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 velocity and tangential speed 7(b)–(c)
Question 1
A turntable rotates at 120 revolutions per minute. Find its frequency, angular velocity and the speed of a point 0.25 m from the axis.
Check the model response
120 rpm = 2.00 Hz. Hence ω = 2πf = 4π = 12.6 rad s⁻¹ and v = rω = 0.25(4π) = 3.14 m s⁻¹.
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 velocity and tangential speed 7(b)–(c)
Check this idea
Misconception: Every point on a rigid rotating body has the same tangential speed.
Repair: All points share angular velocity, but v = rω means a point farther from the axis has a greater tangential speed.
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 velocity and tangential speed 7(b)–(c)
Model 1
A centrifuge rotates at 1800 rpm. Find its angular velocity and the speed of a sample 0.12 m from the axis.
Check the model response
1800 rpm = 30.0 Hz, so ω = 2πf = 60π = 188 rad s⁻¹. The tangential speed is v = rω = 0.12(60π) = 22.6 m s⁻¹.
guided practice
4. Guided practice
Use each hint only to choose the signed angle, angular relation or inward resultant.
Angular velocity and tangential speed 7(b)–(c)
Question 1
A fan rotates at 3.0 Hz. Find its angular velocity and the speed of a blade tip 0.20 m from the axis.
Hint: Use ω = 2πf before v = rω.
Check the model response
ω = 2π(3.0) = 18.8 rad s⁻¹. Hence v = 0.20(18.8) = 3.77 m s⁻¹.
independent practice
5. Independent practice
Solve without repair notes and show the angle conversion, rotation-rate relation and radial equation.
Angular velocity and tangential speed 7(b)–(c)
Question 1
Two points are 0.10 m and 0.40 m from the axis of the same rigid disc rotating at 15 rad s⁻¹. Compare their angular velocities and tangential speeds.
Check the model response
Both points have angular velocity 15 rad s⁻¹. Their speeds are v₁ = 0.10(15) = 1.5 m s⁻¹ and v₂ = 0.40(15) = 6.0 m s⁻¹.
Practice exit check
6. Practice assessment
Use this as extra closed-book practice, then complete the separate recorded assessment in your plan.
Angular velocity and tangential speed 7(b)–(c)
Question 1
A wheel rotates at 300 rpm. Find its frequency, angular velocity and the speed of a point 0.30 m from the axis.
Check the model response
300 rpm = 5.00 Hz. Thus ω = 2πf = 10π = 31.4 rad s⁻¹ and v = rω = 0.30(10π) = 9.42 m s⁻¹.
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 velocity and tangential speed 7(b)–(c)
Question 1
A laboratory rotor spins at 2400 rpm. Find its angular velocity and the speed of a sample 0.075 m from the axis.
Check the model response
2400 rpm = 40.0 Hz, so ω = 2πf = 80π = 251 rad s⁻¹. The sample speed is v = rω = 0.075(80π) = 18.8 m s⁻¹.