The shape of the lesson
By the end of the lesson, learners can
- Going round, steadily: A point on a spinning disc sweeps out equal angles in equal times.
- Radian, period, and v = rω: One radian is the angle whose arc length equals the radius, so a whole turn is 2π rad.
- Angular speed: ω = 2πf = v / r.
- Centripetal acceleration: a = v² / r = rω².
- Centripetal force: F = mv² / r = mrω².
One picture every idea returns to
For an object moving in a circle, the angle in radians is arc divided by radius, the angular speed is ω = 2π / T = 2πf, and the linear speed is v = rω.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Going round: angle, time, speed.. Circular motion starts with measuring angle in radians and rate of turn as angular speed. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 17 min | Teach and check: Going round, steadily | A point on a spinning disc sweeps out equal angles in equal times. Project tab 1 of the lab (Horizontal circle) and let a student drive it while the class predicts. | Whole class, then pairs |
| 17 to 30 min | Teach and check: Radian, period, and v = rω | One radian is the angle whose arc length equals the radius, so a whole turn is 2π rad. | Frayer Model |
| 30 to 43 min | Check and plenary | The quick-check questions from the student lesson, answered on mini whiteboards one at a time; re-teach the one most get wrong, then a random call on the last. | Individual, then whole class |
| 43 to 45 min | Exit and homework | Set the worksheet below (5 questions, 54 marks) from the topic practice set. Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Radian, period, and v = rω, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Frayer Model
Pairs fill a four-box grid for the key term: definition, characteristics, examples and non-examples. The non-examples box is where the misconceptions surface, so give it the most time.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
No examiner trap is recorded for this lesson. Use the quiz key below to spot the distractor most of the class picks.
Angular quantities: topic worksheet
5 questions, 54 marks. Written in the style of the papers; no past paper question is reproduced. These questions cover the whole topic, so sibling lessons share them.
A horizontal turntable rotates at a constant rate of 45 revolutions per minute. A small coin rests on the turntable a distance 0.12 m from the axis, as shown in Fig. 1.1.
(a) State what is meant by the angular velocity of a rotating object. [1]
(b) For the turntable:
(i) show that its angular velocity is about 4.7 rad s⁻¹, [2]
(ii) calculate the linear speed of the coin, [2]
(iii) calculate the centripetal acceleration of the coin. [2]
(c) The coin has a mass of 8.0 g.
(i) State the origin of the centripetal force acting on the coin. [1]
(ii) Calculate the magnitude of this force. [2]
(iii) The coin is moved to a larger distance from the axis, with the rate of rotation unchanged. Explain why there is a maximum distance beyond which the coin cannot remain on the turntable. [3]
A flywheel rotates at 300 revolutions per minute; its rim is at a radius of 0.40 m.
(a) Define the radian. [1]
(b) Show that 300 revolutions per minute is an angular speed of about 31 rad s⁻¹. [2]
(c) Calculate the frequency and period of the rotation. [2]
(d) Calculate the linear speed of a point on the rim. [2]
(e) Calculate the linear speed of a point at radius 0.20 m. [1]
(f) State and explain how the angular speed of the inner point compares with that of the rim. [2]
A turntable completes one revolution every 1.8 s. Point P lies 12 cm from the centre and point Q lies 24 cm from the centre.
(a) Calculate the angular speed of the turntable. [2]
(b) Calculate the linear speed of P and of Q. [2]
(c) State and explain how the angular speeds of P and Q compare. [2]
(d) Calculate the angle, in radians, turned through in 0.50 s. [2]
(e) Calculate the centripetal acceleration of Q. [2]
A centrifuge spins a 5.0 g sample at a radius of 8.0 cm, rotating at 4000 revolutions per minute. This question links the angular quantities of Q4 and Q5 with the centripetal force ideas of Q6 to Q8.
(a) Calculate the angular speed in rad s⁻¹. [2]
(b) Calculate the linear speed of the sample. [1]
(c) Calculate the centripetal acceleration of the sample. [2]
(d) Express this acceleration as a multiple of g. [2]
(e) Calculate the centripetal force on the sample. [2]
(f) State the direction of this force. [1]
In a fairground "rotor" ride, riders stand against the inside wall of a vertical cylinder of radius 3.0 m. The cylinder spins, the floor drops away, and friction holds the riders up, as shown in Fig. 10.1. The coefficient of friction between clothing and wall is 0.40.
(a) State what provides the centripetal force on a rider. [1]
(b) State the direction of the centripetal force. [1]
(c) Write the condition for a rider not to slip down the wall. [2]
(d) Show that the minimum angular speed is ω = √(g/(μr)). [3]
(e) Calculate the minimum angular speed and the linear speed of a rider at the wall. [3]
(f) State why the rider’s mass does not affect the minimum speed. [1]
Answer key and marking notes
- (a)Rate of change of angular displacement (angle swept out per unit time), measured in rad s⁻¹ ✓
- (b)(i)ω = 2π × (45/60) = 2π × 0.75 = 4.71 rad s⁻¹ ✓✓
- (b)(ii)v = rω = 0.12 × 4.71 = 0.57 m s⁻¹ ✓✓
- (b)(iii)a = rω² = 0.12 × 4.71² = 2.66 m s⁻² (or v²/r) ✓✓
- (c)(i)Friction between the coin and the turntable surface ✓
- (c)(ii)F = ma = 0.0080 × 2.66 = 0.021 N ✓✓
- (c)(iii)The required force F = mω²r grows as r increases; friction has a maximum value; beyond a certain radius the needed force exceeds the maximum friction, so the coin slides outwards (slips) ✓✓✓
- (a)The angle subtended at the centre of a circle by an arc equal in length to the radius ✓
- (b)ω = 300 × 2π / 60 ✓ = 31 rad s⁻¹ ✓
- (c)f = 300/60 = 5.0 Hz ✓; T = 1/f = 0.20 s ✓
- (d)v = rω = 0.40 × 31.4 ✓ = 13 m s⁻¹ ✓
- (e)v = 0.20 × 31.4 = 6.3 m s⁻¹ ✓
- (f)The angular speed is the same (it is one rigid body) ✓; the linear speed is proportional to radius, so the inner point is slower ✓
- (a)ω = 2π/T = 2π/1.8 ✓ = 3.5 rad s⁻¹ ✓
- (b)vₚ = rω = 0.12 × 3.49 = 0.42 m s⁻¹ ✓; vₜ = 0.24 × 3.49 = 0.84 m s⁻¹ ✓
- (c)Equal ✓: both complete a revolution in the same time, so they share the same angular speed ✓
- (d)θ = ωt = 3.49 × 0.50 ✓ = 1.7 rad ✓
- (e)a = rω² = 0.24 × 3.49² ✓ = 2.9 m s⁻² ✓
- (a)ω = 4000 × 2π/60 ✓ = 4.2 × 10² rad s⁻¹ ✓
- (b)v = rω = 0.080 × 418.9 = 34 m s⁻¹ ✓
- (c)a = rω² = 0.080 × 418.9² ✓ = 1.4 × 10⁴ m s⁻² ✓
- (d)a/g = 1.4 × 10⁴ / 9.81 ≈ 1.4 × 10³ g ✓✓
- (e)F = ma = 0.0050 × 1.40 × 10⁴ ✓ = 70 N ✓
- (f)Toward the axis of rotation (the centre) ✓
- (a)The normal (contact) force from the wall ✓
- (b)Horizontally, toward the axis of the cylinder ✓
- (c)The friction must at least support the weight: μN ≥ mg, with N = mrω² ✓✓
- (d)μ(mrω²) ≥ mg ✓; ω² ≥ g/(μr) ✓; so ω_min = √(g/(μr)) ✓
- (e)ω = √(9.81/(0.40 × 3.0)) = √8.18 = 2.9 rad s⁻¹ ✓✓; v = rω = 3.0 × 2.86 = 8.6 m s⁻¹ ✓
- (f)The mass cancels: both the friction needed and the weight are proportional to m ✓
The lesson's own checks as a quiz
The quick-check and exam-style questions from the student lesson, printable with the key.
One radian is the angle subtended at the centre of a circle by an arc whose length equals:
- Athe radius
- Bthe diameter
- Cthe circumference
- Dhalf the radius
A wheel turns at 2.0 revolutions per second. Its angular speed ω = 2πf is:
- Aabout 13 rad s⁻¹
- B2.0 rad s⁻¹
- C6.3 rad s⁻¹
- D0.50 rad s⁻¹
A point 0.15 m from the axis of a disc turning at ω = 8.0 rad s⁻¹ has a linear speed v = rω of:
- A1.2 m s⁻¹
- B8.0 m s⁻¹
- C53 m s⁻¹
- D0.019 m s⁻¹
On a rigid rotating disc, compared with a point near the axis, a point near the rim has:
- Athe same angular speed but a greater linear speed
- Ba greater angular speed but the same linear speed
- Ca greater angular speed and a greater linear speed
- Dthe same angular speed and the same linear speed
Quiz key
By definition, 1 rad is the angle for which the arc length equals the radius, so a full circle (arc = 2πr) is 2π rad.
ω = 2πf = 2π × 2.0 = 12.6 rad s⁻¹, about 13 rad s⁻¹.
v = rω = 0.15 × 8.0 = 1.2 m s⁻¹.
Every point of a rigid body shares one angular speed ω, but v = rω grows with r, so the rim moves faster.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.