15 - E xe rc ises 1. cos ( θ + 5 π /3), sin( θ + π /6)...

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Unformatted text preview: E xe rc ises 1. cos ( θ + 5 π /3), sin( θ + π /6) 12. (a) F = k 1 x 1 = k 2 x 2 = ( x 1 + x 2 )k e f f ; F/k e f f = F /k 1 + F /k 2 , s o T = 2 π (m/k e f f ) 1 / 2 = 2 π [m(k 1 + k 2 )/k 1 k 2 ] 1 / 2 . ( b) & (c) : F = k 1 (x + x o 1 ) – k 2 (x o 2 – x ) = (k 1 + k 2 ) = k e f f x 14. ω = ( k/m) 1 / 2 = 2 5.3 rad/s, A =0.2 m (a) K = 1/2 mv 2 = 1/2 m[– ω Asin(2 π /5)] 2 = 0 .579 J U = 1 /2 kx 2 = 1 /2 k[Acos (2 π /5)] 2 = 6 .11 × 1 0 – 2 J (b) cos 2 ( ω t) = 1/4, sin 2 ( ω t) = 3/4 K = 1/2 m ( ω A) 2 s in 2 ( ω t) =480 mJ U = 1 /2 kA 2 cos 2 ( ω t) = 160 mJ (c) K = U when ω t = (2n +1) π /4 , t hus t = ( 2n +1) T/8 when T = 2 π / ω =248 ms. 15. If K = U/2, then U = 2E/3 thsu x = ±(2/3) 1 / 2 A. Sin (27.4t)= +(2/3) 1 / 2 , t = 34.8 ms, 79.8 ms, Sin(27.5t) = –(2.3) 1 / 2 , t = 150 ms, 194 ms 21. (a) 1/2 kx 2 + 1 /2 mv 2 = 1 /2 kA 2 ; ths v 2 = ( k/m)(A 2 – x 2 ) 26. T = 2 π (I/k) 1 / 2 = 0 . 32. Say θ = θ o sin( ω t) , where ω = π r ad/s. Given +0.5 = sin( π t), thus t = +1/6 s. The time to go from – θ o /2 to is + θ o /2 is 1/3 s. Problems 1. ω = 8 r ad/s. E = K m a x = 1 J = 1 /2 kA 2 , thus A =1/4 m x = 0 .25 sin(8t) m. 4. ω 2 = k /(M +m). From ∑F x a nd ∑ F y f ind a = μ g for m. a max = ω 2 A = μ g, thus μ = 0 .136 Identical to simple pendulum. Could also use τ = I α : – (mg s in θ )R = (MR 2 )(d 2 /dt 2 ) leads to ω = (g/R) 1 / 2 . (m/F o )dA/d ω = – 1/2 [ ] – 3/2[2( ω o 2 – ω 2 )(–2 ω ) + 2 γ 2 ω /m 2 ] = 0 F ind ω 2 = ω o 2 – γ 2 /2m 2 5. 7. 9. (a) dK = 1/2 (m dx/L) (xv/L) 2 , thus K = ∫ d K = mv 2 /6. (b) E = (M/2 + m/6)v 2 + 1 /2 kx 2 , set dE/dt = 0 to find d 2 x/dt 2 + ( k/(M + m/3)) x = 0, t hus T = 2 π [ (M + m/3)/k] 1 / 2 1 1. (a) 1.00430 s; (b) 1.01738 s ; (c) 1.03963 s; (d) 1.07129 s 13. F x = – (mgrR) sin θ w here sin θ = x /r. From F x = m a, find ω = ( g/R) 1 / 2 a nd T = 2 π / ω ≈ s 84.4 min. 14. τ = I α : – (kx)L = I d 2 θ /dt 2 . Note that x = L θ a nd I = ML 2 /3 F ind d 2 θ /dt 2 + 3 k θ /M = 0, thus T = 2 π (M/3k) 1 / 2 ...
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This note was uploaded on 05/01/2011 for the course ELECTRIC MAT351 taught by Professor Ahmettantuni during the Spring '10 term at Abant İzzet Baysal University.

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15 - E xe rc ises 1. cos ( θ + 5 π /3), sin( θ + π /6)...

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