Short on time? This page is the recap. For explanations, the diagram and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Every point on a rotating rigid body moves on a circle centred on the axis.
- All points turn through the same angle, so they share Δθ, ω and α.
- A point at distance r from the axis travels farther, faster and with more tangential acceleration the larger r is.
- A point’s velocity is tangent to its circle. Its direction is described as clockwise or counterclockwise about the axis.
- The tangential acceleration changes the point’s speed. The centripetal acceleration (Topic 2.9) changes its direction.
Key relationships
| Relationship | Units | Use it to |
|---|---|---|
| Δs = rΔθ | m (θ in rad) | turn an angle into an arc length, or a string length into an angle |
| v = rω | m/s (ω in rad/s) | find a point’s linear speed, or ω from a belt or string speed |
| a_T = rα | m/s² | find tangential acceleration, or α from a string’s acceleration |
| a_c = v²/r = ω²r | m/s² | centripetal acceleration of a point on the body |
| r₁ω₁ = r₂ω₂ | pulleys or gears linked without slipping | |
| 1 rev = 2π rad; arc for one turn = 2πr | conversions |
Assumptions behind the numbers
- The body is rigid and turns about a fixed axis, so each point’s r stays the same.
- Belts, chains and strings do not slip or stretch.
- A string wound on a drum is thin, so the drum radius does not change as it unwinds.
Mistakes to avoid
- Different ω for points on one body. Only v and a_T change with r.
- Same ω for belt-linked pulleys. They share v, not ω.
- rpm or degrees in v = rω. Convert to rad/s first.
- Calling rα the total acceleration. Add the perpendicular centripetal part ω²r.
- Using a_c = v²/r to claim inner points have more centripetal acceleration. On one body, a_c = ω²r is bigger farther out.
Quick self-check
- A wheel of radius 0.40 m turns through 5.0 rad. How far does a rim point travel? (2.0 m)
- A disc spins at 12 rad/s. Find the speed of a point 0.25 m from the axis. (3.0 m/s)
- A disc has α = 3.0 rad/s². Find a_T for a point 0.50 m from the axis. (1.5 m/s²)
- Point B is three times as far from the axis as point A on the same disc. Find v_B / v_A and ω_B / ω_A. (3 and 1)
Next: practice questions.