Short on time? This page is the recap. For diagrams, derivations and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- A rolling object translates (its centre of mass moves) and rotates (it spins about its centre of mass). Its kinetic energy is the sum of both kinds.
- Rolling without slipping: the object turns through Δθ while its centre moves rΔθ. The contact point is momentarily at rest; the top moves at 2v_cm.
- In ideal rolling without slipping, static friction does no work, so mechanical energy is conserved. Friction’s torque shares gravity’s energy between translation and rotation.
- Rolling while slipping: the contact point slides, so v_cm ≠ rω. Kinetic friction opposes the sliding, changes v_cm and ω until rolling starts, and dissipates energy while it does.
- Rolling friction (rolling resistance) is outside the course.
Key relationships
| Relationship | When it applies | Use it to |
|---|---|---|
| K_total = ½Mv_cm² + ½I_cm ω² | any object that moves and spins | add the two kinds of kinetic energy |
| Δx_cm = rΔθ, v_cm = rω, a_cm = rα | rolling without slipping only | swap rotational and translational quantities |
| K_total = ½(1 + β)Mv_cm², with I_cm = βMr² | rolling without slipping | energy in one step |
| v = √(2gh / (1 + β)) | rolling from rest down height h, no slipping | speed at the bottom (no M or r) |
| a_cm = g sin θ / (1 + β) | rolling down a ramp at angle θ | acceleration and time |
| h = (1 + β)v² / (2g) | rolling up a ramp until it stops | greatest height |
Shape factors (given in questions): hoop 1, hollow sphere ⅔, disc or solid cylinder ½, solid sphere ⅖. Lower β means a larger share of energy in translation and a faster roll.
Assumptions behind the numbers
- Objects are rigid and surfaces are hard: no rolling friction, no air resistance.
- “Rolls without slipping” means static friction is large enough. You do not use μ_s F_N unless slipping is about to start.
- g = 9.8 m/s².
Mistakes to avoid
- Leaving out ½Iω². Mgh = ½Mv² gives the frictionless-block speed, which is too high.
- Saying friction removes energy in ideal rolling. The contact point is at rest, so static friction does no work.
- Using v = rω for a skidding wheel.
- Thinking heavier or bigger means faster. Only β matters.
- Assuming friction always points backwards. It opposes sliding of the contact point: up a slope for a ball rolling down; forwards for a wheel spinning too fast.
- Justifying with a law’s name only. “Because of conservation of energy” is not enough. Explain where the energy goes.
Quick self-check
- A wheel of radius 0.25 m rolls without slipping at 4.0 m/s. What are its angular speed and the speed of its top point? (16 rad/s; 8.0 m/s)
- What fraction of a rolling disc’s kinetic energy is rotational? (1/3)
- A hoop rolls from rest down a 0.45 m drop. What is its speed at the bottom? (√(gh) = 2.1 m/s; a sliding frictionless block would reach 3.0 m/s.)
Next: practice questions.