Rotation and torque formulas

From rest: ω₀ = 0. Comes to a stop: ω = 0. Slowing down: α is negative. Given rpm? Type rpm*2π/60 to get rad/s (300 rpm: 300*2π/60). Given rev/s (Hz)? Type f*2π. One turn every T s? Type 2π/T. No time given? Use ω² = ω₀² + 2αΔθ. Rope or rim accel. a given? Type a/r for α.

Use when you know the time and α. From rest: ω₀ = 0. Slowing down: α is negative. Steady spin (no speed-up): α = 0, so Δθ = ω₀t. Given revolutions? Type rev*2π (12 rev: 12*2π). Given degrees? Type deg*π/180. Want revolutions? Divide the Δθ answer by 2π. Given rpm? Type rpm*2π/60. Rope unwound or distance s on the rim: Δθ = s/r. Two answers for t? Use the smaller positive one.

Use when no time is given. From rest: ω₀ = 0. Stops: ω = 0 (α comes out negative). Revolutions: type rev*2π for Δθ (12 rev: 12*2π). Degrees: type deg*π/180. Given rpm? Type rpm*2π/60. Two answers for ω (+ and −)? Same speed, two spin directions: use + unless it turned back.

Use when α is not given and not asked. From rest: ω₀ = 0. Stops: ω = 0. Want revolutions? Divide the Δθ answer by 2π (6.283). Given revolutions: type rev*2π. Given rpm: type rpm*2π/60. (ω₀ + ω)/2 is the average ω.

Speed of a point on the rim, or of a belt, rope or rolling wheel. ω in rad/s: rpm → type rpm*2π/60 (900 rpm: 900*2π/60). One turn every T s → type 2π/T. Same formula, other uses: s = rθ: put θ (rad) in the ω box, answer is s aₜ = rα: put α in the ω box, answer is aₜ ac = rω²: put ω² in the ω box (or ac = vₜ²/r) Total accel.: a = √(aₜ² + ac²)

r = distance from the pivot (axis) to where F pushes. θ = angle between r and F. Push at a right angle (like a door handle): θ = 90. Push straight toward the pivot: θ = 0, no torque. Type τ as a positive size here. Add the sign after: counterclockwise +, clockwise −. Several forces: add the torques (net torque Στ).

Net torque: add the torques, counterclockwise +, clockwise −. Force on the rim at a right angle: τ = FR, type F*R (20 N on 0.3 m: 20*0.3). Type I right in, e.g. solid disk ½MR²: 1/2*4*0.3² Rope on a wheel: α = a/R. Rod on an end pivot, held level: weight acts at the center, type Στ = M*9.8*L/2 and I = 1/3*M*L². Then use ω = ω₀ + αt etc. for speed and turns. Block hanging on the wheel? Use a = mgR²/(I + mR²).

c by shape (type a fraction so sig figs stay right): Point mass, hoop, ring, thin pipe: 1/1 Solid disk, cylinder, pulley: 1/2 Solid sphere (ball): 2/5 Hollow sphere: 2/3 Rod about its center: 1/12 (L in the R box) Rod about one end: 1/3 (L in the R box) Axis a distance d from the center: add Md². Many parts: add them, I = I₁ + I₂ + … Point masses: I = m₁r₁² + m₂r₂² + … Type a sum right in any I box: 2*0.5²+3*0.2²

A block hangs from a rope on a pulley or wheel and falls. Disk pulley: type I as 1/2*M*R² (2 kg, 0.1 m: 1/2*2*0.1²). Then: tension T = m(g − a), α = a/R. Fall h from rest: v = √(2ah), t = √(2h/a) Also pulls a block on a smooth table? Add its m₂R² to I. Two hanging blocks (m₁ heavier): type m₁−m₂ for m and I+2*m₂*R² for I. Tensions: T₁ = m₁(g − a), T₂ = m₂(g + a)

Spin energy. ω in rad/s. No friction: energy at start = energy at end. Only spinning, and its center drops h: K = Mgh, type M*g*h (2 kg, 0.5 m: 2*9.8*0.5). Rod swinging down from level on an end pivot: h = L/2. Work by a torque: W = τΔθ, and W = K − K₀ (from rest: K = W). Rolling (moving + spinning)? Use the Rolling topic.

Power of a motor or engine. ω in rad/s. Given rpm? Type rpm*2π/60 for ω (3000 rpm: 3000*2π/60). Work by a steady torque: W = τΔθ (Δθ in rad). Power is also P = W/t (work / time). 1 hp = 746 W.