Short on time? This page is the recap for the calculus-based course (separate from Physics 1, which uses no calculus). For the derivations, a labelled diagram and worked examples, use the full study guide.
Recap
- Each point of a rigid system moves on a circle of fixed radius r about the axis.
- The radian is defined by s = rθ, so θ must be in radians in every formula below.
- Differentiating with r constant gives v = rω (speed along the circle, velocity tangent) and a_T = rα (the component that changes the speed).
- Every rotating point also has a_c = v²/r = ω²r toward the axis, even if ω is constant.
- All points share θ, ω and α. Linear quantities are proportional to r.
- A string, hose or belt that does not slip moves with the rim: v = Rω, a = Rα.
Key relationships
| Relationship | Units | Notes |
|---|---|---|
| s = rθ, Δs = rΔθ | m | θ in rad; distance along the arc |
| v = ds/dt = rω | m/s | tangent to the circle |
| a_T = dv/dt = rα | m/s² | along v if speeding up, against v if slowing |
| a_c = v²/r = ω²r | m/s² | toward the axis (Topic 2.10) |
| |a| = √(a_T² + a_c²) | m/s² | components are perpendicular |
| v_string = Rω, a_string = Rα | m/s, m/s² | no slipping, no stretching; R where it leaves the rim |
Assumptions behind the numbers
- Fixed axis; rigid system, so r is constant for each point.
- Strings and belts do not slip or stretch; layers thin enough that R stays constant unless stated.
- Rotation directions given as clockwise or counterclockwise only.
Mistakes to avoid
- Degrees or rpm in v = rω. Convert to rad/s.
- Bigger ω for outer points. ω is shared; v grows with r.
- Taking a_T as the whole acceleration. Add a_c at right angles.
- Zero acceleration at constant ω. a_c = ω²r is still there.
- Wrong radius for a string. Use the radius where it leaves the rim.
- Centripetal acceleration for the straight string. Only parts on the circle have one.
- Proportional reasoning with a_c = v²/r across points. v also changes with r; use ω²r to see a_c ∝ r at fixed ω.
Quick self-check
- A point 0.50 m from the axis turns through 3.0 rad. How far does it travel? (s = rθ = 1.5 m)
- ω = 12 rad/s, r = 0.25 m. Find v and a_c. (v = 3.0 m/s; a_c = 144 × 0.25 = 36 m/s²)
- A rope on a drum of radius 0.15 m has acceleration 0.60 m/s² and does not slip. Find α. (α = a/R = 4.0 rad/s²)
Next: practice questions.