1

θ = srω = dθdt,  α = dωdt = d²θdt²ω = ω₀ + αtθ = θ₀ + ω₀t + (1/2)αt²ω² = ω²₀ + 2α(θ − θ₀)

2

s = θrv = ωrT = 2πrv = 2πωa_t = αra_r = v²r = ω²r

3

K = Σ(1/2)m_iv_i² = (1/2)(Σm_ir_i²)ω²I = Σm_ir_i²K = (1/2)Iω²I = ∫r² dm

4

I_{rod, center} = ∫^L/2_−L/2 x²MLdx = 112ML²

5

I′ = ∫(r + Δr) · (r + Δr) dm = I + 2Δr · ∫r dm + mΔr²I = I_com + Md² ≥ I_com

6

|τ| = rF sin φτ = r × F = (yF_z − zF_y, zF_x − xF_z, xF_y − yF_x)τ = Σr_iF_i = (Σm_ir_i²)α = Iα

7

mg − T = maTR = (1/2)MR²αa = Rαa = 2mM + 2mgT = MmM + 2mg

8

W = ∫θfθi τ dθΔK = (1/2)Iω2f − (1/2)Iω2i = WP = dWdt = τωτ dθ = F cos φ′ ds = F · ds

9

s = θRv_com = dsdt = ωRI_P = I_com + MR²K = (1/2)I_comω² + (1/2)Mv²_com

10

Mg sin θ − f_s = Ma_comRf_s = I_comα = I_coma_comRa_com = g sin θ1 + I_com/(MR²)

11

ℓ = r × p,  |ℓ| = rp sin θτ = dℓdtτ_net = dLdtL = IωI_iω_i = I_fω_f

12

L = Iωτ = dLdt,  Mgr = dLdtdL = L dϕΩ = dϕdt = MgrIω