Short on time? This page is the recap. For explanations, diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- A spinning body is a rigid system: it keeps its shape, but its points move in different directions. It cannot be treated as one point while its rotation matters.
- If the rotation does not affect the question, you may still model the body as an object at its centre of mass.
- Describe rotation about a stated axis, from a stated reference line, with clockwise or counterclockwise chosen as positive.
- All points on a rigid body share the same Δθ, ω and α.
- Instantaneous ω (or α) is the average over a very short interval, or the tangent slope on the graph.
Key relationships
| Relationship | Units | Use it to |
|---|---|---|
| Δθ = θ − θ₀ | rad | find angular displacement (signed) |
| ω_avg = Δθ / Δt | rad/s | find average angular velocity |
| α_avg = Δω / Δt | rad/s² | find average angular acceleration |
| ω = ω₀ + αt | constant α, no Δθ | |
| θ = θ₀ + ω₀t + ½αt² | constant α, no final ω | |
| ω² = ω₀² + 2α(θ − θ₀) | constant α, no time | |
| slope of θ–t graph | rad/s | angular velocity |
| slope of ω–t graph; area under it | rad/s²; rad | angular acceleration; angular displacement |
| 1 rev = 2π rad = 360° | converting units |
Assumptions behind the numbers
- The body is rigid and turns about one fixed axis.
- The three equations need constant α. If α changes in stages, treat each stage separately.
- Directions are only clockwise or counterclockwise about the stated axis.
Mistakes to avoid
- No rotation sense stated. A negative ω means nothing without one.
- rpm or degrees in the equations. Convert to rad/s and rad first.
- Negative α = slowing down. Compare the signs of ω and α.
- Different ω for different points on one body. ω is shared; only linear speed changes with radius (Topic 5.2).
- Resetting θ to zero after each turn. Count all the turns.
- Linear scaling for squared quantities. Doubling ω₀ makes the stopping angle four times larger.
Quick self-check
- A fan blade turns 2.5 rev in 0.50 s. What is its average angular velocity? (10π ≈ 31 rad/s)
- Counterclockwise positive. ω changes from +6.0 rad/s to −3.0 rad/s in 1.5 s. Find α_avg. (−6.0 rad/s²)
- Convert 45 rpm to rad/s. (about 4.7 rad/s)
- ω = −8.0 rad/s, α = +2.0 rad/s². Speeding up or slowing down? (Slowing down: opposite signs.)
Next: practice questions.