ω θ ω = Δθ Δt, θ ω v v = rω ω = v r, r ω α α = Δω Δt,
Δω Δt (rad/s)/s rad/s 2 ω α ω α rad/s 2 87.3 rad/s 2 α = Δω Δt Δω Δt α = Δω Δt = 250 rpm 5.00 s. Δω rad/s 2 Δω α Δω = 250 min rev 2π rad rev 60 1 min sec = 26.2 rad s. α = Δω Δt = 26.2 rad/s 5.00 s = 5.24 rad/s 2. Δt Δt = Δω α. 26.2 rad/s Δω 26.2 rad/s α 87.3 rad/s 2 Δt = 26.2 rad/s 87.3 rad/s 2 = 0.300 s. a t
a a a t a c a t a c a t α a c a t α a t = Δv Δt. v = rω a t = Δ(rω). Δt r Δ(rω) = r(δω) α = Δω Δt a t = r Δω Δt. a t = rα, α = a t r. α
a t α = a t r a t = Δv Δt = 30.0 m/s 4.20 s = 7.14 m/s 2. a t r α = a t r α = a t r = 7.14 m/s2 0.320 m = 22.3 rad/s 2. θ, ω α x, v a θ x θ = x r ω v ω = v r α a α = a t r
α rad/s 2 θ ω α α t ω θ ω α t v = v 0 + at (constant a) a = a t a a α a = rα v = rω a = rα rω = rω 0 + rαt. r ω = ω 0 + at (constant a), ω 0 ω α t θ = ω t x = v - t ω = ω 0 + αt v = v 0 + at α a θ = ω 0 t + 1 2 αt2 x = v 0 t + 1 2 at2 α a ω 2 = ω 0 2 + 2αθ v 2 = v 0 2 + 2ax α a
θ 0 x 0 t 0 ω - v - ω = ω 0 + ω 2 and v = v 0 + v. 2 a α 110 rad/s 2 α t ω ω = ω 0 + αt ω = ω 0 + αt. ω 0 = 0 ω = 0 + 110 rad/s 2 (2.00s) = 220 rad/s. ω v v = rω, r v = (0.0450 m)(220 rad/s) = 9.90 m/s. m rad = m 1 rev = 2π rad θ α t ω 0 θ θ = ω 0 t + 1 2 αt2 θ = ω 0 t + 1 2 αt2 = 0 + (0.500) 110 rad/s 2 (2.00 s) 2 = 220 rad.
θ = (220 rad) 2π 1 rev = 35.0 rev. rad x θ x = rθ = (0.0450 m)(220 rad) = 9.90 m. 300 rad/s 2 t ω 0 = 220 rad/s ω α = 300 rad/s 2 ω = ω 0 + αt, ω = ω 0 + αt. t = ω ω 0 α = 0 220 rad/s = 0.733 s. 2 300 rad/s 0.250 rad/s 2 x ω v θ r α
x θ = x r. x x = rθ. θ = (200 rev) 2π rad = 1257 rad. 1 rev x = rθ x = rθ = (0.350 m)(1257 rad) = 440 m. t ω ω 2 = ω 0 2 + 2αθ ω ω 2 = ω 2 0 + 2αθ ω = 0 + 2(0.250 rad/s 2 )(1257 rad) 1 / 2 = 25.1 rad/s. v ω v = rω = (0.350 m)(25.1 rad/s) = 8.77 m/s. θ x θ = ω t θ ω - θ = ω t
θ = ω - t = 6.0 rpm (2.0 min) = 12 rev. x θ θ = (12 rev) 2π rad 1 rev = 75.4 rad. x θ x = rθ = (0.15 m)(75.4 rad) = 11 m. F m r r a = F m F F = ma a = rα F = ma F = mrα. F r τ = Fr r rf = mr 2 α τ = mr 2 α. F = ma mr 2 mr 2 m r
F r r I mr 2 I = mr 2 I m r I MR 2 M R M R m r I I kg m 2 net τ = Iα α = net I τ, τ τ = Iα, α = net I τ
net τ = Iα, α = net I τ
α = net τ I α = τ I. α τ I τ = rf sin θ = (1.50 m)(250 N) = 375 N m. 1 2 MR2, M = 50.0 kg R = 1.50 m I = (0.500)(50.0 kg)(1.50 m) 2 = 56.25 kg m 2. α = τ I = 375 N m 56.25 kg m 2 = 6.67rad s 2. I I c I c = MR 2 = (18.0 kg)(1.25 m) 2 = 28.13 kg m 2. I I = 28.13 kg m 2 + 56.25 kg m 2 = 84.38 kg m 2. α α = τ I = 375 N m 84.38 kg m 2 = 4.44rad s 2.
net W = (net F)Δs. r net W = (r net F) Δs r. r net F = net τ Δs / r = θ net W = (net τ)θ. net W = (net τ)θ net τ = Iα net W = Iαθ. (net F)Δs
αθ ω 2 = ω 0 2 + 2αθ. αθ αθ = ω2 2 ω 0. 2 W net W = 1 2 Iω2 1 2 Iω 0 2. 1 2 Iω2 KE rot I ω KE rot = 1 2 Iω2. I m ω v KE rot KE rot 1.00 rad(57.3º) net W = (net τ)θ KE rot = 1 2 Iω2 net W = (net τ)θ, τ (rf) r θ 1 N m = 1 J net W = rfθ = (0.320 m)(200 N)(1.00 rad) = 64.0 N m. net W = 64.0 J.
ω ω 2 = ω 0 2 + 2αθ. ω 0 = 0 α ω = (2αθ) 1 / 2. α = net I τ, net τ = rf = (0.320 m)(200 N) = 64.0 N m. I = 1 2 MR2 = 0.5 85.0 kg (0.320 m) 2 = 4.352 kg m 2. α α = 64.0 N m 4.352 kg m 2 = 14.7rad s 2. θ ω ω = (2αθ) 1 / 2 = 2 14.7rad s 2 (1.00 rad) 1 / 2 = 5.42 rad s. KE rot = 1 2 Iω2. I ω KE rot = (0.5) 4.352 kg m 2 (5.42 rad/s) 2 = 64.0 J.
KE i + PE i = KE f + PE f. KE i KE f OE KE rot = 1 2 Iω2. KE rot ω ω = 300 rev 2π rad 1.00 min = 31.4 1.00 min 1 rev 60.0 s rad s. I I = 4 Ml2 3 = 4 50.0 kg (4.00 m) 2 3 = 1067 kg m 2. ω I KE rot = 0.5(1067 kg m 2 )(31.4 rad/s) 2 = 5.26 10 5 J KE trans = 1 2 mv2 = (0.5) 1000 kg (20.0 m/s) 2 = 2.00 10 5 J. 2.00 10 5 J 5.26 10 5 J = 0.380. KE rot = PE grav 1 2 Iω2 = mgh.
h h = 1 2 Iω2 mg = 5.26 10 5 J = 53.7 m. 1000 kg 9.80 m/s 2 KE PE grav KE KE KE trans KE rot KE
PE i = KE f. PE grav = KE trans + KE rot mgh = 1 2 mv2 + 1 2 Iω2. mgh I ω = 0 v mgh = 1 2 mv2 + 1 2 Iω2. v I v ω ω = v / R R m v mgh = 1 2 mv2 + 1 2 1 2 mr2 v 2 R 2. gh = 1 2 v2 + 1 4 v2 = 3 4 v2. v = 4gh 3 1 / 2. v = 4 9.80 m/s 2 1 / 2 (2.00 m) = 5.11 m/s. 3 m R v = 4 3 gh 1 / 2 1 2 mv 2 = mgh v = (2gh) 1 / 2 (4gh / 3) 1 / 2
L L = Iω. p = mv kg m/s kg m 2 /s I ω L = Iω I = 2MR2 5 L = Iω = 2MR2 ω. 5 M 5.979 10 24 kg R 6.376 10 6 m ω ω L ω
L = 0.4 5.979 10 24 kg 6.376 10 6 m 2 1 rev d = 9.72 10 37 kg m 2 rev/d. 2π 1 8.64 10 4 s L = 9.72 10 37 kg m 2 2π rad/rev 8.64 10 4 (1 rev/d) s/d = 7.07 10 33 kg m 2 /s. L net τ = ΔL Δt. F = Δp / Δt net τ = ΔL Δt net τ = ΔL Δt net τ = ΔL Δt ΔL ΔL = L ω L L = Iω net τ = ΔL Δt ΔL ΔL = (net τ)δt. r net τ = rf L = rfδt = (0.260 m)(2.50 N)(0.150 s) = 9.75 10 2 kg m 2 / s. L = Iω.
ω ω = L I = L 1 2 MR2. ω = 9.75 10 2 kg m 2 /s = 0.721 rad/s. (0.500) 4.00 kg (0.260 m) 1.25 kg m 2 57.3º F r α = net τ / I I α α α = net I τ. net τ = r F = (0.0220 m)(2000 N) = 44.0 N m. α α = 44.0 N m 1.25 kg m 2 = 35.2 rad/s2. ω 2 = ω 0 2 + 2αθ ω 2 = 2αθ
KE rot = 1 2 Iω2 ω 2 KE rot = 0.5 1.25 kg m 2 70.4 rad 2 / s 2. = 44.0 J ΔL = (net τ)δt net τ = ΔL Δt netτ = 0 ΔL Δt = 0. ΔL L = constant (net τ = 0) L = L (netτ = 0). F r τ L = L. Iω = I ω, I ω
2.34 kg m 2 0.363 kg m 2 ω KE rot = 1 2 Iω2. Iω = I ω L = L Iω = I ω ω ω = I I ω = 2.34 kg m2 0.363 kg m 2 (0.800 rev/s) = 5.16 rev/s. KE rot = 1 2 Iω2. KE rot = (0.5) 2.34 kg m 2 (0.800 rev/s)(2π rad/rev) 2 = 29.6 J. KE rot = 1 2 I ω 2. KE rot = (0.5) 0.363 kg m 2 (5.16 rev/s)(2π rad/rev) 2 = 191 J.
r
Iω L = L, L = Iω, I ω I = mr 2 ω = v / r L = mr 2v r = mvr. L = I ω. ω I ω = mvr. ω = mvr, I I I = Mr 2 / 3 I = mr 2 + Mr2 3 = m + M 3 r2. I = 0.0500 kg + 0.667 kg (1.20 m) 2 = 1.032 kg m 2. I ω ω = mvr I = 0.0500 kg (30.0 m/s)(1.20 m) 1.032 kg m 2 = 1.744 rad/s 1.74 rad/s.
KE = 1 2 mv2 = (0.500) 0.0500 kg (30.0 m/s) 2 = 22.5 J KE = 1 2 I ω 2 = (0.5) 1.032 kg m 2 1.744rad s 2 = 1.57 J. p = mv = 0.0500 kg (30.0 m/s) = 1.50 kg m/s. v = rω v CM = 2 r ω = v 2 p = mv + Mv CM = mv + Mv 2. p = m + M 2 p = m + M 2 v rω. p = 1.050 kg (1.20 m)(1.744 rad/s) = 2.20 kg m/s.
L ω
L = Iω L ω ω L net τ = ΔL Δt. ΔL τ L L L L ΔL L r F F
ΔL ΔL L L L = ΔL ΔL ΔL L
I = mr 2 KE 1 KE 2 W ω = Δθ Δt α = Δω Δt a t = Δv Δt v = rω a t = Δ(rω). Δt Δ(rω) = rδω Δω / Δt = α a t = r Δω Δt. a t = rα α = a t r. x 0 t 0 ω - v - ω = ω 0 + ω 2 and v = v 0 + v. 2 F m r r a = F/m F F = ma, a = rα F=ma F=mrα
F r τ = rf r rf = mr 2 α τ = mr 2 α. I MR 2 I = mr 2. τ = Iα α = net I τ KE rot I ω KE rot = 1 2 Iω2. net W = 1 2 Iω2 1 2 Iω 0 2. L L = Iω. p = mv net τ = ΔL Δt. L L = Iω net τ = ΔL Δt. L = constant (net τ = 0) L = L (net τ = 0) r F F L L = ΔL
ML 2 /3 M L / 2 ML 2 /4 M R M R
rad/s 2 m/s 2 g 100 rad/s 2 0.700 rad/s 2 7.00 m/s 2 95.0 rad/s 1.50 m/s 2 2.00 10 3 N 120 rad/s 2 30.00 rad/s 2 0.750 kg m 2
0.500 kg m 2 4.00 10 7 N I = Ml 2 / 3 I = Ml 2 / 12 0.050 kg m 2 3.75 kg m 2 0.900 kg m 2 20.0º 60.0º 0.250 kg m 2 MR 2 M R
0.720 kg m 2 0.400 kg m 2
L