Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For derivations, diagrams and worked examples, use the full study guide.
Recap
- Any moving, spinning rigid body: K_tot = K_trans + K_rot, with I about the centre of mass in K_rot.
- Rolling without slipping: the arc that touches the ground equals the distance moved, Δx_cm = rΔθ. Differentiate once and twice for the speed and acceleration links.
- The contact point is momentarily at rest; the top moves at 2v_cm.
- Static friction supplies the torque that keeps ω matched to v_cm/r. It acts at a point that is not moving, so it does no work.
- Slipping: v_cm ≠ rω. Use F_net = Ma_cm and τ_net = I_cm α separately, with kinetic friction μ_k N opposing the sliding of the contact point. Slipping stops when v_cm = rω.
Key relationships
| Relationship | When it applies |
|---|---|
| K_tot = ½Mv_cm² + ½I_cm ω² | any moving, spinning rigid body |
| Δx_cm = rΔθ, v_cm = rω, a_cm = rα | rolling without slipping only |
| K = ½M(1 + β)v_cm², with I_cm = βMr² | rolling without slipping |
| a_cm = g sin θ ÷ (1 + β); f = βMg sin θ ÷ (1 + β) | rolling down a slope |
| μ_s ≥ β tan θ ÷ (1 + β) | condition to roll without slipping on a slope |
| v = √(2gh ÷ (1 + β)) | rolling from rest through a drop h |
| energy dissipated = f_k × (distance the contact point slides) | slipping |
| β: hoop 1, spherical shell 2/3, solid cylinder 1/2, solid sphere 2/5 | about the central axis |
Assumptions behind the numbers
- Rigid bodies on rigid surfaces: no rolling friction (outside the course).
- Angular quantities in radians; g = 9.8 m/s².
- Choose +x and the positive sense of rotation so that a_cm = rα with matching signs.
Mistakes to avoid
- v_cm = rω while slipping. Only after slipping stops.
- Friction “wastes” energy in rolling. Not static friction on an ideal rolling body.
- f = μ_s N automatically. That is the maximum; find f from Newton’s laws.
- Dissipated energy from the centre’s distance. Use the sliding distance of the contact point.
- Double counting. ½I_P ω² about the contact point is already the total.
- Assuming friction’s direction. Let the sign of your answer decide.
Quick self-check
- A 2.0 kg hoop rolls without slipping at 3.0 m/s. What is its total kinetic energy? (β = 1, so K = Mv² = 18 J)
- A solid sphere rolls from rest down a slope; its centre drops 0.70 m. How fast is it moving at the bottom? (v = √(2 × 9.8 × 0.70 ÷ 1.4) = 3.1 m/s)
- A wheel of radius 0.30 m rolls at 6.0 m/s. Find ω and the speed of the top of the wheel. (20 rad/s; 12 m/s)
Next: practice questions.