Page 1 14. Wave Motion Introductory Exercise 14.1 1. A func tion, f can rep re sent wave equa tion, if it sat isfy ¶ ¶ = ¶ ¶ 2 2 2 2 2 f t v f x For, y a t = sin , w ¶ ¶ = - = - 2 2 2 2 y t a t y w w w sin but, ¶ ¶ = 2 2 0 y x So, y do not represent wave equation. 2. y x t ae bx et ( , ) ( ) = - - 2 = - - ae kx t ( ) w 2 Þ k b = and w = e Þ v k c b = = w 3. y x t x t ( , ) ( ) = + + 1 1 4 2 w rep r e sent the given pulse, where, y x k x x ( , ) 0 1 1 1 1 2 2 2 = + = + Þ k = 1 and y x z x x ( , ) ( ) ( ) = + - = + - 1 1 2 1 1 1 2 2 w Þ w = 1 2 \ v k = = = w 1 2 1 / 0.5 m/s 4. y x t a b kx t = + + = + + 10 5 2 2 2 ( ) ( ) w Amplitude, y a b max = = = 10 5 2 m and k = 1; w = 2 v k = = w 2 m/s and is travelling in (–) x direction. 5. y kx t = - + 10 2 2 ( ) w y x k x x ( , ) 0 10 2 10 2 2 2 2 = + = + Þ k = 1 w = = vk 2 m/s ´ - 1 1 m = 2 rad/s Þ y x t = - + 10 2 2 2 ( ) Introductory Exercise 14.2 1. y x t ( , ) = 0.02 sin x t 0.05 0.01 + æ è ç ö ø ÷ m = + A kx t sin ( ) w m Þ A = 0.02 m, k = 1 0.05 m -1 , w = - 1 001 1 . s (a) v k = = w 0.05 0.01 m/s = 5 m/s (b) v y t A kx t p = ¶ ¶ = + w w cos ( ) v p ( , ) 0.2 0.3 0.02 0.01 = ´ 1 cos 0.2 0.5 0.3 0.01 + æ è ç ö ø ÷ = + 2 4 30 cos ( ) = 2 34 cos = - 2( 0.85) = - 1.7 m/s 2. Yes, ( ) ( ) max v A Ak k Ak v p = = × = w w 3. l = 4 cm, v = 40 cm/s (given) (a) n l = = = v 40 4 10 cm/s cm Hz Page 2 14. Wave Motion Introductory Exercise 14.1 1. A func tion, f can rep re sent wave equa tion, if it sat isfy ¶ ¶ = ¶ ¶ 2 2 2 2 2 f t v f x For, y a t = sin , w ¶ ¶ = - = - 2 2 2 2 y t a t y w w w sin but, ¶ ¶ = 2 2 0 y x So, y do not represent wave equation. 2. y x t ae bx et ( , ) ( ) = - - 2 = - - ae kx t ( ) w 2 Þ k b = and w = e Þ v k c b = = w 3. y x t x t ( , ) ( ) = + + 1 1 4 2 w rep r e sent the given pulse, where, y x k x x ( , ) 0 1 1 1 1 2 2 2 = + = + Þ k = 1 and y x z x x ( , ) ( ) ( ) = + - = + - 1 1 2 1 1 1 2 2 w Þ w = 1 2 \ v k = = = w 1 2 1 / 0.5 m/s 4. y x t a b kx t = + + = + + 10 5 2 2 2 ( ) ( ) w Amplitude, y a b max = = = 10 5 2 m and k = 1; w = 2 v k = = w 2 m/s and is travelling in (–) x direction. 5. y kx t = - + 10 2 2 ( ) w y x k x x ( , ) 0 10 2 10 2 2 2 2 = + = + Þ k = 1 w = = vk 2 m/s ´ - 1 1 m = 2 rad/s Þ y x t = - + 10 2 2 2 ( ) Introductory Exercise 14.2 1. y x t ( , ) = 0.02 sin x t 0.05 0.01 + æ è ç ö ø ÷ m = + A kx t sin ( ) w m Þ A = 0.02 m, k = 1 0.05 m -1 , w = - 1 001 1 . s (a) v k = = w 0.05 0.01 m/s = 5 m/s (b) v y t A kx t p = ¶ ¶ = + w w cos ( ) v p ( , ) 0.2 0.3 0.02 0.01 = ´ 1 cos 0.2 0.5 0.3 0.01 + æ è ç ö ø ÷ = + 2 4 30 cos ( ) = 2 34 cos = - 2( 0.85) = - 1.7 m/s 2. Yes, ( ) ( ) max v A Ak k Ak v p = = × = w w 3. l = 4 cm, v = 40 cm/s (given) (a) n l = = = v 40 4 10 cm/s cm Hz (b) D D f p l = 2 x = ´ 2 4 p cm 2.5 cm = 5 4 p rad (c) D D D t T = = 2 1 2 p f pn f = ´ ´ 1 2 10 3 p p = 1 60 s (d) v v p p = ( ) max = - = - A A w p n 2 = - ´ ´ - 2 2 10 1 p cm s = - 40 p cm/s = - 1.26 cm/s 4. (a) y A t kx = - sin ( ) w = × - æ è ç ö ø ÷ A v t x sin 2 2 p l p l = ´ - æ è ç ö ø ÷ 0.05 0.4 0.4 sin 12 2 2 p p t x = - 0.05 sin ( ) 60 5 p p t x (b) y ( , 0.25 0.15) = ´ - ´ 0.05 0.15 0.25) sin (60 5 p p = - 0.05 .25 sin ( ) 9 1 p p = 0.05 7.75 sin ( ) p = 0.05 1.75 sin ( ) p = - 0.0354 m = - 3.54 cm (c) D D Df t T = = = 2 60 p f w p p 0.25 = 1 240 s = 4.2 ms Introductory Exercise 14.3 1. v T T m l Tl m = = = m / = ´ = 500 2 100 5 3 0.06 = 129.1 m/s 2. v T T A = = × m r = ´ ´ = - 0.98 9.8 10 10 10 3 6 m/s Introductory Exercise 14.4 1. I P r = = ´ = 4 1 4 1 1 4 2 2 p p p W m W m 2 ( ) / 2. For line source, I rl = r p 2 Þ I r µ 1 and as I A µ 2 Þ A r µ 1 2 | Waves & Motion x y Page 3 14. Wave Motion Introductory Exercise 14.1 1. A func tion, f can rep re sent wave equa tion, if it sat isfy ¶ ¶ = ¶ ¶ 2 2 2 2 2 f t v f x For, y a t = sin , w ¶ ¶ = - = - 2 2 2 2 y t a t y w w w sin but, ¶ ¶ = 2 2 0 y x So, y do not represent wave equation. 2. y x t ae bx et ( , ) ( ) = - - 2 = - - ae kx t ( ) w 2 Þ k b = and w = e Þ v k c b = = w 3. y x t x t ( , ) ( ) = + + 1 1 4 2 w rep r e sent the given pulse, where, y x k x x ( , ) 0 1 1 1 1 2 2 2 = + = + Þ k = 1 and y x z x x ( , ) ( ) ( ) = + - = + - 1 1 2 1 1 1 2 2 w Þ w = 1 2 \ v k = = = w 1 2 1 / 0.5 m/s 4. y x t a b kx t = + + = + + 10 5 2 2 2 ( ) ( ) w Amplitude, y a b max = = = 10 5 2 m and k = 1; w = 2 v k = = w 2 m/s and is travelling in (–) x direction. 5. y kx t = - + 10 2 2 ( ) w y x k x x ( , ) 0 10 2 10 2 2 2 2 = + = + Þ k = 1 w = = vk 2 m/s ´ - 1 1 m = 2 rad/s Þ y x t = - + 10 2 2 2 ( ) Introductory Exercise 14.2 1. y x t ( , ) = 0.02 sin x t 0.05 0.01 + æ è ç ö ø ÷ m = + A kx t sin ( ) w m Þ A = 0.02 m, k = 1 0.05 m -1 , w = - 1 001 1 . s (a) v k = = w 0.05 0.01 m/s = 5 m/s (b) v y t A kx t p = ¶ ¶ = + w w cos ( ) v p ( , ) 0.2 0.3 0.02 0.01 = ´ 1 cos 0.2 0.5 0.3 0.01 + æ è ç ö ø ÷ = + 2 4 30 cos ( ) = 2 34 cos = - 2( 0.85) = - 1.7 m/s 2. Yes, ( ) ( ) max v A Ak k Ak v p = = × = w w 3. l = 4 cm, v = 40 cm/s (given) (a) n l = = = v 40 4 10 cm/s cm Hz (b) D D f p l = 2 x = ´ 2 4 p cm 2.5 cm = 5 4 p rad (c) D D D t T = = 2 1 2 p f pn f = ´ ´ 1 2 10 3 p p = 1 60 s (d) v v p p = ( ) max = - = - A A w p n 2 = - ´ ´ - 2 2 10 1 p cm s = - 40 p cm/s = - 1.26 cm/s 4. (a) y A t kx = - sin ( ) w = × - æ è ç ö ø ÷ A v t x sin 2 2 p l p l = ´ - æ è ç ö ø ÷ 0.05 0.4 0.4 sin 12 2 2 p p t x = - 0.05 sin ( ) 60 5 p p t x (b) y ( , 0.25 0.15) = ´ - ´ 0.05 0.15 0.25) sin (60 5 p p = - 0.05 .25 sin ( ) 9 1 p p = 0.05 7.75 sin ( ) p = 0.05 1.75 sin ( ) p = - 0.0354 m = - 3.54 cm (c) D D Df t T = = = 2 60 p f w p p 0.25 = 1 240 s = 4.2 ms Introductory Exercise 14.3 1. v T T m l Tl m = = = m / = ´ = 500 2 100 5 3 0.06 = 129.1 m/s 2. v T T A = = × m r = ´ ´ = - 0.98 9.8 10 10 10 3 6 m/s Introductory Exercise 14.4 1. I P r = = ´ = 4 1 4 1 1 4 2 2 p p p W m W m 2 ( ) / 2. For line source, I rl = r p 2 Þ I r µ 1 and as I A µ 2 Þ A r µ 1 2 | Waves & Motion x y AIEEE Corner ¢ Sub je c tive Ques ti ons (Level 1) 1. y x t ( , ) cos = 6.50 mm 2p p 28.0 cm 0.0360 s - æ è ç ç ö ø ÷ ÷ t = - æ è ç ö ø ÷ A x t T cos 2p l Þ A = 6.50 mm, l = 28.0 cm, n = = = - 1 1 1 T 0.036 s 27.78 Hz v = = ´ = - nl 28.0 cm 27.78s cm/s 1 778 = 7.78 m/s The wave is travelling along ( ) + ve x-axis. 2. y t x = - æ è ç ö ø ÷ 5 30 240 sin p = - æ è ç ö ø ÷ 5 30 8 sin p p t x = - A t kx sin ( ) w (a) y( , ) sin 2 0 5 3 0 8 2 = ´ - ´ æ è ç ö ø ÷ p p = - = - = - 5 4 5 2 35 sin p 3.5 cm (b) l p p p / = = = 2 2 8 16 k cm (c) v k = = = w p p 30 8 240 / cm/s (d) n w p p p = = = 2 30 2 15 Hz 3. y x t = - - - 3 314 1 1 cm 3.14 cm s sin ( ) = = - - 3 100 1 1 cm cm s sin ( ) p p x t = - A kx t sin ( ) w (a) ( ) max v A p = = ´ - w p 3 100 1 cm s = = = 300 3 p p cm/s m/s 9.4 m/ s (b) a y = - = - ´ - w p 2 1 2 100 3 ( ) s cm sin ( ) 6 111 p p - = - - = 300 105 0 p p sin ( ) 4. (a) D D D x v = f = = ´ ´ l p n p f p p 2 2 350 500 2 3 / = = = 7 60 7 50 p p m m 0.166 (b) D D D f p pn = = 2 2 T t t = ´ ´ - 2 500 10 3 p = = ° p 180 5. y x t kx t ( , ) ( ) = + + 6 3 2 w y x k x x ( , ) 0 6 3 6 3 2 2 2 = + = + Þ k = - 1 1 m Þ w = = ´ = - vk 4.5 m/s 4.5 rad/s 1 1 m Þ y x t x t ( , ) ( ) = - + 6 3 2 4.5 6. y x t = - æ è ç ö ø ÷ 1.0 2.0 0.01 sin p = - æ è ç ö ø ÷ 1.0 sin 4.0 0.02 2p x t = - æ è ç ö ø ÷ A x t T sin 2p l (a) A = 1.0 mm, l = = 4.0 cm, 0.02 T s (b) v y t A x t T p = = - - æ è ç ö ø ÷ ¶ ¶ w p l cos 2 = - - æ è ç ö ø ÷ 2 2 p p l A T x t T cos = - ´ - æ è ç ç ö ø ÷ ÷ 2 2 p p 1.0 mm 0.02 4.0 0.02 s s cos x t = - - æ è ç ç ö ø ÷ ÷ p p 10 m/s 2.0 cm 0.01 cos x t s v p ( ) 1.0 cm, 0.01s = - - æ è ç ö ø ÷ p p 10 1 2 m/s 0.01 0.01 cos = - = p p 10 2 0 m/ s m/ s cos (c) v p ( ) 3.0, 0.01 = - - æ è ç ö ø ÷ p p 1 0 3 2 1 c os = 0 m/ s Waves & Motion | 3 Page 4 14. Wave Motion Introductory Exercise 14.1 1. A func tion, f can rep re sent wave equa tion, if it sat isfy ¶ ¶ = ¶ ¶ 2 2 2 2 2 f t v f x For, y a t = sin , w ¶ ¶ = - = - 2 2 2 2 y t a t y w w w sin but, ¶ ¶ = 2 2 0 y x So, y do not represent wave equation. 2. y x t ae bx et ( , ) ( ) = - - 2 = - - ae kx t ( ) w 2 Þ k b = and w = e Þ v k c b = = w 3. y x t x t ( , ) ( ) = + + 1 1 4 2 w rep r e sent the given pulse, where, y x k x x ( , ) 0 1 1 1 1 2 2 2 = + = + Þ k = 1 and y x z x x ( , ) ( ) ( ) = + - = + - 1 1 2 1 1 1 2 2 w Þ w = 1 2 \ v k = = = w 1 2 1 / 0.5 m/s 4. y x t a b kx t = + + = + + 10 5 2 2 2 ( ) ( ) w Amplitude, y a b max = = = 10 5 2 m and k = 1; w = 2 v k = = w 2 m/s and is travelling in (–) x direction. 5. y kx t = - + 10 2 2 ( ) w y x k x x ( , ) 0 10 2 10 2 2 2 2 = + = + Þ k = 1 w = = vk 2 m/s ´ - 1 1 m = 2 rad/s Þ y x t = - + 10 2 2 2 ( ) Introductory Exercise 14.2 1. y x t ( , ) = 0.02 sin x t 0.05 0.01 + æ è ç ö ø ÷ m = + A kx t sin ( ) w m Þ A = 0.02 m, k = 1 0.05 m -1 , w = - 1 001 1 . s (a) v k = = w 0.05 0.01 m/s = 5 m/s (b) v y t A kx t p = ¶ ¶ = + w w cos ( ) v p ( , ) 0.2 0.3 0.02 0.01 = ´ 1 cos 0.2 0.5 0.3 0.01 + æ è ç ö ø ÷ = + 2 4 30 cos ( ) = 2 34 cos = - 2( 0.85) = - 1.7 m/s 2. Yes, ( ) ( ) max v A Ak k Ak v p = = × = w w 3. l = 4 cm, v = 40 cm/s (given) (a) n l = = = v 40 4 10 cm/s cm Hz (b) D D f p l = 2 x = ´ 2 4 p cm 2.5 cm = 5 4 p rad (c) D D D t T = = 2 1 2 p f pn f = ´ ´ 1 2 10 3 p p = 1 60 s (d) v v p p = ( ) max = - = - A A w p n 2 = - ´ ´ - 2 2 10 1 p cm s = - 40 p cm/s = - 1.26 cm/s 4. (a) y A t kx = - sin ( ) w = × - æ è ç ö ø ÷ A v t x sin 2 2 p l p l = ´ - æ è ç ö ø ÷ 0.05 0.4 0.4 sin 12 2 2 p p t x = - 0.05 sin ( ) 60 5 p p t x (b) y ( , 0.25 0.15) = ´ - ´ 0.05 0.15 0.25) sin (60 5 p p = - 0.05 .25 sin ( ) 9 1 p p = 0.05 7.75 sin ( ) p = 0.05 1.75 sin ( ) p = - 0.0354 m = - 3.54 cm (c) D D Df t T = = = 2 60 p f w p p 0.25 = 1 240 s = 4.2 ms Introductory Exercise 14.3 1. v T T m l Tl m = = = m / = ´ = 500 2 100 5 3 0.06 = 129.1 m/s 2. v T T A = = × m r = ´ ´ = - 0.98 9.8 10 10 10 3 6 m/s Introductory Exercise 14.4 1. I P r = = ´ = 4 1 4 1 1 4 2 2 p p p W m W m 2 ( ) / 2. For line source, I rl = r p 2 Þ I r µ 1 and as I A µ 2 Þ A r µ 1 2 | Waves & Motion x y AIEEE Corner ¢ Sub je c tive Ques ti ons (Level 1) 1. y x t ( , ) cos = 6.50 mm 2p p 28.0 cm 0.0360 s - æ è ç ç ö ø ÷ ÷ t = - æ è ç ö ø ÷ A x t T cos 2p l Þ A = 6.50 mm, l = 28.0 cm, n = = = - 1 1 1 T 0.036 s 27.78 Hz v = = ´ = - nl 28.0 cm 27.78s cm/s 1 778 = 7.78 m/s The wave is travelling along ( ) + ve x-axis. 2. y t x = - æ è ç ö ø ÷ 5 30 240 sin p = - æ è ç ö ø ÷ 5 30 8 sin p p t x = - A t kx sin ( ) w (a) y( , ) sin 2 0 5 3 0 8 2 = ´ - ´ æ è ç ö ø ÷ p p = - = - = - 5 4 5 2 35 sin p 3.5 cm (b) l p p p / = = = 2 2 8 16 k cm (c) v k = = = w p p 30 8 240 / cm/s (d) n w p p p = = = 2 30 2 15 Hz 3. y x t = - - - 3 314 1 1 cm 3.14 cm s sin ( ) = = - - 3 100 1 1 cm cm s sin ( ) p p x t = - A kx t sin ( ) w (a) ( ) max v A p = = ´ - w p 3 100 1 cm s = = = 300 3 p p cm/s m/s 9.4 m/ s (b) a y = - = - ´ - w p 2 1 2 100 3 ( ) s cm sin ( ) 6 111 p p - = - - = 300 105 0 p p sin ( ) 4. (a) D D D x v = f = = ´ ´ l p n p f p p 2 2 350 500 2 3 / = = = 7 60 7 50 p p m m 0.166 (b) D D D f p pn = = 2 2 T t t = ´ ´ - 2 500 10 3 p = = ° p 180 5. y x t kx t ( , ) ( ) = + + 6 3 2 w y x k x x ( , ) 0 6 3 6 3 2 2 2 = + = + Þ k = - 1 1 m Þ w = = ´ = - vk 4.5 m/s 4.5 rad/s 1 1 m Þ y x t x t ( , ) ( ) = - + 6 3 2 4.5 6. y x t = - æ è ç ö ø ÷ 1.0 2.0 0.01 sin p = - æ è ç ö ø ÷ 1.0 sin 4.0 0.02 2p x t = - æ è ç ö ø ÷ A x t T sin 2p l (a) A = 1.0 mm, l = = 4.0 cm, 0.02 T s (b) v y t A x t T p = = - - æ è ç ö ø ÷ ¶ ¶ w p l cos 2 = - - æ è ç ö ø ÷ 2 2 p p l A T x t T cos = - ´ - æ è ç ç ö ø ÷ ÷ 2 2 p p 1.0 mm 0.02 4.0 0.02 s s cos x t = - - æ è ç ç ö ø ÷ ÷ p p 10 m/s 2.0 cm 0.01 cos x t s v p ( ) 1.0 cm, 0.01s = - - æ è ç ö ø ÷ p p 10 1 2 m/s 0.01 0.01 cos = - = p p 10 2 0 m/ s m/ s cos (c) v p ( ) 3.0, 0.01 = - - æ è ç ö ø ÷ p p 1 0 3 2 1 c os = 0 m/ s Waves & Motion | 3 v p ( , ) cos 5.0 cm 0.01 ms s = - - æ è ç ö ø ÷ p p 10 5 2 1 = 0 m/s v p ( ) cos 7.0 cm, 0.01s m/s = - - æ è ç ö ø ÷ p p 10 7 2 1 = 0 m/s (d) v p ( ) 1.0 cm, 0.011s = - p 10 m/s cos p 1 2 - æ è ç ö ø ÷ 0.011 0.01 = - - æ è ç ö ø ÷ p p 10 1 12 cos 1.1 = - p p 10 cos 0.6 = - = p p 10 3 5 cos 9.7 cm/ s v p ( ) 1.0 cm, 0.012 s = - - æ è ç ö ø ÷ p 10 1 2 m / s 0 . 0 1 2 0 . 0 1 cos = - - p p 10 cos ( ) 0.5 1.2 = - = p p 10 cos 0.7 18.5 cm/s v p ( ) 1.0 cm, 0.013 s = - p 10 m/s cos cos p p p 1 2 10 - æ è ç ö ø ÷ = - 0.013 0.01 0.8 = 25.4 cm/s 7. (a) k = = = - 2 2 40 20 1 p l p p cm cm = 0.157 rad/cm T = = = 1 1 8 n s s 0.125 w pn p = = 2 16 rad/s = 50.26 rad/s v = = ´ = - nl 8 40 320 1 s cm cm/ s (b) y x t A kx t ( , ) cos ( ) = - w = - 15.0 cm 0.157 50.3 cos ( ) x t 8. A = 0.06m and 2.5 cm l = 20 Þ l = = 20 8 2.5 cm cm n l = = v 300 8 m/s cm = 3750 Hz y A kx t = - sin ( ) w = 0.06m sin 2 2 3750 p p 0.08 x t - ´ æ è ç ö ø ÷ = - - - 0.06 78.5 23561.9 m m s sin ( ) 1 1 x t 9. (a) n l = = = v 8.00 m/s 0.32 m 25 Hz T = = = 1 1 15 n s 0.043 Hz k = = = 2 2 p l p 0.32 19.63 rad/m m (b) y A kx t A x t T = + = + æ è ç ö ø ÷ cos ( ) cos w p l 2 = + æ è ç ç ö ø ÷ ÷ 0.07 0.32 0.04 s m m cos 2p x t (c) y = + æ è ç ö ø ÷ 0.07 0.36 0.32 0.15 0.04 m cos 2p = + æ è ç ö ø ÷ 0.07 m cos 2 9 8 30 8 p = 0.07 m cos 39 4 p = - æ è ç ö ø ÷ 0.07 m cos 10 4 p p = 0.07 m cos p 4 = 0.0495 m (d) D D D t T = f = f = + ´ 2 2 4 2 25 p pn p p p / = = 3 200 0.015 s s 10. v T T A Mg A = = = m r r = ´ ´ ´ ´ - 2 8920 10 3 2 9.8 3.14 1.2 ( ) = ´ ´ ´ ´ = 2 10 22 4 9.8 89.2 3.14 1.44 m/s 11. l n µ µ µ T M Þ l l 2 1 2 1 = M M = = = 8 2 4 2. Þ l l 2 1 2 = = 0.12 m. 4 | Waves & Motion Page 5 14. Wave Motion Introductory Exercise 14.1 1. A func tion, f can rep re sent wave equa tion, if it sat isfy ¶ ¶ = ¶ ¶ 2 2 2 2 2 f t v f x For, y a t = sin , w ¶ ¶ = - = - 2 2 2 2 y t a t y w w w sin but, ¶ ¶ = 2 2 0 y x So, y do not represent wave equation. 2. y x t ae bx et ( , ) ( ) = - - 2 = - - ae kx t ( ) w 2 Þ k b = and w = e Þ v k c b = = w 3. y x t x t ( , ) ( ) = + + 1 1 4 2 w rep r e sent the given pulse, where, y x k x x ( , ) 0 1 1 1 1 2 2 2 = + = + Þ k = 1 and y x z x x ( , ) ( ) ( ) = + - = + - 1 1 2 1 1 1 2 2 w Þ w = 1 2 \ v k = = = w 1 2 1 / 0.5 m/s 4. y x t a b kx t = + + = + + 10 5 2 2 2 ( ) ( ) w Amplitude, y a b max = = = 10 5 2 m and k = 1; w = 2 v k = = w 2 m/s and is travelling in (–) x direction. 5. y kx t = - + 10 2 2 ( ) w y x k x x ( , ) 0 10 2 10 2 2 2 2 = + = + Þ k = 1 w = = vk 2 m/s ´ - 1 1 m = 2 rad/s Þ y x t = - + 10 2 2 2 ( ) Introductory Exercise 14.2 1. y x t ( , ) = 0.02 sin x t 0.05 0.01 + æ è ç ö ø ÷ m = + A kx t sin ( ) w m Þ A = 0.02 m, k = 1 0.05 m -1 , w = - 1 001 1 . s (a) v k = = w 0.05 0.01 m/s = 5 m/s (b) v y t A kx t p = ¶ ¶ = + w w cos ( ) v p ( , ) 0.2 0.3 0.02 0.01 = ´ 1 cos 0.2 0.5 0.3 0.01 + æ è ç ö ø ÷ = + 2 4 30 cos ( ) = 2 34 cos = - 2( 0.85) = - 1.7 m/s 2. Yes, ( ) ( ) max v A Ak k Ak v p = = × = w w 3. l = 4 cm, v = 40 cm/s (given) (a) n l = = = v 40 4 10 cm/s cm Hz (b) D D f p l = 2 x = ´ 2 4 p cm 2.5 cm = 5 4 p rad (c) D D D t T = = 2 1 2 p f pn f = ´ ´ 1 2 10 3 p p = 1 60 s (d) v v p p = ( ) max = - = - A A w p n 2 = - ´ ´ - 2 2 10 1 p cm s = - 40 p cm/s = - 1.26 cm/s 4. (a) y A t kx = - sin ( ) w = × - æ è ç ö ø ÷ A v t x sin 2 2 p l p l = ´ - æ è ç ö ø ÷ 0.05 0.4 0.4 sin 12 2 2 p p t x = - 0.05 sin ( ) 60 5 p p t x (b) y ( , 0.25 0.15) = ´ - ´ 0.05 0.15 0.25) sin (60 5 p p = - 0.05 .25 sin ( ) 9 1 p p = 0.05 7.75 sin ( ) p = 0.05 1.75 sin ( ) p = - 0.0354 m = - 3.54 cm (c) D D Df t T = = = 2 60 p f w p p 0.25 = 1 240 s = 4.2 ms Introductory Exercise 14.3 1. v T T m l Tl m = = = m / = ´ = 500 2 100 5 3 0.06 = 129.1 m/s 2. v T T A = = × m r = ´ ´ = - 0.98 9.8 10 10 10 3 6 m/s Introductory Exercise 14.4 1. I P r = = ´ = 4 1 4 1 1 4 2 2 p p p W m W m 2 ( ) / 2. For line source, I rl = r p 2 Þ I r µ 1 and as I A µ 2 Þ A r µ 1 2 | Waves & Motion x y AIEEE Corner ¢ Sub je c tive Ques ti ons (Level 1) 1. y x t ( , ) cos = 6.50 mm 2p p 28.0 cm 0.0360 s - æ è ç ç ö ø ÷ ÷ t = - æ è ç ö ø ÷ A x t T cos 2p l Þ A = 6.50 mm, l = 28.0 cm, n = = = - 1 1 1 T 0.036 s 27.78 Hz v = = ´ = - nl 28.0 cm 27.78s cm/s 1 778 = 7.78 m/s The wave is travelling along ( ) + ve x-axis. 2. y t x = - æ è ç ö ø ÷ 5 30 240 sin p = - æ è ç ö ø ÷ 5 30 8 sin p p t x = - A t kx sin ( ) w (a) y( , ) sin 2 0 5 3 0 8 2 = ´ - ´ æ è ç ö ø ÷ p p = - = - = - 5 4 5 2 35 sin p 3.5 cm (b) l p p p / = = = 2 2 8 16 k cm (c) v k = = = w p p 30 8 240 / cm/s (d) n w p p p = = = 2 30 2 15 Hz 3. y x t = - - - 3 314 1 1 cm 3.14 cm s sin ( ) = = - - 3 100 1 1 cm cm s sin ( ) p p x t = - A kx t sin ( ) w (a) ( ) max v A p = = ´ - w p 3 100 1 cm s = = = 300 3 p p cm/s m/s 9.4 m/ s (b) a y = - = - ´ - w p 2 1 2 100 3 ( ) s cm sin ( ) 6 111 p p - = - - = 300 105 0 p p sin ( ) 4. (a) D D D x v = f = = ´ ´ l p n p f p p 2 2 350 500 2 3 / = = = 7 60 7 50 p p m m 0.166 (b) D D D f p pn = = 2 2 T t t = ´ ´ - 2 500 10 3 p = = ° p 180 5. y x t kx t ( , ) ( ) = + + 6 3 2 w y x k x x ( , ) 0 6 3 6 3 2 2 2 = + = + Þ k = - 1 1 m Þ w = = ´ = - vk 4.5 m/s 4.5 rad/s 1 1 m Þ y x t x t ( , ) ( ) = - + 6 3 2 4.5 6. y x t = - æ è ç ö ø ÷ 1.0 2.0 0.01 sin p = - æ è ç ö ø ÷ 1.0 sin 4.0 0.02 2p x t = - æ è ç ö ø ÷ A x t T sin 2p l (a) A = 1.0 mm, l = = 4.0 cm, 0.02 T s (b) v y t A x t T p = = - - æ è ç ö ø ÷ ¶ ¶ w p l cos 2 = - - æ è ç ö ø ÷ 2 2 p p l A T x t T cos = - ´ - æ è ç ç ö ø ÷ ÷ 2 2 p p 1.0 mm 0.02 4.0 0.02 s s cos x t = - - æ è ç ç ö ø ÷ ÷ p p 10 m/s 2.0 cm 0.01 cos x t s v p ( ) 1.0 cm, 0.01s = - - æ è ç ö ø ÷ p p 10 1 2 m/s 0.01 0.01 cos = - = p p 10 2 0 m/ s m/ s cos (c) v p ( ) 3.0, 0.01 = - - æ è ç ö ø ÷ p p 1 0 3 2 1 c os = 0 m/ s Waves & Motion | 3 v p ( , ) cos 5.0 cm 0.01 ms s = - - æ è ç ö ø ÷ p p 10 5 2 1 = 0 m/s v p ( ) cos 7.0 cm, 0.01s m/s = - - æ è ç ö ø ÷ p p 10 7 2 1 = 0 m/s (d) v p ( ) 1.0 cm, 0.011s = - p 10 m/s cos p 1 2 - æ è ç ö ø ÷ 0.011 0.01 = - - æ è ç ö ø ÷ p p 10 1 12 cos 1.1 = - p p 10 cos 0.6 = - = p p 10 3 5 cos 9.7 cm/ s v p ( ) 1.0 cm, 0.012 s = - - æ è ç ö ø ÷ p 10 1 2 m / s 0 . 0 1 2 0 . 0 1 cos = - - p p 10 cos ( ) 0.5 1.2 = - = p p 10 cos 0.7 18.5 cm/s v p ( ) 1.0 cm, 0.013 s = - p 10 m/s cos cos p p p 1 2 10 - æ è ç ö ø ÷ = - 0.013 0.01 0.8 = 25.4 cm/s 7. (a) k = = = - 2 2 40 20 1 p l p p cm cm = 0.157 rad/cm T = = = 1 1 8 n s s 0.125 w pn p = = 2 16 rad/s = 50.26 rad/s v = = ´ = - nl 8 40 320 1 s cm cm/ s (b) y x t A kx t ( , ) cos ( ) = - w = - 15.0 cm 0.157 50.3 cos ( ) x t 8. A = 0.06m and 2.5 cm l = 20 Þ l = = 20 8 2.5 cm cm n l = = v 300 8 m/s cm = 3750 Hz y A kx t = - sin ( ) w = 0.06m sin 2 2 3750 p p 0.08 x t - ´ æ è ç ö ø ÷ = - - - 0.06 78.5 23561.9 m m s sin ( ) 1 1 x t 9. (a) n l = = = v 8.00 m/s 0.32 m 25 Hz T = = = 1 1 15 n s 0.043 Hz k = = = 2 2 p l p 0.32 19.63 rad/m m (b) y A kx t A x t T = + = + æ è ç ö ø ÷ cos ( ) cos w p l 2 = + æ è ç ç ö ø ÷ ÷ 0.07 0.32 0.04 s m m cos 2p x t (c) y = + æ è ç ö ø ÷ 0.07 0.36 0.32 0.15 0.04 m cos 2p = + æ è ç ö ø ÷ 0.07 m cos 2 9 8 30 8 p = 0.07 m cos 39 4 p = - æ è ç ö ø ÷ 0.07 m cos 10 4 p p = 0.07 m cos p 4 = 0.0495 m (d) D D D t T = f = f = + ´ 2 2 4 2 25 p pn p p p / = = 3 200 0.015 s s 10. v T T A Mg A = = = m r r = ´ ´ ´ ´ - 2 8920 10 3 2 9.8 3.14 1.2 ( ) = ´ ´ ´ ´ = 2 10 22 4 9.8 89.2 3.14 1.44 m/s 11. l n µ µ µ T M Þ l l 2 1 2 1 = M M = = = 8 2 4 2. Þ l l 2 1 2 = = 0.12 m. 4 | Waves & Motion 12. T x L x g v x T x ( ) ( ) , ( ) ( ) = - = m m = - g L x ( ) dx g L x dt ( ) - = ; Let, L x y - = dx dy = - - = ò dy g y t L 0 \ t g y = - 1 12 1 0 / = 2 L g 13. (a) dm R T d w q 2 2 = sin m q w q R d R T d 2 2 2 = Þ w m 2 2 R T = \ Wave speed, v T R R = = = m w w 2 2 (b) Kink remains stationary when rope and kink moves in opposite sence ie . ., if rope is rotating anticlockwise then kink has to move clockwise. 14. x is be ing mea sured from lover end of the string \ m x dm x dx x x ( ) = = = ò ò m m 0 0 0 2 1 2 \ v x T x m x g ( ) ( ) ( ) = = m m = = 1 2 1 2 0 2 0 m m x g x gx Þ dx gx dt l t 1 2 0 0 ò ò = Þ t g l = ´ 2 2 0 \ t l g = 8 0 15. m = = dm dx kx Þ M dm kx dx kL = = = ò ò 0 2 2 1 2 Þ k M L = 2 2 v x T T kx TL Mx dx dt ( ) = = = = m 2 2 \ t dt Mx TL dx M TL L L = = = + ò ò + 2 2 1 2 1 2 0 2 1 2 1 = = 2 3 2 2 3 2 3 2 ML TL ML T 16. (a) v T Mg = = m m = ´ = 1.5 9.8 0.055 16.3 m/s (b) l n = = = v 16.3 m/s 0.136 120 / s m (c) l µ µ µ v T M i e . ., if M is doubled both speed and wavelength increases by a factor of 2 . 17. E I At a vAt = = 2 2 2 2 p n r = 2 2 2 2 p n r a A v t ( ) ( . ) = 2 2 2 2 p n m a l . = 2 2 2 2 p n a m = ´ ´ ´ ´ - 2 120 10 2 2 3 2 ( ) ( ) ( ) 3.14 0.16 ´ ´ - 80 10 3 = ´ - 582 10 6 J = = 582 mJ 0.58 mJ 18. P E t IA a A = = = 2 2 2 2 p n rn = 2 2 2 2 p n m a v = 2 2 2 2 p n m a T = ´ ´ 2 60 2 2 ( ( ) 3.14) ´ ´ ´ ´ - - ( ) 6 10 80 5 10 2 2 2 = ´ ´ = 4 60 2 ( ) 3.14 0.06 511.6 W Waves & Motion | 5 R T T dq dq xRead More
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