Explicit physics instruction
Rotational and linear quantities are connected by s=rθ; v=rω; a_t=rα. Radius is therefore essential when translating an angular description into a linear one.
A point 0.50 m from an axis rotates at 6.0 rad/s. Its linear speed is v=rω=0.50×6.0=3.0 m/s. A point twice as far from the axis on the same rigid body has the same ω but twice the linear speed.
Always check radians before using s=rθ or v=rω.
Core relations: s=rθ; v=rω; a_t=rα.
Worked example
A point 0.50 m from an axis rotates at 6.0 rad/s. Its linear speed is v=rω=0.50×6.0=3.0 m/s. A point twice as far from the axis on the same rigid body has the same ω but twice the linear speed.
Guided question
Two points lie at 0.25 m and 0.75 m on one rigid disk. Compare their linear speeds for the same ω.
Independent work
- Solve three angular-motion calculations.
- Draw or interpret one rotational graph.
- State one model assumption or limitation.
Misconception check
Same angular speed does not mean same linear speed for points at different radii.
Formative check
What quantity must be known, in addition to ω, to determine a point’s linear speed?
Reveal response
Its distance r from the rotation axis.
Mentor evidence conference
Ask Archaemenes to audit radians, graph interpretation, radius dependence, and unit consistency.
Teacher guidance
Foundation: Identify θ, ω, α and use one-step relations.
Core: Connect angular graphs with linear speed.
Extended: Analyze residuals and compare points at multiple radii.