega}{\Delta t} [/katex], so [katex] a_t = r\alpha [/katex]. [Remove Ads](https://nerd-notes.com/pricing/?for=students) [katex] v = r\omega \qquad a_t = r\alpha [/katex] Here [katex] a_t [/katex] is the **tangential** acceleration, the part that changes the point’s speed. The point also has the [centripetal acceleration]({CIRC}) from circular motion, [katex] a_c = \frac{v^2}{r} = \omega^2 r [/katex], which points to the center and only changes the direction. These links only work with angles in radians. Degrees and revolutions break them. Every rotational kinematics question on the AP exam is one of the four types below. | **Problem type** | **The clue** | **The setup** | | --- | --- | --- | | 1. Spin-ups and slow-downs | “rpm”, “revolutions”, “comes to rest” | Convert to rad, then a rotational kinematic equation | | 2. Points at different radii | two points on one disk, “on the rim”, “halfway out” | Same [katex] \omega [/katex] for both, then [katex] v = r\omega [/katex], [katex] a_c = \omega^2 r [/katex] | | 3. Rolling, unwinding and belts | a wheel rolls, a rope unwinds, a belt joins two wheels | The rim and what it touches share one speed: [katex] v = r\omega [/katex] | | 4. Angular motion graphs | a graph of [katex] \theta [/katex], [katex] \omega [/katex] or [katex] \alpha [/katex] against [katex] t [/katex] | Slope and area, exactly like Unit 1 graphs | [Remove Ads](https://nerd-notes.com/pricing/?for=students) Rotational Kinematics Simplified Official course · step-by-step video walkthrough [Watch · 22 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-5-1-rotational-kinematics&utm_content=video-mid#course=94339&module=97778&lesson=99048) ### Problem Type 1: Spin-ups and slow-downs **How to spot it:** Turning rates in rpm or rev/s, “how many revolutions”, or something that “comes to rest” with a constant angular acceleration. **Setup:** Convert to radians and seconds. List [katex] \omega_0 [/katex], [katex] \omega [/katex], [katex] \alpha [/katex], [katex] t [/katex] and [katex] \Delta\theta [/katex], then pick the equation that skips the one you do not know. A ceiling fan spinning at [katex] 180\,\text{rpm} [/katex] is switched off and coasts to a stop in [katex] 12\,\text{s} [/katex] with a constant angular acceleration. Find its angular acceleration and how many revolutions it makes while stopping. Convert: [katex] \omega_0 = 180 \cdot \frac{2\pi}{60} = 6\pi \approx 18.8 \text{ rad/s} [/katex]. Then [katex] \alpha = \frac{0 – 6\pi}{12} = -\frac{\pi}{2} \approx -1.57 \text{ rad/s}^2 [/katex]. The angle is [katex] \Delta\theta = \frac{\omega_0 + \omega}{2}t = \frac{6\pi}{2}(12) = 36\pi \text{ rad} [/katex], and [katex] \frac{36\pi}{2\pi} = 18 [/katex] revolutions. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Plugging rpm or revolutions straight into the equations. Convert to rad/s and rad first. Divide by [katex] 2\pi [/katex] at the end only if the question asks for revolutions. [Open in UBQ](https://nerd-notes.com/ubq/29787/) Question 16 A pulley has an initial angular speed of 12.5 rad/s and a constant angular acceleration of 3.41 rad/s². Through what angle does the pulley turn in… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/29216/) Question 17 A centrifuge accelerates uniformly from rest to 15,000 rpm in 240 s. Through how many revolutions did it turn in this time? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/28834/) Question 18 The tub of a washer goes into its spin-dry cycle, starting from rest and reaching an angular speed of 5.0 rev/s in 8.0 s. At this point, the person… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/122667/) Question 19 A laboratory centrifuge spinning at an initial angular speed ω₀ is powered off and slows down with a constant angular acceleration due to friction… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-5-1-rotational-kinematics&utm_content=mid) ### Problem Type 2: Points at different distances from the axis **How to spot it:** Two points on the same rotating object, or a point “on the rim”, “halfway out”, “at radius r”. **Setup:** Every point shares [katex] \omega [/katex] and [katex] \alpha [/katex]. Only the linear quantities scale with radius: [katex] v = r\omega [/katex], [katex] a_t = r\alpha [/katex], [katex] a_c = \omega^2 r [/katex]. \(v\) \(2v\) \(r\) \(2r\) \(\omega\) \(\text{same } \omega \text{ for A and B}\) \(v = r\omega: \; v_B = 2v_A\) \(a_c = \omega^2 r: \; a_{c,B} = 2a_{c,A}\) *B is twice as far from the axis as A. Same \(\omega\), so B moves twice as fast.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) A merry-go-round turns at [katex] 1.5\,\text{rad/s} [/katex]. Child A sits [katex] 1.0\,\text{m} [/katex] from the center and child B sits [katex] 2.0\,\text{m} [/katex] from the center. Compare their angular speeds, linear speeds and centripetal accelerations. Both turn at [katex] 1.5\,\text{rad/s} [/katex]. [katex] v_A = (1.0)(1.5) = 1.5 \text{ m/s} [/katex] and [katex] v_B = (2.0)(1.5) = 3.0 \text{ m/s} [/katex]. Centripetal: [katex] a_{c,A} = (1.5)^2(1.0) = 2.25 \text{ m/s}^2 [/katex] and [katex] a_{c,B} = 4.5 \text{ m/s}^2 [/katex]. Twice the radius gives twice the speed and twice the centripetal acceleration. If the object is speeding up, a point on it has both accelerations. [katex] a_t = r\alpha [/katex] stays constant when [katex] \alpha [/katex] is constant, but [katex] a_c = \omega^2 r [/katex] grows as [katex] \omega [/katex] grows. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Saying the outer point has a bigger angular velocity. It covers more distance, but the same angle, in the same time. [Open in UBQ](https://nerd-notes.com/ubq/109946/) Question 20 A rigid vinyl record rotates about its center at a constant angular speed ω. Point A is located at a distance r from the center of the record, and… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/111285/) Question 21 A fan with blades of length L rotates at a constant rate about its center. A small speck of dust is located on one of the blades at a distance of… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/109565/) Question 22 A uniform disk is initially at rest and begins to rotate about its center with a constant angular acceleration α. Let aₜ be the magnitude of the… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/122709/) Question 23 A circular grinding wheel rotates about a fixed axis through its center and slows down with constant angular acceleration. At an initial instant t₁… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) Linear vs. Rotational Overview Official course · step-by-step video walkthrough [Watch · 18 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-5-1-rotational-kinematics&utm_content=video-mid2#course=94339&module=97778&lesson=99046) ### Problem Type 3: Rolling, unwinding and belts **How to spot it:** A wheel rolls without slipping, a rope or cable winds onto a spool, or a belt or chain connects two wheels. **Setup:** Whatever touches the rim without slipping moves with the rim: [katex] \Delta x = r\Delta\theta [/katex], [katex] v = r\omega [/katex], [katex] a = r\alpha [/katex]. Two wheels joined by a belt share the rim speed, not [katex] \omega [/katex]. \(v\) \(\omega\) \(v\) \(\omega\) \(\Delta x = r\Delta\theta\) \(v = r\omega \qquad a = r\alpha\) *The string comes off the spool at the rim speed. A rolling wheel moves forward one circumference for each full turn.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) A bucket hangs from a rope wound on a drum of radius [katex] 0.10\,\text{m} [/katex]. The drum turns at [katex] 4.0\,\text{rad/s} [/katex]. How fast does the bucket rise, and how far does it rise in [katex] 5.0\,\text{s} [/katex]? [katex] v = r\omega = (0.10)(4.0) = 0.40 \text{ m/s} [/katex]. In [katex] 5.0\,\text{s} [/katex] the drum turns [katex] \Delta\theta = (4.0)(5.0) = 20 \text{ rad} [/katex], so the rope winds up [katex] \Delta x = r\Delta\theta = (0.10)(20) = 2.0 \text{ m} [/katex]. **AP trap:** With two wheels on one belt, giving them the same [katex] \omega [/katex]. The belt speed is the same at both rims, so [katex] \omega_1 r_1 = \omega_2 r_2 [/katex] and the small wheel spins faster. [Open in UBQ](https://nerd-notes.com/ubq/91138/) Question 24 The downward motion of an elevator is controlled by a cable that unwinds from a cylinder of radius 0.20 m. What is the angular velocity of the… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/39386/) Question 25 A mechanical wheel initially at rest on the floor begins rolling forward with an angular acceleration of 2πrad/s². If the wheel has a radius of 2 m… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/111386/) Question 26 A small motor pulley with a radius of 0.050 m is rotating at 1200 rotations per minute (RPM). The motor pulley is connected by a non-slipping belt to… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Problem Type 4: Angular motion graphs **How to spot it:** A graph of [katex] \theta [/katex], [katex] \omega [/katex] or [katex] \alpha [/katex] against time. **Setup:** Read it like a Unit 1 graph. Slope of [katex] \theta [/katex] vs. [katex] t [/katex] is [katex] \omega [/katex]. Slope of [katex] \omega [/katex] vs. [katex] t [/katex] is [katex] \alpha [/katex]. Area under [katex] \omega [/katex] vs. [katex] t [/katex] is [katex] \Delta\theta [/katex]. Area under [katex] \alpha [/katex] vs. [katex] t [/katex] is [katex] \Delta\omega [/katex]. A wheel speeds up, cruises, then stops nerd-notes.com *Slope = \(\alpha\): \(3\,\text{rad/s}^2\), then \(0\), then \(-4\,\text{rad/s}^2\). Area = \(\Delta\theta\): \(24 + 36 + 18 = 78\,\text{rad}\).* The graph above shows a wheel’s angular velocity. Find its angular acceleration from 0 to [katex] 4\,\text{s} [/katex], and the total angle it turns in [katex] 10\,\text{s} [/katex]. [Remove Ads](https://nerd-notes.com/pricing/?for=students) Slope: [katex] \alpha = \frac{12 – 0}{4} = 3 \text{ rad/s}^2 [/katex]. Area: a triangle [katex] \tfrac{1}{2}(4)(12) = 24 [/katex], a rectangle [katex] (3)(12) = 36 [/katex] and a triangle [katex] \tfrac{1}{2}(3)(12) = 18 [/katex], so [katex] \Delta\theta = 78 \text{ rad} [/katex], about 12.4 revolutions. **AP trap:** Reading the height of the graph instead of its slope. On an [katex] \omega [/katex] vs. [katex] t [/katex] graph, a high flat line means fast and steady, so [katex] \alpha = 0 [/katex]. [Open in UBQ](https://nerd-notes.com/ubq/122697/) Question 27 An industrial flywheel undergoes uniform angular deceleration from an initial angular velocity of 40 rad/s at t = 0 s until it comes to rest at t =… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/28518/) Question 28 A point on the edge of a disk rotates around the center of the disk with an initial angular velocity of 3 rad/s clockwise. The graph shows the… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/133156/) Question 29 A wheel of radius R = 0.50 m rotates about a fixed axle through its center. The angular velocity ω of the wheel as a function of time t is shown in… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/122864/) Question 30 A rigid disc is initially at rest and free to rotate about a fixed axle through its center. A motor exerts a variable torque on the disc, producing… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) 15 Rotational Kinematics Problems Official course · graded, with step-by-step explanations [Practice](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-5-1-rotational-kinematics&utm_content=practice-practice#course=94339&module=97778&lesson=99050) Quiz Lab Rotational Kinematics Spinning, speeding up and slowing down. Connect the rotation to the motion of points on it. 15 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=181) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-5-1-rotational-kinematics&utm_content=end) ### Lesson 5.2 Preview In the [next lesson](https://nerd-notes.com/unit-5-2-torque/) we ask what changes rotation. In Unit 2 a net force changes linear motion. For rotation, that job belongs to **torque**. ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-5-1-rotational-kinematics/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-5-1-rotational-kinematics/