# SOLUTIONS TO CONCEPTS CHAPTER 15

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1 SOLUTIONS TO CONCEPTS CHAPTER v = 40 cm/sec As velocity of a wave is constant location of maximum after 5 sec = 40 5 = 00 cm along negative x-axis. [(x / a) (t / T)]. Given y = Ae a) [A] = [M 0 L 1 T 0 ], [T] = [M 0 L 0 T 1 ] [a] = [M 0 L 1 T 0 ] b) Wave speed, v = /T = a/t [Wave length = a] c) If y = f(t x/v) wave is traveling in positive direction and if y = f( t + x/v) wave is traveling in negative direction So, y = [(x / a) (t / T)] Ae = x (1/ T) t Ae x (1/ T) t a / T v = Ae i.e. y = f{t + (x / v)} d) Wave speed, v = a/t Max. of pulse at t = T is (a/t) T = a (negative x-axis) Max. of pulse at t = T = (a/t) T = a (along negative x-axis) So, the wave travels in negative x-direction. 3. At t = 1 sec, s 1 = vt = 10 1 = 10 cm t = sec, s = vt = 10 = 0 cm t = 3 sec, s 3 = vt = 10 3 = 30 cm 4. The pulse is given by, y = [(a 3 ) / {(x vt) + a }] a = 5 mm = 0.5 cm, v = 0 cm/s At t = 0s, y = a 3 / (x + a ) The graph between y and x can be plotted by taking different values of x. (left as exercise for the student) similarly, at t = 1 s, y = a 3 / {(x v) + a } and at t = s, y = a 3 / {(x v) + a } 5. At x = 0, f(t) = a sin (t/t) Wave speed = v = wavelength = vt (T = Time period) So, general equation of wave Y = A sin [(t/t) (x/vt)] [because y = f((t/t) (x/)) 6. At t = 0, g(x) = A sin (x/a) a) [M 0 L 1 T 0 ] = [L] a = [M 0 L 1 T 0 ] = [L] b) Wave speed = v Time period, T = a/v (a = wave length = ) General equation of wave y = A sin {(x/a) t/(a/v)} = A sin {(x vt) / a} 7. At t = t 0, g(x, t 0 ) = A sin (x/a) (1) For a wave traveling in the positive x-direction, the general equation is given by x t y = f a T Putting t = t 0 and comparing with equation (1), we get g(x, 0) = A sin {(x/a) + (t 0 /T)} g(x, t) = A sin {(x/a) + (t 0 /T) (t/t)} 15.1 y x

2 Chapter 15 As T = a/v (a = wave length, v = speed of the wave) y = = t 0 x t A sin a (a / v) (a / v) x v(t0 t) A sin a x v(t t 0 ) y = A sin a 8. The equation of the wave is given by y = (0.1 mm) sin [(31.4 m 1 )x +(314 s 1 )t] y = r sin {(x / )} + t) a) Negative x-direction b) k = 31.4 m 1 / = 31.4 = /31.4 = 0. mt = 0 cm Again, = 314 s 1 f = 314 f = 314 / = 314 / ( (3/14)} = 50 sec 1 wave speed, v = f = 0 50 = 1000 cm/s c) Max. displacement = 0.10 mm Max. velocity = a = = 3.14 cm/sec. 9. Wave speed, v = 0 m/s A = 0.0 cm = cm a) Equation of wave along the x-axis y = A sin (kx wt) k = / = / = cm 1 T = /v = /000 = 1/1000 sec = 10 3 sec = /T = 10 3 sec 1 So, the wave equation is, y = (0. cm)sin[( cm 1 )x ( 10 3 sec 1 )t] b) At x = cm, and t = 0, y = (0. cm) sin (/) = 0 v = r cos x = cos = 400 = 400 (3.14) = 156 cm/s = 400 cm/s = 4 m/s x t 10. Y = (1 mm) sin cm 0.01sec a) T = 0.01 = 0.0 sec = 0 ms = = 4 cm b) v = dy/dt = d/dt [sin (x/4 t/0.0)] = cos {x/4) (t/0.0)} 1/(0.0) v = 50 cos {(x/4) (t/0.0)} at x = 1 and t = 0.01 sec, v = 50 cos * [(1/4) (1/)] = 0 c) i) at x = 3 cm, t = 0.01 sec v = 50 cos (3/4 ½) = 0 ii) at x = 5 cm, t = 0.01 sec, v = 0 (putting the values) iii) at x = 7 cm, t = 0.01 sec, v = 0 at x = 1 cm and t = sec v = 50 cos {(1/4) (0.011/0.0)} = 50 cos (3/5) = 9.7 cm/sec (similarly the other two can be calculated) 11. Time period, T = 4 5 ms = = 10 s = cm = 4 cm frequency, f = 1/T = 1/( 10 ) = 50 s 1 = 50 Hz Wave speed = f = 4 50 m/s = 000 m/s = m/s 15.

3 Chapter Given that, v = 00 m/s a) Amplitude, A = 1 mm b) Wave length, = 4 cm c) wave number, n = / = ( 3.14)/4 = 1.57 cm 1 (wave number = k) d) frequency, f = 1/T = (6/)/0 = 0/4 = 5 Hz (where time period T = /v) 13. Wave speed = v = 10 m/sec Time period = T = 0 ms = = 10 sec a) wave length, = vt = = 0. m = 0 cm b) wave length, = 0 cm phase diff n = (/) x = ( / 0) 10 = rad y 1 = a sin (t kx) 1.5 = a sin (t kx) So, the displacement of the particle at a distance x = 10 cm. [ = x 10 ] is given by 0 y = a sin (t kx + ) a sin(t kx) = 1.5 mm displacement = 1.5 mm 14. mass = 5 g, length l = 64 cm mass per unit length = m = 5/64 g/cm Tension, T = 8N = dyne 15. V = 5 (T / m) ( )/ cm/s = 3 m/s a) Velocity of the wave, v = 5 (T / m) (16 10 )/ cm/sec Time taken to reach to the other end = 0/000 = 0.01 sec Time taken to see the pulse again in the original position = 0.01 = 0.0 sec b) At t = 0.01 s, there will be a though at the right end as it is reflected. 16. The crest reflects as a crest here, as the wire is traveling from denser to rarer medium. phase change = 0 a) To again original shape distance travelled by the wave S = = 40 cm. Wave speed, v = 0 m/s time = s/v = 40/0 = sec b) The wave regains its shape, after traveling a periodic distance = 30 = 60 cm Time period = 60/0 = 3 sec. c) Frequency, n = (1/3 sec 1 ) n = (1/l) (T / m) m = mass per unit length = 0.5 g/cm 1/3 = 1/( 30) (T / 0.5) T = = 00 dyne = 10 3 Newton. 17. Let v 1 = velocity in the 1 st string v 1 = (T / m 1) Because m 1 = mass per unit length = ( 1 a 1 l 1 / l 1 ) = 1 a 1 where a 1 = Area of cross section v 1 = (T / 1a 1) (1) Let v = velocity in the second string v = (T / m ) v = (T / a ) () 0 cm 30 cm Given that, v 1 = v (T / 1a 1) = (T / a ) (T/a 1 1 ) = 4(T/a ) 1 / = 1/4 1 : = 1 : 4 (because a 1 = a ) 15.3

4 Chapter m = mass per unit length = kg/mt Y = (0.0m) sin [(1.0 m 1 )x + (30 s 1 )t] Here, k = 1 m 1 = / = 30 s 1 = f velocity of the wave in the stretched string v = f = /k = 30/I = 30 m/s v = 4 T / m 30 (T /1.) 10 N) T = N T = Newton. 19. Amplitude, A = 1 cm, Tension T = 90 N Frequency, f = 00/ = 100 Hz Mass per unit length, m = 0.1 kg/mt a) V = T / m = 30 m/s = V/f = 30/100 = 0.3 m = 30 cm b) The wave equation y = (1 cm) cos (t/0.01 s) (x/30 cm) [because at x = 0, displacement is maximum] c) y = 1 cos (x/30 t/0.01) v = dy/dt = (1/0.01)sin {(x/30) (t/0.01)} a = dv/dt = {4 / (0.01) } cos {(x/30) (t/0.01)} When, x = 50 cm, t = 10 ms = s x = ( / 0.01) sin {(5/3) (0.01/0.01)} = (p/0.01) sin ( / 3) = (1/0.01) sin (4/3) = 00 sin (/3) = 00 x ( 3 / ) = 544 cm/s = 5.4 m/s Similarly a = {4 / (0.01) } cos {(5/3) 1} = ½ 10 5 cm/s km/s 0. l = 40 cm, mass = 10 g mass per unit length, m = 10 / 40 = 1/4 (g/cm) spring constant K = 160 N/m deflection = x = 1 cm = 0.01 m T = kx = = 1.6 N = dyne Again v = (T / m) = 4 (16 10 /(1/ 4) Time taken by the pulse to reach the spring t = 40/800 = 1/0 = 0/05 sec. 1. m 1 = m = 3. kg mass per unit length of AB = 10 g/mt = 0.01 kg.mt mass per unit length of CD = 8 g/mt = kg/mt for the string CD, T = 3. g v = (T / m) = = 8 10 cm/s = 800 cm/s 3 (3. 10)/ (3 10 )/ 8 = for the string AB, T = 3. g = 6.4 g = 64 N v = (T / m) = (64 / 0.01) 6400 = 80 m/s. Total length of string =.5 mt Mass per unit length m = T = g = 0 N Wave speed, v = (T / m) = = 10 3 kg/m /(10 ) 10 = 10 m/s = 100 m/s Time taken to reach the pully, t = (s/v) = /100 = 0.0 sec. 3. m = kg/m from the freebody diagram, T 4g 4a = 0 T = 4(a + g) = 48 N wave speed, v = (T / m) = 50 m/s = = 63 m/s 5 cm 4 kg kg B A D C mt m 1 m T g a = m/s 4g 4a 15.4

5 4. Let M = mass of the heavy ball (m = mass per unit length) Wave speed, v 1 = (T / m) = (Mg/ m) (because T = Mg) 60 = (Mg/m) Mg/ m = 60 (1) From the freebody diagram (), v = v = 6 = (T'/m) (Ma) 1/ 4 [(Ma) (Mg) ] m 1/ 1/ 4 [(Ma) (Mg) ] m 1/ (Mg) m (because T = = 6 () Eq(1) + Eq() (Mg/m) [m / g / (Ma) (Ma) (a g ) = g / (a + g ) = (Mg) ) (Mg) ] = 3600 / 3844 (a + 100) = 100 a = = 1.4 a = 1.4 / = a = 3.76 m/s Acce n of the car = 3.7 m/s 5. m = mass per unit length of the string R = Radius of the loop = angular velocity, V = linear velocity of the string Consider one half of the string as shown in figure. The half loop experiences cetrifugal force at every point, away from centre, which is balanced by tension T. Consider an element of angular part d at angle. Consider another T element symmetric to this centrifugal force experienced by the element = (mrd) R. ( Length of element = Rd, mass = mrd) Resolving into rectangular components net force on the two symmetric elements, DF = mr d sin [horizontal components cancels each other] So, total F = / mr sind = mr [ cos] mr 0 Again, T = mr T = mr Velocity of transverse vibration V = T / m = R = V So, the speed of the disturbance will be V. 6. a) m mass per unit of length of string consider an element at distance x from lower end. Here wt acting down ward = (mx)g = Tension in the string of upper part Velocity of transverse vibration = v = T /m = (mgx / m) (gx) b) For small displacement dx, dt = dx / (gx) Total time T = L 0 dx / gx (4L / g) c) Suppose after time t from start the pulse meet the particle at distance y from lower end. t = y 0 dx / gx (4y / g) Distance travelled by the particle in this time is (L y) c Ma d A B Chapter 15 T Mg (Motion) T T Mg (Rest) a (mrd)w R 4xl L-y y x T A T B

6 S ut + 1/ gt L y (1/)g { (4y / g) } {u = 0} L y = y 3y = L y = L/3. So, the particle meet at distance L/3 from lower end. 7. m A = kg/m, T A = 4.8 N V A = T / m = 0 m/s m B = kg/m, T B = 7.5 N V B = T / m = 5 m/s t = 0 in string A t 1 = ms = = 0.0 sec In 0.0 sec A has travelled = 0.4 mt Relative speed between A and B = 5 0 = 5 m/s Time taken for B for overtake A = s/v = 0.4/5 = 0.08 sec 8. r = 0.5 mm = mt f = 100 Hz, T = 100 N v = 100 m/s v = T /m v = (T/m) m = (T/v ) = 0.01 kg/m P ave = mvr f = (3.14) (0.01) 100 ( ) (100) watt = 49 mw. 9. A = 1 mm = 10 3 m, m = 6 g/m = kg/m T = 60 N, f = 00 Hz V = T /m = 100 m/s a) P average = mv A f = 0.47 W b) Length of the string is m. So, t = /100 = 0.0 sec. Energy = mvf A t = 9.46 mj. 30. f = 440 Hz, m = 0.01 kg/m, T = 49 N, r = m a) v = T /m = 70 m/s b) v = f = v/f = 16 cm c) P average = mvr f = 0.67 W. 31. Phase difference = / f and are same. So, is same. y 1 = r sin wt, y = rsin(wt + /) From the principle of superposition y = y 1 + y = r sin wt + r sin (wt + /) = r[sin wt + sin(wt + /)] = r[sin{(wt + wt + /)/} cos {(wt wt /)/}] y = r sin (wt + /4) cos ( /4) Resultant amplitude = r = 4 mm (because r = 4 mm) 3. The distance travelled by the pulses are shown below. t = 4 ms = s s = vt = = mm t = 8 ms = s s = vt = = 4 mm t = 6 ms = s s = 3 mm t = 1 ms = s s = = 6 mm The shape of the string at different times are shown in the figure. 33. f = 100 Hz, = cm = 10 m wave speed, v = f = m/s a) in sec 1 st wave has travelled x = = 0.03 m = path diff n corresponding phase difference, = x/ = { / ( 10 )} 0.03 = 3. b) Path different x = 4 cm = 0.04 m Chapter

7 Chapter 15 = (/)x = {(/ 10 ) 0.04} = 4. c) The waves have same frequency, same wavelength and same amplitude. Let, y 1 = r sin wt, y = r sin (wt + ) y = y 1 + y = r[sin wt + (wt + )] = r sin (wt + /) cos (/) resultant amplitude = r cos / So, when = 3, r = 10 3 m R res = ( 10 3 ) cos (3/) = 0 Again, when = 4, R res = ( 10 3 ) cos (4/) = 4 mm. 34. l = 1 m, V = 60 m/s fundamental frequency, f 0 = V/l = 30 sec 1 = 30 Hz. 35. l = m, f 0 = 100 Hz, T = 160 N f 0 = 1/ l (T / m) m = 1 g/m. So, the linear mass density is 1 g/m. 36. m = (4/80) g/ cm = kg/m T = 50 N, l = 80 cm = 0.8 m v = (T / m) = 100 m/s fundamental frequency f 0 = 1/ l (T / m) = 6.5 Hz First harmonic = 6.5 Hz f 4 = frequency of fourth harmonic = 4f 0 = F 3 = 50 Hz V = f = (v/f 4 ) = 40 cm. 37. l = 90 cm = 0.9 m m = (6/90) g/cm = (6/900) kg/mt f = Hz f = 1/ l (T / m) T = N = 1480 N. 38. First harmonic be f 0, second harmonic be f 1 f 1 = f 0 f 0 = f 1 / f 1 = 56 Hz 1 st harmonic or fundamental frequency f 0 = f 1 / = 56 / = 18 Hz / = 1.5 m = 3m (when fundamental wave is produced) Wave speed = V = f 0 Ql = 384 m/s. 39. l = 1.5 m, mass 1 g m = 1/1.5 g/m = kg/m T = 9 g = 90 N = 1.5 m, f 1 = /l T / m [for, second harmonic two loops are produced] f 1 = f 0 70 Hz. 40. A string of mass 40 g is attached to the tuning fork m = ( ) kg/m The fork vibrates with f = 18 Hz = 0.5 m v = f = = 64 m/s v = T /m T = v m = N 164 N. 41. This wire makes a resonant frequency of 40 Hz and 30 Hz. The fundamental frequency of the wire must be divisible by both 40 Hz and 30 Hz. a) So, the maximum value of fundamental frequency is 80 Hz. b) Wave speed, v = 40 m/s 80 = (1/l) m cm 9 kg 9 kg l

8 Chapter Let there be n loops in the 1 st case length of the wire, l = (n 1 )/ [ 1 = = 4 cm] So there are (n + 1) loops with the nd case length of the wire, l = {(n+1) / [ = 1.6 = 3. cm] (n 1) n 1 / = n 4 = (n + 1) (3.) n = 4 length of the string, l = (n 1 )/ = 8 cm. 43. Frequency of the tuning fork, f = 660 Hz Wave speed, v = 0 m/s = v/f = 1/3 m No.of loops = 3 a) So, f = (3/l)v l = 50 cm b) The equation of resultant stationary wave is given by y = A cos (x/ql) sin (vt/) y = (0.5 cm) cos (0.06 cm 1 ) sin (130 s 1 t) 44. l 1 = 30 cm = 0.3 m f 1 = 196 Hz, f = 0 Hz We know f (1/l) (as V is constant for a medium) f 1 l l = 6.7 cm f l 1 Again f 3 = 47 Hz f 3 l f l l l 3 = 0.4 m =.4 cm and l 3 = 0 cm 45. Fundamental frequency f 1 = 00 Hz Let l 4 Hz be nth harmonic F /F 1 = 14000/00 NF 1 /F 1 = 70 N = 70 The highest harmonic audible is 70 th harmonic. 46. The resonant frequencies of a string are f 1 = 90 Hz, f = 150 Hz, f 3 = 10 Hz a) The highest possible fundamental frequency of the string is f = 30 Hz [because f 1, f and f 3 are integral multiple of 30 Hz] b) The frequencies are f 1 = 3f, f = 5f, f 3 = 7f So, f 1, f and f 3 are 3 rd harmonic, 5 th harmonic and 7 th harmonic respectively. c) The frequencies in the string are f, f, 3f, 4f, 5f,. So, 3f = nd overtone and 3 rd harmonic 5f = 4 th overtone and 5 th harmonic 7f = 6 th overtone and 7 th harmonic d) length of the string is l = 80 cm f 1 = (3/l)v (v = velocity of the wave) 90 = {3/(80)} K K = (90 80) / 3 = 4800 cm/s = 48 m/s. 1 T 1 T1 47. Frequency f = f1 ld l D f 1 l n T l cm l 1.6 cm l Given that, T 1 /T =, r 1 / r = 3 = D 1 /D 1 1 f l D T f l D T So, f 1 : f = : 3 (l 1 = l = length of string) 15.8

9 48. Length of the rod = L = 40 cm = 0.4 m Mass of the rod m = 1. kg Let the 4.8 kg mass be placed at a distance x from the left end. Given that, f l = f r 1 T T l m l m l r T = T l T = 4 (1) r T l r From the freebody diagram, T l + T r = 60 N 4T r +T r = 60 N T r = 1 N and T l = 48 N Now taking moment about point A, T r (0.4) = 48x + 1 (0.) x = 5 cm So, the mass should be placed at a distance 5 cm from the left end. 49. s = 7.8 g/cm 3, A =.6 g/cm 3 m s = s A s = g/cm (m = mass per unit length) m A = A A A = g/cm = kg/m A node is always placed in the joint. Since aluminium and steel rod has same mass per unit length, velocity of wave in both of them is same Chapter 15 v = T /m 500/7 m/x For minimum frequency there would be maximum wavelength for maximum wavelength minimum no of loops are to be produced. maximum distance of a loop = 0 cm wavelength = = 0 = 40 cm = 0.4 m f = v/ = 180 Hz. 50. Fundamental frequency V = 1/l T / m T / m = vl [ T / m = velocity of wave] a) wavelength, = velocity / frequency = vl / v = l and wave number = K = / = /l = /l b) Therefore, equation of the stationary wave is y = A cos (x/) sin (Vt / L) L = A cos (x / l) sin (Vt / L) v = V/L [because v = (v/l)] 51. V = 00 m/s, A = 0.5 m a) The string is vibrating in its 1 st overtone = 1 = m f = v/ = 100 Hz b) The stationary wave equation is given by l = m x Vt y = A cos sin = (0.5 cm) cos [(m 1 )x] sin [(00 s 1 )t] 5. The stationary wave equation is given by y = (0.4 cm) sin [(0.314 cm 1)x] cos [(6.00 s 1 )t] a) = 600 f = 600 f = 300 Hz wavelength, = /0.314 = ( 3.14) / = 0 cm b) Therefore nodes are located at, 0, 10 cm, 0 cm, 30 cm c) Length of the string = 3/ = 3 0/ = 30 cm l d) y = 0.4 sin (0.314 x) cos (600 t) 0.4 sin {(/10)x} cos (600 t) since, and v are the wavelength and velocity of the waves that interfere to give this vibration = 0 cm T l A T l A Steel 40 cm 1N B B 48N C T r 80 cm 60 cm 0 cm T r C Aluminium

10 v= /k = 6000 cm/sec = 60 m/s 53. The equation of the standing wave is given by y = (0.4 cm) sin [(0.314 cm 1 )x] cos [(6.00 s 1 )t] k = = /10 / = /10 = 0 cm for smallest length of the string, as wavelength remains constant, the string should vibrate in fundamental frequency l = / = 0 cm / = 10 cm 54. L = 40 cm = 0.4 m, mass = 3. kg = kg mass per unit length, m = (3.)/(0.4) = kg/m change in length, L = = m strain = L/L = m f = 0 Hz f = 1 T 1 T T = N 3 l' m (0.4005) 8 10 Strain = 48.19/1 mm = Y = stress / strain = N/m 55. Let, density of the block Weight Vg where V = volume of block The same turning fork resonates with the string in the two cases 10 T wvg 11 ( w )Vg f 10 = l m l m As the f of tuning fork is same, f 10 Vg 11 ( w )Vg f l m l m w m 11 (because, w = 1 gm/cc) Chapter = kg/m l = length of rope = m M = mass = 80 gm = 0.8 kg mass per unit length = m = 0.08/ = 0.04 kg/m Tension T = 56 N l = /4 Velocity, V = T /m = 80 m/s Initial position For fundamental frequency, l = /4 = 4l = 8 m f = 80/8 = 10 Hz a) Therefore, the frequency of 1 st two overtones are 1 st overtone = 3f = 30 Hz nd overtone = 5f = 50 Hz b) 1 = 4l = 8 m Final position 1 = V/ f 1 =.67 m = V/f = 1.6 mt so, the wavelengths are 8 m,.67 m and 1.6 m respectively. 57. Initially because the end A is free, an antinode will be formed. So, l = Ql 1 / 4 Again, if the movable support is pushed to right by 10 m, so that the joint is placed on the pulley, a node will be formed there. So, l = / Since, the tension remains same in both the cases, velocity remains same. As the wavelength is reduced by half, the frequency will become twice as that of 10 Hz i.e. 40 Hz. L L rope string

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