AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/5/5-1)
Unit 5 · Topic 5.1
5.1 Rotational Kinematics
Rotational kinematics describes spinning motion with three quantities: angular displacement θ, angular velocity ω and angular acceleration α. They work exactly like displacement, velocity and acceleration in one dimension, so the same equations and graph rules carry over once you swap in the rotational symbols.
Key terms
- angular displacement
- radian
- angular velocity
- angular acceleration
- rigid system
- rotational kinematic equations
Rigid systems and the axis
Until now you've mostly treated things as objects: single points with mass. A spinning wheel can't be treated that way, because different parts of it move in different directions at the same moment. The top of a wheel moves one way while the bottom moves the other.
So for rotation you use a rigid system: something that keeps its shape while it turns, like a wheel, a door or a rod. Every rotation happens about an axis, the line the system spins around. A door's axis runs through its hinges.
If you only care about where something goes, not how it spins, you can still treat it as an object. A spinning football's flight path depends only on how its center of mass moves, so for the path you can ignore the spin.
Angular position and radians
Angular position θ tells you how far something has turned from a reference line. Angular displacement Δθ is the change in angular position.
Physics measures angles in radians (rad). One radian is the angle where the arc length equals the radius, so θ = s/r. One full turn is 2π rad, which is 360° or 1 revolution. A radian is a ratio of two lengths, so it has no real unit, which is why it sometimes seems to disappear from answers.
You pick one direction, clockwise or counterclockwise, as positive. Counterclockwise is the usual choice, but any choice works if you stick with it. In this course you only describe rotation directions as clockwise or counterclockwise.
| Revolutions | Degrees | Radians |
|---|---|---|
| 1 | 360° | 2π ≈ 6.28 |
| 1/2 | 180° | π ≈ 3.14 |
| 1/4 | 90° | π/2 ≈ 1.57 |
Angular velocity and angular acceleration
Average angular velocity is how fast the angle changes: ω_avg = Δθ/Δt, in rad/s. Its sign shows the direction of spin. Angular speed is just its size.
Average angular acceleration is how fast the angular velocity changes: α_avg = Δω/Δt, in rad/s². If ω and α have the same sign, the spin is speeding up. If they have opposite signs, it's slowing down. A negative α alone doesn't mean slowing down.
Speeds are often given in revolutions per minute (rpm). To convert, multiply by 2π rad per revolution and divide by 60 s per minute. For example, 60 rpm = 2π rad/s ≈ 6.28 rad/s.
The rotational kinematic equations
When α is constant, three equations link the quantities. Each one is the linear equation you already know with x → θ, v → ω and a → α. They're on the equation sheet.
| Linear (constant a) | Rotational (constant α) |
|---|---|
| v = v₀ + at | ω = ω₀ + αt |
| x = x₀ + v₀t + ½at² | θ = θ₀ + ω₀t + ½αt² |
| v² = v₀² + 2a(x − x₀) | ω² = ω₀² + 2α(θ − θ₀) |
Graphs of rotation
The graph rules from 1.3 carry over directly. The slope of a θ–t graph is ω. The slope of an ω–t graph is α. The area between an ω–t graph and the time axis is the angular displacement Δθ, and the area under an α–t graph is Δω.
Area below the time axis counts as negative. If ω changes sign, the object reversed its spin, so the angular displacement is smaller than the total angle it turned through.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Spinning up from rest
A wheel starts from rest and turns counterclockwise with a constant angular acceleration of 2.0 rad/s² for 5.0 s. Find its angular velocity at 5.0 s and the angle it turns through, in radians and in revolutions.
Show the solutionHide the solution
- Step 1: Take counterclockwise as positive. Known: ω₀ = 0, α = +2.0 rad/s², t = 5.0 s.
- Step 2: ω = ω₀ + αt = 0 + (2.0)(5.0) = 10 rad/s.
- Step 3: Δθ = ω₀t + ½αt² = 0 + ½(2.0)(5.0)² = 25 rad.
- Step 4: In revolutions: 25 rad ÷ 2π rad/rev ≈ 4.0 rev.
Answer: ω = 10 rad/s counterclockwise; Δθ = 25 rad ≈ 4.0 revolutions
- Example 2Calculator allowed
Converting rpm (classic trap)
A ceiling fan spinning at 1200 rpm is switched off and slows down uniformly, stopping in 8.0 s. Find its angular acceleration and how many revolutions it makes while stopping.
Show the solutionHide the solution
- Step 1: Convert first. 1200 rev/min × 2π rad/rev ÷ 60 s/min = 40π rad/s ≈ 126 rad/s. Plugging in 1200 directly is the trap.
- Step 2: Take the spin direction as positive. α = (ω − ω₀)/t = (0 − 126)/8.0 ≈ −15.7 rad/s². It's negative because it's opposite to ω, so the fan is slowing.
- Step 3: Angular displacement = average ω × time. With constant α the average is (126 + 0)/2 ≈ 62.8 rad/s, so Δθ ≈ 62.8 × 8.0 ≈ 503 rad.
- Step 4: 503 rad ÷ 2π ≈ 80 rev. Check: the average speed is 600 rpm, and 600 rev/min × (8.0/60) min = 80 rev.
Answer: α ≈ −15.7 rad/s² (opposite the spin); about 80 revolutions
- Example 3Calculator allowed
Reading an ω–t graph
Counterclockwise is positive. A disk's angular velocity changes linearly from +8.0 rad/s at t = 0 to −4.0 rad/s at t = 6.0 s. Find (a) its angular acceleration, (b) when it reverses direction, (c) its angular displacement from 0 to 6.0 s and (d) the total angle it turns through.
Show the solutionHide the solution
- Step 1: (a) α is the slope: (−4.0 − 8.0)/(6.0 − 0) = −2.0 rad/s².
- Step 2: (b) It reverses when ω = 0. From ω = 8.0 − 2.0t, that's t = 4.0 s.
- Step 3: (c) Area from 0 to 4.0 s is a triangle above the axis: ½(4.0)(8.0) = +16 rad. Area from 4.0 to 6.0 s is a triangle below: ½(2.0)(4.0) = 4.0 rad, counted as −4.0 rad. Net Δθ = 16 − 4.0 = +12 rad.
- Step 4: (d) Total angle turned adds the sizes: 16 + 4.0 = 20 rad.
Answer: (a) −2.0 rad/s² (b) t = 4.0 s (c) +12 rad, counterclockwise (d) 20 rad
Common mistakes
- Leaving angles in degrees or revolutions, or speeds in rpm. Convert to radians and rad/s first, especially before linking to linear motion with v = rω.
- Deciding an object is slowing down just because α is negative. Compare the signs of ω and α: opposite signs mean slowing down.
- Using the constant-α equations when α changes. If the torque changes, use graphs or energy instead.
- Treating net angular displacement as the total angle turned when the spin reverses. Add the sizes of the areas for the total angle.
On the exam
- Expect graphs of θ, ω or α against time where you find slopes and areas, or sketch one graph from another, just like the linear graphs in 1.3.
- Factor-of-change questions are common: from ω² = ω₀² + 2αΔθ, doubling the starting angular speed while α stays the same makes the stopping angle four times as large.
- Show your sign convention (for example, counterclockwise positive) at the start of free-response work.
Connected topics
Videos
Check yourself
5 questions on 5.1 Rotational Kinematics. Pick an answer to see if you got it, and why.
A wheel starts from rest and turns with a constant angular acceleration of 2.0 rad/s². Through what angle does it turn during the first 3.0 s?
A ceiling fan spinning at 30 rad/s is switched off and slows down uniformly, stopping 6.0 s later. About how many revolutions does it make while stopping?
A disk rotates about a fixed axle. Counterclockwise is chosen as positive.
A graph of the disk's angular velocity ω as a function of time t is a single straight line. It starts at ω = +4.0 rad/s at t = 0, crosses ω = 0 at t = 2.0 s, and ends at ω = −4.0 rad/s at t = 4.0 s.
Described graph
Which of the following best describes the disk's motion from t = 0 to t = 4.0 s?
What is the disk's net angular displacement from t = 0 to t = 4.0 s?
What is the disk's angular acceleration at t = 2.0 s?
0 of 5 answered