Circular motion: radians, angular velocity and inward force
Key idea: Connect the geometry and timing of circular motion to a vector model in which the resultant acceleration and force point towards the centre.
Before you start: Angular Displacement & VelocityNewton's Laws, Vectors & Problem Solving
By the end, you can
- Express signed angular displacement in radians and use s = rθ.
- Use angular velocity and connect period, frequency and tangential speed through v = rω.
- Explain why uniform circular motion has an inward acceleration although its speed is constant.
- Use a = rω², a = v²/r, F = mrω² and F = mv²/r in radial force models.
Starting-point self-check
1. Check your starting point
Attempt all three groups without notes and mark the first angle conversion, motion relation or radial-force decision you could not justify. Use the recorded topic diagnostic above when you want scoring and a personalised repair plan.
Angular displacement in radians 7(a)
Question 1
A point moves clockwise through an arc length of 1.20 m on a circle of radius 0.80 m. Taking anticlockwise as positive, find its angular displacement.
Check the model response
θ = s/r = 1.20/0.80 = 1.50 rad in magnitude. Clockwise is negative, so the angular displacement is −1.50 rad.
repair
2. Repair the six common breaks
Use only the repair matching an error, then redraw the angle, velocity or free-body model before retrying.
Angular displacement in radians 7(a)
Check this idea
Misconception: An angle in degrees can be substituted directly into s = rθ.
Repair: The direct arc-length relation requires θ in radians. Convert with θ(rad) = θ(°)π/180 before using s = rθ.
Check this idea
Misconception: Angular displacement has no direction because radians are dimensionless.
Repair: Declare a positive sense of rotation. Angular displacement is signed; use rad to communicate the angle measure even though it is dimensionless in SI base units.
worked example
3. Follow three worked models
Follow how each solution fixes the angle unit, converts the rotation rate and identifies the inward resultant before calculating.
Angular displacement in radians 7(a)
Model 1
A wheel of radius 0.35 m turns anticlockwise through 135°. Find the angular displacement and distance travelled by a point on its rim.
Check the model response
θ = 135π/180 = 3π/4 = 2.36 rad. The rim distance is s = rθ = 0.35(3π/4) = 0.825 m.
guided practice
4. Guided practice
Use each hint only to choose the signed angle, angular relation or inward resultant.
Angular displacement in radians 7(a)
Question 1
A point on a wheel of radius 0.40 m makes five-eighths of a revolution clockwise. Taking anticlockwise as positive, find its angular displacement and distance travelled.
Hint: One revolution is 2π rad; distance travelled is a positive path length.
Check the model response
θ = −(5/8)(2π) = −5π/4 = −3.93 rad. The distance travelled is r|θ| = 0.40(5π/4) = 1.57 m.
independent practice
5. Independent practice
Solve without repair notes and show the angle conversion, rotation-rate relation and radial equation.
Angular displacement in radians 7(a)
Question 1
A radius turns anticlockwise from 30° to 250° on a circle of radius 0.45 m. Find its angular displacement and the arc length swept out.
Check the model response
The displacement is 220° = 11π/9 = 3.84 rad. The arc length is s = rθ = 0.45(11π/9) = 1.73 m.
Practice exit check
6. Practice assessment
Use this as extra closed-book practice, then complete the separate recorded assessment in your plan.
Angular displacement in radians 7(a)
Question 1
A drum of radius 0.25 m winds in 1.10 m of cable clockwise. Taking anticlockwise as positive, find its angular displacement.
Check the model response
θ = s/r = 1.10/0.25 = 4.40 rad in magnitude. Clockwise is negative, so θ = −4.40 rad.
Re-test practice
7. Delayed re-test practice
Return after at least three days and solve these fresh contexts without reopening earlier responses. The recorded plan enforces the delay and uses a separate re-test family for selected-response skill-group evidence.
Angular displacement in radians 7(a)
Question 1
A reel turns 3.25 revolutions anticlockwise. Find its angular displacement and the length unwound from a radius of 0.18 m.
Check the model response
θ = 3.25(2π) = 6.5π = 20.4 rad. The length is s = rθ = 0.18(6.5π) = 3.68 m.