Resolução Lista 1 Cinematica 1

1 \"Velocidad\" = frecuencia = 1800 rpm = 1800 = 30 Hz 60 A velocidad angular \\( \\omega = 2\\pi f \\) = 2\\pi(30) = 188.49 rad/s. Como \\( \\alpha \\) es cte.: \\[ \\omega^2 = \\omega_0^2 + 2\\alpha \\Delta \\theta \\] \\( 65.(2\\pi) \\) ? \\( 188.49 \\) O^2 = (188.49)^2 + 2.\\alpha.[ 408.40 ] \\( \\alpha = -43.49 rad/s^2. \\) \"Velocidad Nominal\" \\( = frecuencia = 1800 rpm = 30 \\) a) \\( \\omega = \\omega_0 + \\alpha t. \\) \\( 188.49 = 0 + \\alpha(5) \\) \\( \\alpha = 37.698 rad/s^2 \\) b) Lo igualo: \\( \\omega^2 = \\omega_0^2 + 2\\alpha \\Delta \\Theta \\) \\( (188.49)^2 = 0 + 2.(37.698)\\Delta \\Theta \\) \\( \\Delta \\Theta = 252.47 rad \\approx 15.66 voltas \\) \\( \\Delta \\Theta = 16.266.80 \\approx 270.71 voltas. \\) \\( \\alpha = 0.5 rad/s^2 \\) \\( t = 0s. \\) \\( \\omega = \\omega_0 + \\alpha t \\) \\( \\omega = 0 \\) \\( \\alpha t. \\) \\( a_f = 0.5.10^{-2} = 1.10^{-2} m/s^2. \\) \\( a_n = \\omega^2.R \\) \\( a_n = 0 \\) \\( = 1.10^{-2} m/s^2. \\) \\( t = 2s \\) \\( \\omega = \\omega_0 + \\alpha(2) \\) \\( \\omega = 1 rad/s \\) \\( a_f = 0.5.10^{-2} = 1.10^{-2} m/s^2. \\) Aún t = 2s \\( a_{res}^2 = (1.10^{-2})^2 + (2.10^{-2})^2 \\) \\( a_{res} = 2.23.10^{-2} m/s^2 \\) \\( t = 5s \\) \\( \\omega = \\omega_0 + \\alpha t \\rightarrow \\omega = 0 + 0.5(5) \\rightarrow \\omega = 2.5 rad/s \\) \\( a_r = 1.10^{-2} m/s^2 \\) \\( a_n = \\omega^2.R \\) \\( (2.5)^2(0.02) \\) \\( a_n = 6.25.0.02 \\) \\( a_{nr} = \\omega^2.R = 1^3.0.02^2 = 0.02 m \\) \\( = 2.10^{-3} m/s^2. \\) α = (t + 2t²)\nΔω = ∫ dω = α dt\nω = ω₀ + αΔt\nΔω = ∫α dt\nω - ω₀ = ∫(t + 2t²) dt\nΔΘ = ∫(t²/3 + t³/6) dt\nΔΘ = 1,299,99 ≈ 2000 rad α = Δω Δt = 78,57 - 31,41 3\nα A = 15,70 rad/s²\n1° de voltas\nω² = ω₀² + 2 * α * ΔΘ\n(78,57)² = (31,41)² + 2 * (15,70) * ΔΘ\n6165,4 = 386,58 + 341, ΔΘ\nΔΘ = 1649,93 rad ≈ 26 vol.\n\nα B = 47,15 - 18,86 3\nα B = 9,413 rad/s²\n1° de voltas\nω² = ω₀² + 2 * α * ΔΘ\n(47,15)² = (18,86)² + 2 * (9,413) * ΔΘ\n220,31 = 355,69 + 1886. ΔΘ\nΔΘ = 99,01 rad ≈ 15,76 vol.\n\nRelais entre os dados: 1 √5\n\nTambor 25 para 45 rpm → W inicial = 2,61 = 2,61\nω = ω₀ + 2 * α * ΔΘ\n(4,71)² = (2,61)² + 2 * α * (2π)\n22,20 = 6,81 + α * 150,79\nα = 0,1 rad/s²\n5 vezes mais. ∫ dω = ∫ α dt\n t=0\nω - ω₀ = ∫(2t + 2t²) dt\n t=0\nω - ω₀ = [ t²/2 + 2/3 t³ ] t=10 t=0\nω - ω₀ = 10² * 2/3\nω = 466,66 rad/s\nα = 2t + 2t² → p/t=10s\nα = 220 rad/s² α = 10, θ 1/3\n\nProcura uma relação entre (α, ω, θ)\n\nα = dω\ndt\n\nω = dθ\ndt\n\ndω\nα = dθ\nω\n\nIntegrando dos dois lados\n\n∫ ω dω = ∫ α dθ = ∫ 10.θ 1/3 dθ\n\nω2/2 = [10.3.θ 4/3\n4\n]\n\nω 0 = 0\n\nω2 = 2.10.3.θ 4/3 = 15.θ 4/3\n\nω = (15.θ 4/3) 1/2 ω = 387.θ 2/3.\n\nMas ω = dθ\ndt\ndθ = 3.87.θ 2/3 = dθ\ndt\n\n3.87dt = dθ\n\n∫ θ -2/3 dθ = ∫ 3.87dt\nt = 0\n\nθ 1/3 = 387.t\nθ -1/3 = (1/387)t\n\n(θ) 3 = [1/3.387.t]3 → θ = 2.14t 3 Quando t = 4\nθ = 2.14(4) 3 = 137.000\n\n[ω = dθ\ndt = 2.14.(3).t 2 = 6.42t 2\n\nQuando t = 11\nω = 102.000/3\n\n9)\na)\nA×B =\n| 1 3 k |\n| 50 20 2 |\n| -20 2 15 |\n= [284i - 790j + 500k]\n\nb)\nA × B =\n| 1 3 k |\n| 0 2 2 |\n| -2 2 5 |\n= [541i - 449j + 4k]\n\nc)\nA × B =\n| 5 2 2 |\n| 10 4 4 |\n= [0i + 0j + 0k] d) \\( \\mathbf{A} \\times \\mathbf{B} = \\begin{vmatrix} \\hat{i} & \\hat{j} & \\hat{k} \\\\\\ 5 & 1 & 2 \\\\\\ 2 & 15 \\end{vmatrix} = -138\\hat{i} - 73\\hat{j} + 9\\hat{k} \\\n\\\n l) \\mathbf{A} \\times \\mathbf{B} = \\begin{vmatrix} \\hat{i} & \\hat{j} & \\hat{k} \\\\\\ 8 & 2 & -42 \\\\\\ 0 & 2 & 15 \\end{vmatrix} = 1114\\hat{i} - 120\\hat{j} + 16\\hat{k} \\\n\\\n Example: \\mathbf{r}_A = (A - O) = (x_A - x_O, y_A - y_O, z_A - z_O) \\\n\\mathbf{r}_A = 0\\hat{i} + 200\\hat{j} + 120\\hat{k} \\text{ mm} \\\n\\mathbf{r}_B = 300\\hat{i} + 200\\hat{j} + 120\\hat{k} \\text{ mm} \\\n\\mathbf{r}_C = 300\\hat{i} + 0\\hat{j} + 620\\hat{k} \\text{ mm} \\\n\\mathbf{r}_D = 300\\hat{i} + 0\\hat{j} + 0\\hat{k}