Rolling and angular momentum formulas

Rolling without slipping: center speed v = ωR. The point touching the ground is not moving; the top moves at 2v. Same idea for a = αR and distance rolled = Rθ: use aₜ = rα and s = rθ in Rotation. rpm given: type ω as rpm*2π/60, e.g. 300*2π/60. Revolutions: θ = 2πN (Rotation).

Speed after rolling down h, no slipping (v₀ = 0 from rest). c = shape, typed as a fraction: hoop/pipe 1/1, solid disk/cylinder 1/2, solid sphere 2/5, hollow sphere 2/3. Other I: c = I/(mR²). Rolling up: h negative (−2 = up 2 m). Ramp length d: h = d*sin(θ); d unknown: a = g sinθ/(1 + c), then Motion topic. Slides, no friction: c = 0. Block on pulley: c = (I/R² + table block m)/hanging m (no table block: I/(mR²)).

Acceleration rolling down a ramp, no slipping. θ = ramp angle. c = shape, typed as a fraction: hoop/pipe 1/1, solid disk/cylinder 1/2, solid sphere 2/5, hollow sphere 2/3. Other I: type I/(mR²) for c. Yo-yo falling: θ = 90, c = I/(mr²), r = axle radius. Block on a pulley: θ = 90, c = (I/R² + table block m)/hanging m. No table block: c = I/(mR²). Time or distance: use this a in Motion & projectiles (rolling up: −a).

Total KE of a rolling object (moving + spinning). c = shape, typed as a fraction: hoop/pipe 1/1, solid disk/cylinder 1/2, solid sphere 2/5, hollow sphere 2/3. Other I: type I/(mR²) for c, e.g. 0.8/(5*0.3²). Moving part only (½mv²): c = 0. Spinning part only: K = ½Iω² in Rotation. Fraction that is spinning = c/(1 + c), e.g. solid sphere 0.4/1.4 = 0.29.

Smallest static friction coefficient that lets it roll down the ramp without slipping. Less than this: it slides. c = shape, typed as a fraction: hoop/pipe 1/1, solid disk/cylinder 1/2, solid sphere 2/5, hollow sphere 2/3. Steepest ramp it can roll on: leave θ empty. Friction force on it: f = μN (Forces) with the μₛ found here and N = m*9.8*cos(θ).

Angular momentum of a spinning object. Type I right in the box: disk 0.5*m*R², hoop or point m*R², sphere 0.4*m*R², rod about its middle m*length²/12, e.g. 0.5*2*0.3². rpm given: type ω as rpm*2π/60, e.g. 300*2π/60. Period T given: type ω as 2π/T (Earth spin: 2π/86400). Counterclockwise +, clockwise −. Rolling: ω = v/R. Final ω after parts join: L = total starting L (person: m*v*r), I = final total I.

No outside torque, so L stays the same: skater pulls arms in, person jumps on or walks across a turning disk, two disks stick together. Add the parts: I₂ = 0.5*100*2²+40*2² (disk + person at the edge; person = mr²). Turning opposite ways: one ω is negative. rpm in gives rpm out, no change needed. Parts at different ω at the start, or a person running on: use L = Iω. KE lost: K = ½Iω² (Rotation).

Angular momentum of a small object about a point. φ = angle between r (point to object) and v. Moving in a circle, or r = closest distance to its straight path: φ = 90. Moving straight at the point: L = 0. Person running onto a merry-go-round edge: L = mvr (φ = 90). r and v as x,y parts: L = m(x·vᵧ − y·vₓ), + = counterclockwise.

Net torque = change in angular momentum / time. ΔL = L after − L before. One object: type I*(ω after − ω before), e.g. 4*(0−30). Slowing down: ΔL and Στ come out negative. Know α instead of time? Use Στ = Iα in Rotation.

Gyroscope or spinning top: how fast its axis swings around in a circle (rad/s). r = pivot to center of mass, M = mass. I and ω = its spin about its own axis, e.g. disk I = 0.5*M*R². rpm given: type ω as rpm*2π/60. rev/s given: rev/s*2π, e.g. 40*2π. Time for one swing around: T = 2π/ωₚ. T given: type ωₚ as 2π/T.