Answers to test yoursel questions Topi.1 Osilliations 1 a A n osillation is any motion in whih the displaement o a partile rom a ixed point keeps hanging diretion and there is a periodiity in the motion i.e. the motion repeats in some way. In simple harmoni motion, the displaement rom an equilirium position and the aeleration are proportional and opposite eah other. It is an osillation sine we may deine the displaement o the partile rom the middle point and in that ase the displaement hanges diretion and the motion repeats. The motion is not simple harmoni however sine there is no aeleration that is proportional (and opposite) to the displaement. 3 It is an osillation sine the motion repeats. The motion is not simple harmoni however sine the aeleration is onstant and is not proportional (and opposite) to the displaement. a T he aeleration is opposite to the displaement so every time the partile is displaed there is a ore towards the equilirium position. The aeleration is not proportional to the displaement; i it were the graph would e a straight line through the origin. 5 a i It was not intended to ask aout the mass apologies! ii The period is. s; the partile is at one extreme position at t and again at t. s. This is hal a period. EP / J. 1.5 1..5. 1 3 5 7 t /s. Travelling waves The delay time etween you seeing the person next to you stand up and you standing up and the numer density o the people i.e. how many people per unit meter. For a ixed delay time, the loser the people are the aster the wave. 7 There is a disturane that travels through the lie o dominoes just as a disturane travels through a medium when a wave is present. You an inrease the speed y plaing them loser together. An experiment to investigate this might e to plae a numer o dominoes on a line o ixed length suh that the dominoes are a ixed distane d apart. We must give the same initial push to the irst domino (or example using a pendulum that is released rom a ixed height and strikes the domino at the same plae. We then measure time orm when the irst domino is hit until the last one is hit. Dividing the ixed distane y the time taken gives the speed o the pulse. We an then repeat with a dierent domino separation and see how the speed depends on the separation d. physis or the IB Diploma Camridge University Press 15 ANSWERS TO TEST YOURSELF QUESTIONS 1
a d e Wavelength the length o a ull wave; the distane etween two onseutive rests or troughs Period the time needed to produe one ull osillation or wave Amplitude the largest value o the displaement rom equilirium o an osillation Crest a point on a wave o maximum displaement Trough a point on a wave o minimum displaement Displaement /m λ amplitude A.5 1. 1.5. Distane /m period T Displaement /m 1 Time / ms 9 a I n wave motion displaement reers to the dierene in the value o a quantity suh as position, pressure, density et when the wave is present and when the wave is asent. In a transverse wave the displaement is at right angles to the diretion o energy transer, in a longitudinal it is parallel to the energy transer diretion. The alling stone imparts kineti energy to the water at the point o impat and so that water moves. It will ontinue moving (reating many ripples) until the energy is dissipated. d We must reall that the intensity o a wave is proportional to the square o the amplitude. The amplitude will derease or two reasons: irst, some energy is ound to e dissipated as the wave moves away and so the amplitude has to derease. Seond, even in the asene o any energy losses, the amplitude will still derease eause the waveronts get igger as they move away rom the point o impat o the ripple. The energy arried y the wave is now distriuted on a longer waveront and so the energy per unit waveront length dereases. The amplitude must then derease as well. 1 a From let to right: down, down, up. From let to right: up, up, down. 11 1 a λ λ v 33 1.9 m. 5 v 33 1.3 1 m. 5 1 3 ANSWERS TO TEST YOURSELF QUESTIONS physis or the IB Diploma Camridge University Press 15
13 a A wave in whih the displaement is parallel to the diretion o energy transerred y the wave. i x/m ii At x. m i 9 1 3 5 7 x/m ii The ompression is now at x 5. m. v 3 1 a 5 Hz λ. i A ompression ours at x.3 m. Moleules just to the let o this point have positive displaement and so move to the right. Moleules just to the right move to the let reating the ompression at x.3 m. ii By similar reasoning x.1 m is a point where a rareation ours..3 Wave harateristis 15 Adding the pulses point y point gives the ollowing diagram. 1 17 Adding the pulses point y point gives the ollowing diagram. t.5 s t 1. s t 1.5 s 1 unit 1 m 1 m 1 unit units 1 unit m 1 m 1 We add the pulses point y point. For example at x oth waves have zero displaement and so we get zero displaement or the sum. At x 1 m, the lue pulse has y.5 m and the red pulse has y.75 m. The sum is 1.5 m. At x m, the lue pulse has y and the red pulse has y 1. m. The sum is 1. m. At x 3 m, the lue pulse has y.5 m and the red pulse has y.7 m. The sum is. m and so on. 19 a A waveront is a surae on whih all points have the same phase. y waveronts ray x z λ λ physis or the IB Diploma Camridge University Press 15 ANSWERS TO TEST YOURSELF QUESTIONS 3
A ray is the diretion normal to waveronts that orresponds to the diretion o energy transer. soure o disturane point soure a a Polarised light is light in whih the eletri ield osillates on the same plane. Light an e polarised y passage through a polariser and y reletion o a n on-metalli surae. 1 In a polarised wave the displaement must e on the same plane. In a longitudinal wave the displaement is along the diretion o energy transer and so elongs to an ininity o planes at the same time. Hene it annot e polarised. a The light is not polarised. In the ase o unpolarised light inident on an analyser, the intensity o the transmitted light would e hal the inident intensity and so onstant as required in the question. Sine there is an orientation (all it X) o the analyser that makes the transmitted intensity zero, it ollows that the inident light was polarised in a diretion at right angles to the diretion X. Sine the intensity never eomes zero the light was not polarised. Sine the intensity varies however, it ollows that the inident light has unequal omponents in various diretions so it is partially polarised. 3 a This relates the transmitted intensity I to the inident intensity I when polarised light is inident and then transmitted through an analyser. The relation is I I os θ where θ is the angle etween the transmission axis and the diretion o the inident eletri ield. I os θ os 5. I a The light transmitted through the irst polariser will e polarised in a given diretion. The seond polariser s axis is at right angles to this diretion so the eletri ield has zero omponent along the axis o the seond polariser. Hene no light gets transmitted. Light will e transmitted sine now there will e a omponent o the eletri ield along the seond polariser s axis. The situation is now idential to a and so no light goes through.. Wave ehaviour 5 a From 1. sin 3 1. 53 sinθ we ind sinθ. 3. 1 n g 1. 9 1 m s g n 1. 53 The requeny in water is the same as that in air and so λ g s 3. a t 1. s 3. In this time, 1.. 1. 3. λ. 1 1. sin 3 1 θ sin. 39. 9. 1 53 λ. 1 1. 53 a n 7 7. 3 1 m. ull waves have een emitted. (Or, the wavelength is 7 3. 5. m and in a length o 3. m we an it. ull waves.) 7 5. ANSWERS TO TEST YOURSELF QUESTIONS physis or the IB Diploma Camridge University Press 15
7 First we ind the angle o reration (angle θ in the diagram).. m θ φ x d. 1. sin 1.5 sin θ, hene θ.3. This means that x 3. m. os. Now ϕ. 3 13. 7 and so d. sin 13. 7 1. m. Let θ e the angle o inidene rom air. The angle o reration will e larger than θ and so as θ inreases the angle o reration will eome 9 and so will not enter water. This happens when sinθ sin 9 1 3 θ sin 13. 1. 3 15 15 9 The diagram must e similar to the one elow. wavelength λ a 3 There is no appreiale diration here; the wave ontinues straight through the opening. 31 There is poor reeption eause o destrutive intererene etween the waves reahing the antenna diretly and those releting o the mountain. The path dierene is doule the distane etween the house and the mountain. The wave releting o the mountain will suer a phase dierene o π and so the ondition or destrutive intererene is d nλ. The smallest d (other than zero) orresponds to n 1 and so d m..5 Standing waves 3 A standing wave is a speial wave ormed when two idential traveling waves moving in opposite diretions meet and then superpose. This wave, unlike a traveling wave, has nodes i.e. points where the displaement is always zero. The antinodes, points where the displaement is the largest do not appear to e moving. A standing wave diers rom a traveling wave in that it does not transer energy and that the amplitude is variale. In a standing wave points in etween onseutive nodes have the same phase whereas in a travelling wave the phase hanges rom zero to π ater a distane o one wavelength. 33 A standing wave is ormed when two idential traveling waves moving in opposite diretions meet and then superpose. 3 a A node is a point in the medium where the displaement is always zero. An antinode is a point in the medium where the displaement, at some instant, will assume its maximum value. Speed reers to the speed o the travelling waves whose superposition gives the standing wave. physis or the IB Diploma Camridge University Press 15 ANSWERS TO TEST YOURSELF QUESTIONS 5
35 a We must distur the string with a requeny that is equal to the requeny o the seond harmoni. 3 The wavelength o the wave will remain the same (and equal to twie the length o the string). Sine the speed inreases y the requeny must do the same and so is 35 Hz. 37 The irst harmoni has wavelength L (L is the length o the string) and the seond a wavelength L. The ratio o the requenies is then sine the speed is the same. v 3 a The wavelength o the undamental is L 1. m. The requeny is then 5 Hz L The sound produed y the virations o the string will have the same requeny i.e. 5 Hz and so the 3 wavelength o sound will e λ 1. 51 m. 5 39 3 The wavelength o sound is λ 1. 11 m. Standing waves have wavelength given y λ L 3 n with n 1, 3, 5,. Thereore L 1. 11 n 1. 11 m L. This gives. m and 1.m or n 3 and n 5. n L. 1 a The wavelength is given y λ n n and also y λ 7. Hene. 7. 3 1 m s. The answer makes physial sense only i n 1 (the irst harmoni 7 n n n 1 is estalished) in whih ase 3 m s. The next harmoni will have wavelength L. n. L. n. With n 3 we get n L. m. a The wavelengths in the open tue are given y λ L n. The requenies o two onseutive harmonis are then n L, 3 n λ L and 3 ( n + 1). This means that L ( n + 1) 3 1 L n + 1. n + 1 1. n. n 1 n 5; we have the ith and sixth harmonis. 3 n n L 3 5 We get 3 L. 33. m. L 3 The two harmonis have the same requeny and hene the same wavelength. The wavelength o the irst harmoni in the open-open pipe is λ L X. The wavelength o the irst harmoni in the losed-open pipe is LX λ L Y. Hene LX LY. L Y ANSWERS TO TEST YOURSELF QUESTIONS physis or the IB Diploma Camridge University Press 15
With one step per seond you shake the up with a requeny o aout 1 Hz. In the irst harmoni mode the wavelength would e aout twie the diameter o the up i.e. 1 m (we have antinodes at eah end). This gives a 1 speed o v 1 1 1 m s. 5 a A standing wave is made up o two traveling waves. The speed o energy transer o the traveling waves is taken to e the speed o the standing wave. From y 5. os( 5π t) we dedue that the requeny o osillation o point P and hene also o the wave is 5π v 1. 5 Hz. The wavelength is then λ. m. Sine the diagram shows a seond harmoni π. 5 this is also the length o the string. The phase dierene is π and so y 5. os( 5πt + π ) 5. os( 5π t). a The hit reates a longitudinal wave that travels down the length o the rod and relets o the end. The releted waves pushes the hammer ak. s. 1 v 1. 3 1 m s 3 t. 1 1 We assume ree-ree end points and so the wavelength is given y. m. The requeny is then 1. 3 1 λ. 5. khz. physis or the IB Diploma Camridge University Press 15 ANSWERS TO TEST YOURSELF QUESTIONS 7