Circular Motion · 6.1
Circular Motion Foundations
Radians connect distance around a circle to angle, while period and frequency connect one turn to time.
Build the model
Choose the axis, then connect direction and rate.
- Simple definition
- Circular motion is movement along a circular path while staying the same distance from a centre.
- Example
- A point on a bicycle wheel travels around the axle, completing one full turn through 2π radians.
A rotating point can be described by its radius and angular position. One revolution is 2π radians. Angular speed tells how rapidly the angle changes; multiplying by radius converts that angular rate into tangential speed.
Distance travelled along a circle: radius times the angle turned (measured in radians).
θ must be in radians
How fast something spins: angle turned per second, or one full circle (2π) divided by the time for one lap.
rad/s
The actual speed along the circle: spin rate times radius — the outer edge of a wheel moves faster than points near the axle.
Tangent to the path
Radians are natural
An angle of 1 radian cuts off an arc whose length equals the radius.
Period and frequency
Period T is time per revolution; frequency f is revolutions per second, so f = 1/T.
Same ω, different v
Points on one rigid object share angular speed, but points farther from the axis move faster.
Change one variable at a time
See the equation become a graph.
- Angular speed
- 1.26 rad/s
- Tangential speed
- 3.14 m/s
- Frequency
- 0.2 Hz
Catch the common trap
Predict before calculating.
Two points share one rigid turntable. Which point has the larger tangential speed?
Choose an answer, then compare the explanation with your model.
Worked examples
Name the model, substitute, then check direction.
EasyConvert 90° to radians, and state the arc length it subtends on a circle of radius 2.0 m.
- 90° = π/2 ≈ 1.57 rad.
- s = rθ = 2.0 × 1.57 ≈ 3.1 m.
Answerθ ≈ 1.57 rad; s ≈ 3.1 m
MediumA wheel of radius 0.40 m makes 3.0 revolutions each second. Find ω and the rim speed.
- f = 3.0 Hz.
- ω = 2πf = 6π rad/s.
- v = ωr = (6π)(0.40).
Answerω = 18.8 rad/s and v = 7.54 m/s
HardA hard drive platter spins at 7200 rpm. Find ω in rad/s and the speed of a point 4.0 cm from the axis.
- 7200 rpm = 120 rev/s.
- ω = 120 × 2π ≈ 754 rad/s.
- v = ωr = 754 × 0.040 ≈ 30 m/s.
Answerω ≈ 754 rad/s; v ≈ 30 m/s
ChallengingTwo gears mesh: gear A (radius 3.0 cm) drives gear B (radius 9.0 cm). If A turns at 1200 rpm, find B's angular speed and explain which quantity the meshing point shares.
- Meshed teeth share the same rim (tangential) speed: v = ω_A r_A = ω_B r_B.
- ω_B = ω_A (r_A/r_B) = 1200 × (3/9) = 400 rpm.
- Angular speed divides by the radius ratio; the linear speed at contact is common.
Answerω_B = 400 rpm — the contact point's linear speed is shared, not ω