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Sol-Ch11 - CHAPTER 11 E xercises 1 33 1/3 rpm = 3.49 rd/s a...

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Exercises 1. 33 1/3 rpm = 3.49 rd/s. (a) α = 1.75 rad/s 2 ; (1.16) (b) θ 1 = 1/2 α t 2 = 3.49 rad, θ 2 = 3 ×   3.49 rad , θ = 2.22 rev. (c) Δ θ 2 = 9.08 rad, Δt 2 = 2.6 s, Δt = 4.5 s; (d) a r = ω 2 r = 0.46 m/s 2 ; a t = α r = 0.26 m/s 2 . (e) a r = 1.83 m/s 2 ; a t = 0 2. (a) v = ω r = (45π/30) (0.145) = 0.683 m/s; (b) v = ω r = 0.292 m/s; (c) Δr/Δt = (–8.3 cm/20 min) = –6.92 × 10 –5 m/s (d) In 20 min there are 900 revs and the needle moves radially by 8.2 cm, thus it moves (8.3 cm/900 rev) = 9.22 × 10 –5 m, which would be the average groove width. 3. (a) ω = 2π/1 d = 7.27 × 10 –5 rad/s ; (b) ω = 2π/1 y = 1.99 ×   10 –7 rad/s ; (c) v s = ω r = 463 m/s ; v o rb = 29.7 km/s. Thus, v C = 29.2 km/s, v F = 30.2 km/s. 4. (a) ω = 2π/1 d = 7.27 × 10 –5 rad/s, v = ω R = 463 m/s; (b) v = ω Rcos 41 ° = 349 m/s 5. (a) θ 1 = 31 rad, θ 2 = 9 rad, ω a v = 11 rad/s; (b) v = ω R = (–5 + 8r)(0.06) = 0.66 m/s; (c) a r = ω 2 r = 7.26 m/s 2 , a t = α R = 0.48 m/s 2 . 6. a = (a r 2 + a t 2 ) 1/2 = ( ω 4 r 2 + α 2 r 2 ) 1/2 7. ω 0 = 5.24 rad/s , Δr = Δ ω / α = 10.5 s. Δ θ 1 = ω 0 t + 1/2 α t 2 = 82.6 rad, When α = 0, Δ θ 2 = (10.47)(9.5) = 99.5 rad. Δ θ = 182 rad, or 29 rev. 8. (a) ω = (30 m/s)/(0.3 cm) = 100 rad/s ; α = 10 rad/s 2 ; (b) θ = 1/2 α t 2 = 500 rad or 79.6 rev; (c) a r = ω 2 r = (10 4 )(0.3) = 3 × 10 3 m/s 2 9. Bullet makes 60/25 = 2.4 rev as it passes through barrel. Δt = 0.6/850 = 7.06 × 10 –4 s, CHAPTER 11
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Rotational frequency = 2.4 rev/Δt = 2.04 × 10 5 rpm. 10. (a) ω = 2π /60 = π/30 rad/s; (b) v = ω r = 0.838 cm/s 11. (a) ω = v/r = (4.8 cm/s) /0.75 cm) = 6.4 rad/s. (b) ω = 4.8/1.75 = 2.74 rad/s. (c) Δt =86.4 m/(4.8 × 10 –2 m/s) = 1.8 × 10 3 s, α = Δ ω /Δt – 2.03 × 10 –3 rad/s 2 . 12. (a) ω = 2 – 10r + 8t 3 , α = –10 + 24t 2 = 14 rad/s 2 ; (b) Δ ω /Δt = (46 – 0)/1 = 46 rad/s 2 ; (c) ω av = Δ θ /Δt = [16 – (–1)]/1 = 17 rad/s 13. (a) Δ θ = ω 0 t + 1/2 = α t 2 , thus (90 ×   2π) = 4π(60) + 1/2 α (60) 2 , So α = –π/30 = –0.105 rad/s 2 . (b) ω = ω o + α t = 4π – (π/30) (60) = 2π rad/s. For last leg v o = ω r = 0.4π m/s, a = –0.021 m/s 2 : 0 = v o 2 + 2aΔx, Δx = 37.7 m. 14. α = (–750π/30 rad/s) 25 s = –π rad/s 2 , a = α r = –0/7865 m/s 2 At 500 rpm, v = (500π/30) (0.25) = 13.1 m/s. Δx = (13.1) 2 /2(0.785) = 109 m 15. (a) v o = 27.8 m/s, a = –v o 2 /2Δx = –7.73 m/s 2 And α = a/r = –30.9 rad/s 2 . (b) 50 m = (2πr)N tthus N = 31.8 rev. 16. (a) a = 3 m/s 2 , α = a/r = 12 rad/s 2 , so a t = α r = 3 m/s 2 . a r = v 2 /r = 16 m/s 2 (down), thus a = 3i – 16 j   m/s 2 ; (b) a t = α (2r) = 6 m/s 2 , a r = (2v) 2 /2r = 32 m/s 2 So a = 6i – 32 j   m/s 2 17. θ = ω t – 1/2 α t 2 : 80π = 50π/3 – 12.5 α , so α = –15.9 rad/s 2 .
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