Press Next (or Play) to walk through each story one small step at a time. The numbers come last.
1. How far did the merry-go-round turn?
2. How fast does it turn?
Quick check: a wheel turns one whole turn every 2 seconds, steadily. What is ω?
More: Priya's upside-down bike
More: Sam on the spinning stool
3. Faster and faster
4. The ceiling fan
Read the story as text
You flip the switch on a ceiling fan. At first the blades barely move. Each second they spin a bit faster, until they hum along at full speed. Later you switch it off, and the blades slowly coast to a stop.
The fan never goes anywhere. Its centre stays put. Yet it clearly "moves", speeds up and slows down.
How do we describe how far something has turned, how fast it turns, and how fast that rate changes?
Quick check: a fan spins at +8 rad/s and has α = −2 rad/s². What happens?
More: Grandpa's pottery wheel
More: Lena's washing machine
5. Check yourself
Think of your answer first, then tap to see it.
a) Mia rides half a turn. How many degrees is that? How many radians?
Show answer
Half of 360° is 180°. Half of 2π rad is
π rad, about 3.14 rad.
b) A merry-go-round makes one whole turn every 4 seconds, at a steady rate. How much does it turn each second?
Show answer
A quarter turn each second: 90°, or 2π ÷ 4 ≈ 1.57 rad.
So ω ≈ 1.57 rad/s.
c) Mia sits at the edge. Her little brother sits near the middle of the same merry-go-round. Who turns through the bigger angle?
Show answer
Neither: the same angle. The whole merry-go-round turns
together, so every rider turns through the same angle in the same time. (Mia travels a longer path, but that is 5.2.)
Already know this?
Three quick questions. Get all three right first time and you can skip ahead.
One whole turn, in radians, is:
A wheel speeds up steadily from rest to 12 rad/s in 4.0 s. Its angular acceleration is:
Two kids ride the same merry-go-round, one at the edge and one near the middle. What is the same for both?
Angle, angular velocity and angular acceleration: the same three ideas as x, v and a, but for things that spin.
Angles instead of positions
In pictures
Angle θ: how far the merry-go-round has turned. A quarter turn, half a turn, a whole turn.
Angular velocity ω: how much it turns each second (each tick of the stopwatch).
Angular acceleration α: how much faster it turns each second, like Dad pushing and pushing.
The middle never moves. Every rider turns through the same angle together.
Below is the same idea in words, and then with numbers.
For something spinning about a fixed axis, we track one angle, θ (theta). Every point on a rigid object turns through the same angle, so one number describes the whole object.
Angle θ is measured in radians: θ = s / r, the arc length divided by the radius. One full turn is 2π rad (about 6.28 rad) = 360°.
Angular velocity ω (omega) is how fast the angle changes: ω = Δθ / Δt, in rad/s.
Angular acceleration α (alpha) is how fast ω changes: α = Δω / Δt, in rad/s².
Sign rule: we call counterclockwise positive and clockwise negative (state it on every problem). Just like in 1D motion, if ω and α have the same sign the object speeds up; if they have opposite signs it slows down.
When α is constant, the equations are the 1D motion equations with new letters (x → θ, v → ω, a → α):
ω = ω₀ + α t
θ = θ₀ + ω₀ t + ½ α t²
ω² = ω₀² + 2 α (θ − θ₀)
The graphs work the same way too: the slope of θ-t is ω, the slope of ω-t is α, and the area under ω-t is the angular displacement Δθ.
angle θangular velocity ωangular acceleration α
With numbers: the fan
The fan starts from rest. While switched on it has α = +2.0 rad/s² (counterclockwise positive) for 5.0 s. Then it is switched off and has α = −0.50 rad/s² until it stops.
Show all steps as text
Speed after 5.0 s: ω = ω₀ + α t = 0 + (2.0 rad/s²)(5.0 s) = 10 rad/s.ω grows by 2.0 rad/s every second, for 5 seconds.
Angle turned while speeding up: θ = ½ α t² = ½ (2.0)(5.0)² = 25 rad.It starts at rest, so the ω₀ t term is zero.
In turns: 25 rad ÷ 2π rad/turn ≈ 4.0 turns.One turn is 2π rad.
Time to stop after switching off: 0 = 10 + (−0.50) t, so t = 20 s.Final ω is zero when it stops.
Angle while stopping: 0 = 10² + 2(−0.50)Δθ, so Δθ = 100 / 1.0 = 100 rad (about 15.9 turns).We have ω₀, ω and α but not t needed, so we use the ω² equation. Check: average ω × time = 5 rad/s × 20 s = 100 rad.
Spinning wheel
Set the starting angular velocity ω₀ and the angular acceleration α. Watch the red spoke and the three graphs. Counterclockwise is positive. Try ω₀ = +4 rad/s with α = −1 rad/s²: the wheel slows, stops at t = 4 s, then turns back the other way.
Things to notice: θ-t is a curve whenever α is not zero. ω-t is always a straight line here, with slope α. The ω line crossing zero is the moment the wheel reverses.
Worked examples
Record player basic
A record spins at 45 revolutions per minute (rpm). What is its angular velocity in rad/s?
Show solutionShow all steps as text
Turns per second: 45 rev / 60 s = 0.75 rev/s.One minute is 60 s.
Radians per second: 0.75 × 2π = 4.7 rad/s.Each turn is 2π rad.
Bike wheel braking medium
A bike wheel spinning at 12 rad/s slows steadily to 4.0 rad/s in 4.0 s. Find α and the angle it turns.
Show solutionShow all steps as text
α = Δω / Δt = (4.0 − 12) / 4.0 = −2.0 rad/s².Negative because it is slowing while turning in the positive direction.
Average ω = (12 + 4.0) / 2 = 8.0 rad/s.With constant α, the average is halfway between start and end.
Δθ = 8.0 rad/s × 4.0 s = 32 rad (about 5.1 turns).Check with θ = ω₀t + ½αt² = 48 − 16 = 32 rad.
Grinding wheel spin-up medium
A grinding wheel starts from rest and reaches 20 rad/s after turning 30 revolutions. What is its angular acceleration (constant)?
No time is given, so use ω² = ω₀² + 2αΔθ: 20² = 0 + 2α(188.5).This equation has no t in it.
α = 400 / 377 = 1.06 rad/s².Positive: it speeds up in the positive direction.
Reading an ω-t graph AP
A turntable's ω-t graph: ω = +6.0 rad/s from t = 0 to 2.0 s, then a straight line down to ω = −2.0 rad/s at t = 6.0 s. (a) Find α from 2 to 6 s. (b) When does it reverse? (c) Find its angular displacement from 0 to 6 s.
Show solutionShow all steps as text
(a) α = slope = (−2.0 − 6.0) / (6.0 − 2.0) = −2.0 rad/s².α is the slope of ω-t.
(b) It reverses when ω = 0: 6.0 − 2.0 (t − 2.0) = 0, so t = 5.0 s.ω changes sign there.
(c) Areas: 0-2 s rectangle 6.0 × 2.0 = 12 rad; 2-5 s triangle ½ × 3.0 × 6.0 = 9.0 rad; 5-6 s triangle ½ × 1.0 × (−2.0) = −1.0 rad.Area under ω-t is Δθ; area below the axis counts negative.
Δθ = 12 + 9.0 − 1.0 = +20 rad. (Total angle turned, ignoring direction, is 22 rad.)Displacement keeps the signs; "total turned" adds sizes, like distance vs displacement.
Practice
A disk spins counterclockwise at a steady 3.0 rad/s. What is its angular acceleration?
Show answer
Zero. α is the rate of change of ω. ω is steady, so α = 0. (Points on the rim still have centripetal acceleration; that is a different idea, see 5.2.)
A wheel starts from rest with α = 4.0 rad/s². Through what angle does it turn in the first 3.0 s?
Show answer
θ = ½ α t² = ½ (4.0)(3.0)² = 18 rad. 36 rad forgets the ½; 12 rad is ω at 3 s.
The θ-t graph of a wheel is a parabola that opens upward and is flat (zero slope) at t = 0. Which is true?
Show answer
Zero slope at t = 0 means ω₀ = 0. A parabola means constant α; opening upward means α > 0. The slope keeps growing, so it speeds up.
A wheel turns counterclockwise (positive) at 10 rad/s and has α = −2.0 rad/s². What is it doing at t = 6.0 s?
Show answer
ω = 10 + (−2.0)(6.0) = −2.0 rad/s. It stopped at 5.0 s and then started turning the other way (clockwise) because α stayed negative.
A point on the rim and a point halfway to the centre of the same spinning wheel. Which quantity is the same for both?
Show answer
On a rigid object every point turns through the same angle in the same time, so ω (and θ and α) are shared. Speed, distance and centripetal acceleration all grow with radius.
A motor turns at 120 rpm. Its angular velocity is closest to:
Short answer. A turntable starts from rest and speeds up steadily to 3.5 rad/s in 2.0 s, then spins at that rate until t = 6.0 s. Describe the ω-t graph and find the angle turned from 0 to 6.0 s.
Show answer
Graph: a straight line from (0, 0) up to (2.0 s, 3.5 rad/s), then a flat line at 3.5 rad/s to 6.0 s.
Angle = area: triangle ½ × 2.0 × 3.5 = 3.5 rad, plus rectangle 4.0 × 3.5 = 14 rad. Total 17.5 rad (about 2.8 turns).
Short answer. A friend says "α is negative, so the wheel must be slowing down." Give a case where they are wrong and explain.
Show answer
If ω is also negative (spinning clockwise), a negative α makes ω more negative, so the wheel spins clockwise faster and faster. Speeding up or slowing down depends on whether ω and α have the same sign (speeding up) or opposite signs (slowing down), not on the sign of α alone.
AP question types for this topic
Four short free response questions, one of each AP Physics 1 type: Mathematical Routines, Translation Between Representations, Experimental Design and Analysis, and Qualitative/Quantitative Translation. Try each one on paper before you open the worked answer.
1. Mathematical Routines (7 points)
Maya lifts her foot off her pottery wheel while it spins at angular speed ω0. Friction then slows it with a steady angular acceleration of size α until it stops.
Derive expressions for the time t the wheel takes to stop and the angle Δθ it turns while stopping, in terms of ω0 and α.
ω0 = 6.0 rad/s and α = 1.5 rad/s². Calculate t, and Δθ in radians and in revolutions.
Next time the wheel starts twice as fast, with the same α. By what factor does Δθ change? Use your answer to (a).
Show worked answer and scoring
(a) Take the spin direction as positive, so the angular acceleration is −α. ω = ω0 − αt and ω = 0 at the stop, so t = ω0/α. Then ω² = ω0² − 2αΔθ with ω = 0 gives Δθ = ω0²/(2α).
(b) t = 6.0/1.5 = 4.0 s. Δθ = 6.0²/(2 × 1.5) = 36/3.0 = 12 rad. In turns: 12/(2π) = 1.9 rev.
(c) Δθ grows with ω0², so doubling ω0 makes Δθ 4 times larger: 48 rad. In words: it spins twice as fast on average and takes twice as long to stop.
Scoring (7 points): (a) 3 points: 1 point for using ω = 0 at the stop, 1 point for t = ω0/α, 1 point for Δθ = ω0²/(2α). (b) 2 points: 1 point for 4.0 s, 1 point for 12 rad and 1.9 rev with units. (c) 2 points: 1 point for factor 4, 1 point for linking it to ω0² in (a).
2. Translation Between Representations (6 points)
Theo switches on a ceiling fan at t = 0. From rest it speeds up steadily for 3.0 s until ω = 6.0 rad/s, then keeps turning at 6.0 rad/s.
ABCD
In words: Four angle versus time graphs. A is one straight line from the origin. B starts flat, curves upward, then carries on as a straight line. C curves upward the whole time. D curves up steeply, bends over, then goes flat.
Which θ-t graph (A to D) best matches the fan from t = 0 to 5.0 s? Justify your choice using the slope of the graph.
Describe the α-t graph from 0 to 5.0 s, with values.
Find the angle the fan turns from 0 to 5.0 s, and say what feature of the ω-t graph this angle is.
Show worked answer and scoring
(a)B. The slope of θ-t is ω. It starts at 0 (flat at the origin), grows steadily for 3.0 s (curving up), then stays at 6.0 rad/s (straight line). A has constant slope from the start, C never stops curving, D slows down.
(b) α = 6.0/3.0 = 2.0 rad/s² as a flat line from 0 to 3.0 s, then a drop to a flat line at 0 from 3.0 to 5.0 s.
(c) Δθ is the area under the ω-t graph: triangle ½ × 3.0 × 6.0 = 9.0 rad plus rectangle 2.0 × 6.0 = 12 rad, total 21 rad (about 3.3 turns).
Scoring (6 points): (a) 2 points: 1 point for B, 1 point for a slope argument (zero, growing, then constant). (b) 2 points: 1 point for 2.0 rad/s², 1 point for zero after 3.0 s. (c) 2 points: 1 point for using area under ω-t, 1 point for 21 rad.
3. Experimental Design and Analysis (6 points)
Ines wants to measure the constant angular acceleration of a bicycle wheel that a small motor spins up from rest. She sticks a piece of tape on the rim and films the wheel with a phone next to a stopwatch.
Describe how she gets the times from the video, and one way to reduce uncertainty.
What should she plot so the graph is a straight line? What does the slope mean?
Use the data to find α.
Show worked answer and scoring
(a) Start the video before switching on; step frame by frame and note the stopwatch time each time the tape passes the start position (n = 1, 2, 3 ...). Reduce uncertainty: use slow-motion frames, repeat the run 3 times and average each time, and turn n into radians, θ = 2πn.
(b) From rest with constant α, θ = ½αt². Plot θ (rad) on the vertical axis against t² on the horizontal axis: a straight line through the origin with slope = α/2, so α = 2 × slope.
Scoring (6 points): (a) 2 points: 1 point for a valid way to read the times, 1 point for repeats or slow motion. (b) 2 points: 1 point for θ against t², 1 point for slope = α/2. (c) 2 points: 1 point for correct θ and t² values, 1 point for α ≈ 0.80 rad/s².
A merry-go-round starts from rest and speeds up with constant α. Aiko claims: “In the second 2 seconds it turns three times as far as in the first 2 seconds.”
Without equations, explain why it must turn farther in the second 2 s than in the first 2 s.
Derive the ratio of the two angles and say whether Aiko is right.
Connect (a) and (b): use average angular velocity in each interval to explain the number you found.
Show worked answer and scoring
(a) It speeds up the whole time, so during the second 2 s it is always turning faster than at any moment in the first 2 s. Same time, faster turning, so more angle.
(b) θ = ½αt². First 2 s: ½α(2)² = 2α. First 4 s: ½α(4)² = 8α. Second 2 s: 8α − 2α = 6α. Ratio 6α/2α = 3. Aiko is right.
(c) With constant α the average ω is halfway between start and end. First interval: ω goes 0 to 2α, average α. Second: 2α to 4α, average 3α. Same time, three times the average ω, so three times the angle: the “faster turning” of (a) is exactly 3 times faster on average.
Scoring (6 points): (a) 2 points: 1 point for “speeds up the whole time”, 1 point for linking faster turning in equal time to more angle. (b) 2 points: 1 point for both angles from ½αt², 1 point for ratio 3. (c) 2 points: 1 point for the average ω values, 1 point for linking them to the ratio.
Common mistakes
The mistake: putting revolutions or degrees straight into the equations.
Why it is wrong: the rotational equations (and s = rθ later) only work with radians.
How to spot it: a number like "30 rev" or "90°" in your formula. Convert first: 1 rev = 2π rad, 180° = π rad.
The mistake: "negative α means slowing down".
Why it is wrong: slowing down means ω and α point opposite ways. A clockwise wheel with negative α speeds up.
How to spot it: always compare the signs of ω and α together.
The mistake: giving different ω to different points of the same wheel.
Why it is wrong: a rigid object turns as one piece. Only the linear speed v = rω changes with radius.
How to spot it: the question says "same wheel", "same disk" or "rigid".
The mistake: reading the height of an ω-t graph as the angle.
Why it is wrong: the angle turned is the area under ω-t; the height is ω itself.
How to spot it: ask "slope, area or height?" before reading any graph.