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Simple Harmonic Motion Worksheet: Displacement, Velocity, Acceleration Calculations
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9 Wave phenomena 9.1 Simple harmonic motion Name: ………………………………. Date: ………………………………. Displacement, velocity, and acceleration calculations 1 Assume simple harmonic motion in each example. 1 A spring-mass system is oscillating. A stopwatch is started as it travels through equilibrium. If its frequency is 5.0 Hz and its amplitude is 0.12 m, find: a its maximum displacement ........................................................................................................................................................................... ........................................................................................................................................................................... b its maximum velocity ........................................................................................................................................................................... ........................................................................................................................................................................... c its maximum acceleration ........................................................................................................................................................................... ........................................................................................................................................................................... d its equation of motion ........................................................................................................................................................................... ........................................................................................................................................................................... e its velocity at t = 0.15 s ........................................................................................................................................................................... ........................................................................................................................................................................... 2 A 0.45 kg object causes a vertical spring to stretch 0.16 m. What is the period of this combination when it is oscillating? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 A pendulum has a period of 1.23 s. Determine the length of this pendulum if is located near the surface of the earth. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 1 1 9.1 Simple harmonic motion 4 An object is vibrating with a frequency of 40.2 Hz and amplitude of 0.148 m. When its displacement is 0.111 m determine its velocity. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 5 An object is vibrating at a frequency of 45 Hz. When a stopwatch is started, the object is vibrating at its maximum displacement of 0.022 m. a State the equation of motion. ....................................................................................................................................................................... b What is the object’s displacement at 0.014 seconds? ....................................................................................................................................................................... ....................................................................................................................................................................... ....................................................................................................................................................................... 6 A spring’s constant is 12.2 N/m. The spring is attached to a vertical wall and rests on a horizontal, frictionless table. A mass of 1.25 kg is attached to the other end of the spring. When the mass is pulled back a maximum displacement of 0.0888 m and let go it oscillates back and forth, find: a the maximum velocity ....................................................................................................................................................................... ....................................................................................................................................................................... b the maximum acceleration ....................................................................................................................................................................... ....................................................................................................................................................................... 2 2 9 Wave phenomena 9.1 Simple harmonic motion Name: ………………………………. Date: ………………………………. Displacement, velocity, and acceleration calculations 2 Look at this graph and answer the questions below. 1 State the equation of the wave above. .................................................................................................................................................................................. 2 What is the amplitude? .................................................................................................................................................................................. 3 What is the period? .................................................................................................................................................................................. 4 What is the maximum speed? .................................................................................................................................................................................. 5 What is the maximum acceleration? .................................................................................................................................................................................. 6 What is the equation of the velocity graph? .................................................................................................................................................................................. 7 Sketch the corresponding acceleration-time graph. 3 1 9.1 Simple harmonic motion 8 Using the pendulum as an example, describe how you would set up an experiment to obtain a displacement of 0 m when time = 0 seconds. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 4 2 9 Wave phenomena 9.1 Simple harmonic motion Name: ………………………………. Date: ………………………………. Energy transfer 1 A system executing simple harmonic motion has a potential energy of 0.80 J when the displacement is at a maximum. a What is the kinetic energy of this system at equilibrium? ........................................................................................................................................................................... ........................................................................................................................................................................... b What is the speed of this system when the potential energy is 0.80 J? ........................................................................................................................................................................... ........................................................................................................................................................................... 2 The amplitude of a spring-mass system is 2.0 cm. When the displacement is 0.5 cm, what is the potential energy divided by the kinetic energy? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 If the amplitude of a spring-mass system is tripled, by how much does the total energy increase? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 a Which of the waves above shows speed-time and which shows kinetic energy-time for a body displaying simple harmonic motion? ........................................................................................................................................................................... ........................................................................................................................................................................... ........................................................................................................................................................................... ........................................................................................................................................................................... 5 1 9.1 Simple harmonic motion b When t = 0.3 seconds, what is the potential energy of the system? (ignore units) ....................................................................................................................................................................... ....................................................................................................................................................................... ....................................................................................................................................................................... ....................................................................................................................................................................... c Sketch on the axes above the potential and total energies as a function of time. ....................................................................................................................................................................... ....................................................................................................................................................................... ....................................................................................................................................................................... ....................................................................................................................................................................... 6 2 9 Wave phenomena 9.2 Single-slit diffraction Name: ………………………………. Date: ………………………………. Single-slit diffraction 1 1 A screen is placed 1.52 m in front of a single-slit that is 0.0000125 m wide. If the distance from the centre of the central maximum to the first minimum is 0.0652 m, what is the wavelength of the monochromatic light that was used as a source? .................................................................................................................................................................................. .................................................................................................................................................................................. 2 What is the angular width of the central maximum in question 1? .................................................................................................................................................................................. .................................................................................................................................................................................. 3 The angular width of the central maximum is 5.22o when light of wavelength 511 nm falls on a single-slit. What is the distance from the centre of the central maximum to the second minimum if the screen is 1.76 m from the slit? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 The central maximum is measured to be 0.0765 m wide when the screen is 1.84 m away from the slit. If the wavelength of the light source is 606 nm, what is the width of the slit? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 5 Look at the diagram below. The angle from the second minimum on the right to the second minimum on the left is 14.2o when the wavelength of the source is 488 nm as shown below. What is the width of the slit? .................................................................................................................................................................................. 7 1 9 Wave phenomena 9.2 Single-slit diffraction Name: ………………………………. Date: ………………………………. Single-slit diffraction 2 1 A monochromatic source has a wavelength of 422 nm. The light passes through a single-slit which is 1.27 m away from a screen. The width of the slit is 2.84 × 10–6 m. a Find the distance from the centre of the central maximum to the first minimum. ........................................................................................................................................................................... ........................................................................................................................................................................... b Find the width of the central maximum in metres. ........................................................................................................................................................................... ........................................................................................................................................................................... c Find the width of the central maximum in radians. ........................................................................................................................................................................... ........................................................................................................................................................................... d If the wavelength is increased by a factor of 1.50 does the distance from the centre of the central maximum to the first minimum go up by the same factor? Calculate this distance just to make sure. ........................................................................................................................................................................... ........................................................................................................................................................................... 2 The distance from the centre of the central maximum to the third minimum is 0.0128 m. If the width of the slit is 3.42 × 10–5 m and the wavelength of the light is 555 nm, find the distance from the slit to the screen. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 The same slit as above is 2.65 m away from the screen. What is the difference in the width of the central maximum (in metres) when light of 502 nm is replaced with light of 602 nm? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 8 1 9.2 Single-slit diffraction 4 What is the maximum width (in degrees) of the central maximum? .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 5 A light source shines through a single-slit of width 5.94 × 10–7 m. Determine the minimum wavelength that will produce no minimum fringes on the screen. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 6 A light source of wavelength 631 nm shines through a single-slit of width 2.94 × 10–6 m. What is the maximum number of minimum fringes on one side of the central maximum if the screen is 3.00 m away from the slit? .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 9 2 9 Wave phenomena 9.2 Single-slit diffraction Name: ………………………………. Date: ………………………………. Single-slit diffraction 3 1 The wavelength of a light source shining through a single-slit of width 2.34 × 10–5 m is 625 nm. If the screen is 4.44 m away, how far is it from the second minimum on the right to the third minimum on the left? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 2 What is the ratio of the width of the central maximum to the distance between the first and third minimum fringes on one side? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 The wavelength of a light source shining through a single-slit of width 4.34 × 10–5 m is 525 nm. If the screen is 2.44 m away, how far is it from the centre of the central maximum to the second minimum? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 The screen for problem three is shown below. Mark on the screen the locations for the first two minima on each side of the central maximum. 5 The angular width of the central maximum is 28.4°. If a light source of λ = 616 nm shines through the single-slit determine the width of the slit. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 10 1 9 Wave phenomena 9.3 Interference Name: ………………………………. Date: ………………………………. Diffraction grating worksheet 1 Light of wavelength 444 nm is incident on a diffraction grating that has 8560 lines per cm. What is the first order angle? .................................................................................................................................................................................. .................................................................................................................................................................................. 2 What is the second order angle in question 1? .................................................................................................................................................................................. .................................................................................................................................................................................. 3 What is the maximum number of orders in question 1? .................................................................................................................................................................................. .................................................................................................................................................................................. 4 Light of wavelength 712.0 nm produces a first order maximum at an angle of 41.40°. How many lines per cm are on this diffraction grating? .................................................................................................................................................................................. .................................................................................................................................................................................. 5 Light of wavelength 652 nm is incident upon a diffraction grating whose slits are 1.64 × 10–3 cm apart. If the grating is 1.47 m from the screen, what is the distance from the central dot to the first order dot on the screen? .................................................................................................................................................................................. .................................................................................................................................................................................. 6 Light of wavelength 631 nm is incident upon a diffraction grating. It produces a second order dot on the screen that is 1.46 m from the central dot when the screen is 1.98 m from the diffraction grating. A different wavelength produces a first order dot that is 0.562 m from the central dot on the screen when the screen is the same distance away from the diffraction grating. What is the wavelength of the second source? .................................................................................................................................................................................. .................................................................................................................................................................................. 7 How many lines per cm are on the diffraction grating in question 6? .................................................................................................................................................................................. .................................................................................................................................................................................. 11 1 9 Wave phenomena 9.3 Interference Name: ………………………………. Date: ………………………………. Thin film interference 1 A thin film of oil (n = 1.42) rests on water (n = 1.33). The viewer looking from above views a wavelength of 451 nm. Calculate the minimum film thickness. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 2 White light is incident upon a soap bubble of thickness 114 nm and index of refraction 1.36. What wavelength will be at the centre? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 Light of wavelength 503 nm is viewed directly above a soap film whose refractive index is 1.36. How thick is the film? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 Light of wavelength 484 nm is normally incident on an air film that is 605 nm thick between two glass plates. Show that a bright area will be seen. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 12 1 9 Wave phenomena 9.4 Resolution Name: ………………………………. Date: ………………………………. Resolution worksheet 1 1 The Hubble Space Telescope has a primary mirror of diameter 2.40 m. Assuming that the wavelength of a light source is 555 nm, determine the angle of resolution for the lens. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 2 If two objects are 2.28 × 1013 m away, what is their minimum separation if we are to see them with the Hubble Space Telescope? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 Two distant objects are 3760 m apart. What is the maximum distance from the Hubble Space Telescope if they are just resolved with 555 nm light? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 What is the angular resolution limit for a telescope that has a mirror diameter of 2.54 m if the wavelength of light is 525 nm? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 13 1 9.4 Resolution 5 Two twins, Monique and Unique have pupil diameters of 3.00 mm and x mm respectively. A vehicle has headlights 1.62 m apart and is coming towards them at night with the lights on. If there is a 122 m difference between when the twins can resolve the two headlights, what is the diameter of Unique’s pupils? Assume that the wavelength of light is 555 nm. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 6 What are the two possible distances that Unique resolves the headlights in question 5? .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 14 2 9 Wave phenomena 9.4 Resolution Name: ………………………………. Date: ………………………………. Resolution worksheet 2 1 Two identical yellow spheres each of radius 80.0 × 10–6 m are shown below. If the wavelength of the light is 575 nm and the spheres are just resolved, how far away from the spheres are you? Assume that your pupil diameter is 4.00 mm. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 2 If the spheres in question 1 are blue with a wavelength of 412 nm, would the distance away be greater? Calculate the answer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 What is the relationship between the two wavelengths and the two distances? Write an equation to show this. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 A student is playing a video game where he is detecting two objects moving towards a base. These objects are 4.08 m apart. The antenna’s radius for the radar detection system is 1.62 m and operates on a wavelength of 0.0222 m. How far away are the objects? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 15 1 9 Wave phenomena 9.5 Doppler effect Name: ………………………………. Date: ………………………………. Doppler effect worksheet 1 The diagram below shows two sources of identical visible wavelengths. The top source is stationary and the bottom source is moving away at 0.75 c. In the time that it takes the top wave to travel eight complete wavelengths to the observer, the bottom source reaches observer 2. Assuming that the amplitude is the same for each wave, sketch the resulting waves after time t has passed. Use the dots to help with your sketch. Explain why the bottom wave would appear to have a reddish tinge. Observer 1 X Source 1 . . . . . . . X Observer 2 X Source 2 . . . . 16 . . . X 1 9.4 Resolution 5 A scientist can detect with the naked eye two objects on a moon of a planet in our solar system that is 5.91 × 1011 m away. If the scientist has pupils of diameter 5.00 mm and the wavelength of the light emitted is 555 nm, how far apart are the objects? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 6 Ganymede is the largest moon in our solar system (diameter = 5.26 × 106 m). Do you think the scientist in question 5 has made a mistake? Explain. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 7 A diffraction grating has x lines per mm. Spectral lines of 615.40 nm and 615.74 nm are resolved in the third order spectrum using a beam with a diameter of 0.1 mm. Solve for x. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 17 2 9 Wave phenomena 9.5 Doppler effect Name: ………………………………. Date: ………………………………. Doppler effect worksheet 2 1 A stationary source gives off a tone of 603 Hz. An observer is approaching in a vehicle travelling at 25.1 ms–1. If the speed of sound is 348 ms–1, determine the apparent frequency heard by the observer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 2 A stationary source gives off a tone of 1640 Hz. An observer is driving away from the source in a vehicle travelling at 18.1 ms–1. If the speed of sound is 356 ms–1, determine the apparent frequency heard by the observer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 A stationary observer hears a car travelling towards her at 26.2 ms–1. If the car is emitting a frequency of 552 Hz and the speed of sound is 339 ms–1, determine the apparent frequency heard by the observer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 A stationary observer hears a car travelling away from her at 16.4 ms–1. If the car is emitting a frequency of 406 Hz and the speed of sound is 351 ms–1, determine the apparent frequency heard by the observer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 5 A car is moving away from a stationary observer at a speed of 23.9 ms–1. It emits a sound of frequency 865 Hz and the observer hears an apparent frequency of 811 Hz. What is the speed of sound in air at that time? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 6 A car is moving towards a stationary observer. The car emits a frequency of 616 Hz and the observer hears an apparent frequency of 659 Hz. If the speed of sound is 347 ms–1, find the speed of the car. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 18 1 9.5 Doppler effect 7 An observer in a vehicle is driving towards a stationary speaker at 17.7 ms–1 that is emitting a frequency. If the speed of sound is 364 ms–1 and the observer hears an apparent frequency of 559 Hz, what is the frequency that the speaker emits? .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 8 An observer in a vehicle hears an apparent frequency that is lower than the stationary source emitted frequency. Is the observer driving towards or away from the source? .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 19 2 9 Wave phenomena 9.5 Doppler effect Name: ………………………………. Date: ………………………………. Doppler effect worksheet 3 1 A stationary source gives off a tone of 851 Hz. An observer is approaching in a vehicle travelling at 34.5 ms–1. If the speed of sound is 342 ms–1, determine the apparent frequency heard by the observer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 2 A stationary source gives off a tone of 1880 Hz. An observer is driving away from the source in a vehicle travelling at 28.1 ms–1. If the speed of sound is 347 ms–1, determine the apparent frequency heard by the observer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 A stationary observer hears a car travelling towards her at 36.2 ms–1. If the car is emitting a frequency of 889 Hz and the speed of sound is 361 ms–1, determine the apparent frequency heard by the observer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 A stationary observer hears a car that is travelling away from her at 27.4 ms–1. If the car is emitting a frequency of 474 Hz and the speed of sound is 348 ms–1, determine the apparent frequency heard by the observer. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 5 A car is moving away from a stationary observer at a speed of 35.9 ms–1. It emits a sound of frequency 704 Hz and the observer hears an apparent frequency of 621 Hz. What is the speed of sound in air at that time? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 6 A race car is moving towards a stationary observer. The car emits a frequency of 446 Hz and the observer hears an apparent frequency of 519 Hz. If the speed of sound is 361 ms–1, find the speed of the car. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 20 1 9.5 Doppler effect 7 An observer in a vehicle is driving towards a speaker (at 19.6 ms–1) that is emitting a frequency. If the speed of sound is 348 ms–1 and the observer hears an apparent frequency of 629 Hz, what is the frequency that the speaker emits? .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 8 An observer in a vehicle hears an apparent frequency that is higher than the stationary source emitted frequency. Is the observer driving away from the source? .............................................................................................................................................................................. .............................................................................................................................................................................. .............................................................................................................................................................................. 21 2 9 Wave phenomena 9.5 Doppler effect Name: ………………………………. Date: ………………………………. Doppler effect worksheet 4 1 An observer is travelling in a car at 26.8 ms–1 towards a source that is moving towards the observer at 15.2 ms–1. If the speed of sound is 361 ms–1 and the frequency of the source is 444 Hz, what is the apparent frequency that the observer hears? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 2 A rotating body has a tangential speed at the edge of 3450 ms–1. If the source is emitting a wavelength of 656 nm, what are the maximum and minimum wavelengths expected due to Doppler shift? (ignore s.f.) .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 3 Microwaves of frequency 1.92 × 109 Hz experience a maximum frequency of 1.92006 × 109 Hz due to the Doppler effect. If these microwaves are incident on a moving object, determine the speed of the moving object. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 4 A transmitter/receiver makes an angle of 28.4° with the direction of blood flow in an organ in the human body in which ultrasound is transmitted. If the frequency of the ultrasound is 30.4 kHz and the speed of sound in body tissue is 1540 ms–1, find the blood flow rate when the change in frequency due to the Doppler effect is 28.9 Hz. .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 5 A galaxy emitting hydrogen at a wavelength of 671 nm is 15 nm greater than the wavelength measured in the laboratory. What is the speed of this galaxy compared with earth? .................................................................................................................................................................................. .................................................................................................................................................................................. .................................................................................................................................................................................. 22 1 QUESTIONS Questions 1 b) Explain why the magnitude of the tension in the string at the midpoint of the oscillation is greater than the weight of the pendulum bob. (IB) The variation with displacement x of the acceleration a of a vibrating object is shown below. 3000 c) The pendulum bob is moved to one side until its centre is 25 mm above its rest position and then released. a/m s−2 point of suspension rigid support 2000 1000 x/mm −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80 m −1000 −2000 25 mm −3000 pendulum bob a) State and explain two reasons why the graph indicates that the object is executing simple harmonic motion. (i) Show that the speed of the pendulum bob at the midpoint of the oscillation is 0.70 m s–1. b) Use data from the graph to show that the frequency of oscillation is 350 Hz. (ii) The mass of the pendulum bob is 0.057 kg. The centre of the pendulum bob is 0.80 m below the support. Calculate the magnitude of the tension in the string when the pendulum bob is vertically below the point of suspension. c) State the amplitude of the vibrations. (9 marks) (IB) a) A pendulum consists of a bob that is suspended from a rigid support by a light inextensible string. The pendulum bob is moved to one side and then released. The sketch graph shows how the displacement of the pendulum bob undergoing simple harmonic motion varies with time over one time period. 3 State the condition (relating to the net force acting on the particle) that is necessary for it to execute simple harmonic motion. displacement 0 (10 marks) (IB) a) A particle of mass m attached to a light spring is executing simple harmonic motion in a horizontal direction. b) The graph shows how the kinetic energy EK of the particle in (a) varies with the displacement x of the particle from equilibrium. 0 0.07 EK /J time 0.06 0.05 0.04 0.03 Copy the sketch graph and on it clearly label 0.02 (i) a point at which the acceleration of the pendulum bob is a maximum. 23 x/m 05 04 0. 03 0. 0. 0. 01 0 0. (ii) a point at which the speed of the pendulum bob is a maximum. 02 0.01 − 0. 05 − 0. 04 − 0. 03 − 0. 02 − 0. 01 2 387 9 WAVE P H E NOM E N A ( A H L) (i) On a copy of the axes above, sketch a graph to show how the potential energy of the particle varies with the displacement x. Q Z ϕW (ii) The mass of the particle is 0.30 kg. Use data from the graph to show that the frequency f of oscillation of the particle is 2.0 Hz. b Y X P (8 marks) 4 (IB) a) Describe what is meant by the diffraction of light. The angle ϕ is small. b) A parallel beam of monochromatic light from a laser is incident on a narrow slit. The diffracted light emerging from the slit is incident on a screen. a) On a copy of the diagram, label the half angular width θ of the central maximum of the diffraction pattern. b) State and explain an expression, in terms of λ, for the path difference ZW between the rays ZP and XP. screen slit parallel light wavelength 620 nm 0.40 mm c) Deduce that the half angular width θ is given by the expression λ θ=_ b d) In a certain demonstration of single slit diffraction, λ = 450 nm, b = 0.15 mm and the screen is a long way from the slits. C 1.9 m The centre of the diffraction pattern produced on the screen is at C. Sketch a graph to show how the intensity I of the light on the screen varies with the distance d from C. c) The slit width is 0.40 mm and it is 1.9 m from the screen. The wavelength of the light is 620 nm. Determine the width of the central maximum on the screen. (8 marks) 5 screen slit (IB) Plane wavefronts of monochromatic light are incident on a narrow, rectangular slit whose width b is comparable to the wavelength λ of the light. After passing through the slit, the light is brought to a focus on a screen. Calculate the angular width of the central maximum of the diffraction pattern on the screen. (8 marks) 6 (IB) Monochromatic parallel light is incident on two slits of equal width and close together. After passing through the slits, the light is brought to a focus on a screen. The diagram below shows the intensity distribution of the light on the screen. I The line XY, normal to the plane of the slit, is drawn from the centre of the slit to the screen. The points P and Q are the first points of minimum intensity as measured from point Y. A The diagram also shows two rays of light incident on the screen at point P. Ray ZP leaves one edge of the slit and ray XP leaves the centre of the slit. 388 24 B distance along the screen a) Light from the same source is incident on many slits of the same width as the widths of the slits above. On a copy of the diagram, draw a possible new intensity distribution of the light between the points A and B on the screen. QUESTIONS a) State the phase change that occurs when light is reflected from A parallel beam of light of wavelength 450 nm is incident at right angles on a diffraction grating. The slit spacing of the diffraction grating is 1.25 × 10–6 m. (i) surface A (ii) surface B. b) Determine the angle between the central maximum and first order principal maximum formed by the grating. The light incident on the plastic has a wavelength of 620 nm. The refractive index of the plastic is 1.4. (4 marks) 7 b) Calculate the minimum thickness of the film needed for the light reflected from surface A and surface B to undergo destructive interference. (IB) Light of wavelength 590 nm is incident normally on a diffraction grating, as shown below. grating 6.0 × 105 lines per metre (5 marks) 9 first order light wavelength 590 nm (IB) The two point sources A and B emit light of the same frequency. The light is incident on a rectangular narrow slit and, after passing through the slit, is brought to a focus on the screen. zero order first order The grating has 6.0 × 10 5 lines per metre. a) Determine the total number of orders of diffracted light, including the zero order, that can be observed. A b) The incident light is replaced by a beam of light consisting of two wavelengths, 590 nm and 589 nm. B point sources slit State two observable differences between a first-order spectrum and a second-order spectrum of the diffracted light. screen a) B is covered. Sketch a graph to show how the intensity I of the light from A varies with distance along the screen. Label the curve you have drawn A. (6 marks) 8 (IB) Monochromatic light is incident on a thin film of transparent plastic as shown below. b) B is now uncovered. The images of A and B on the screen are just resolved. Using your axes, sketch a graph to show how the intensity I of the light from B varies with distance along the screen. Label this curve B. C monochromatic light A c) The bright star Sirius A is accompanied by a much fainter star, Sirius B. The mean distance of the stars from Earth is 8.1 × 1016 m. Under ideal atmospheric conditions, a telescope with an objective lens of diameter 25 cm can just resolve the stars as two separate images. B film The plastic film is in air. Assuming that the average wavelength emitted by the stars is 500 nm, estimate the apparent, linear separation of the two stars. Light is partially reflected at both surface A and surface B of the film. (6 marks) 389 25 9 WAVE P H E NOM E N A ( A H L) a) Explain, using a diagram, any difference between f ’and f. 10 (IB) a) Explain what is meant by the term resolvance with regards to a diffraction grating. b) A grating with a resolvance of 2000 is used in an attempt to separate the red lines in the spectra of hydrogen and deuterium. (i) The incident beam has a width of 0.2 mm. For the first order spectrum, how many lines per mm must the grating have? (ii) Explain whether or not the grating is capable of resolving the hydrogen lines which have wavelengths 656.3 nm and 656.1 nm. (6 marks) 11 A source of sound approaches a stationary observer. The speed of the emitted sound and its wavelength, measured at the source, are v and λ respectively. Compare the wave speed and wavelength, as measured by the observer, with v and λ . Explain your answers. (4 marks) 12 The sound emitted by a car’s horn has frequency f, as measured by the driver. An observer moves towards the stationary car at constant speed and measures the frequency of the sound to be f ’. 390 26 b) The frequency f is 3.00 × 102 Hz. An observer moves towards the stationary car at a constant speed of 15.0 m s–1. Calculate the observed frequency f ’ of the sound. The speed of sound in air is 3.30 × 102 m s-1. (5 marks) 13 (IB) The wavelength diagram shown below represents three lines in the emission spectrum sample of calcium in a laboratory. A B C wavelength A distant star is known to be moving directly away from the Earth at a speed of 0.1c. The light emitted from the star contains the emission spectra of calcium. Copy the diagram and sketch the emission spectrum of the star as observed in the laboratory. Label the lines that correspond to A, B, and C with the letters A*, B*, and C*. Numerical values of the wavelengths are not required. (3 marks) IB Physics Assessment paper: 9 Wave phenomena (AHL) 1. A sound emitting source moves along a straight line with speed v relative to an observer at rest. The speed of sound relative to the medium is c. The observer measures the speed of sound emitted by the source as A. c. B. c + v. C. c – v. D. v – c. (1) 2. A source of sound emits waves of wavelength λ, period T and speed v when at rest. The source moves away from a stationary observer at speed V, relative to the observer. The wavelength of the sound waves, as measured by the observer is A. λ + vT. B. λ – vT. C. λ +VT. D. λ – VT. (1) 3. In order that the interference between the waves emitted by two light sources can be observed, it is essential that the sources must emit waves that A. have the same amplitude. B. are in phase. C. have the same colour. D. have a constant phase difference between them. (1) 4. A police car, sounding its siren, is travelling at constant speed towards a stationary observer. The sound emitted by the siren is of constant frequency. The frequency of the sound as heard by the observer is higher than that heard by the driver of the police car. The reason for this is that A. the wavefronts received by the observer are closer together than the wavefronts received by the driver. B. the speed of the wavefronts is greater as measured by the observer than by the driver. C. the speed of the wavefronts is less as measured by the observer than by the driver. D. the wavefronts received by the observer are further apart than the wavefronts received by the driver. (1) © Oxford University Press 2015 www.oxfordsecondary.co.uk/acknowledgements 27 1 IB Physics Assessment paper: 9 Wave phenomena (AHL) 5. A stationary source emits sound of frequency f0. An observer is moving towards the source at constant speed along the path indicated by the dotted line. The observer passes very close to the source at time T. Which one of the following graphs best shows the variation with time t of the frequency f heard by the observer? (1) 6. A wooden block is at rest on a horizontal frictionless surface. A horizontal spring is attached between the block and a rigid support. The block is displaced to the right by an amount X and is then released. The period of oscillations is T and the total energy of the system is E. X For an initial displacement of which of the following shows the best estimate 2 for the period of oscillations and the total energy of the system? A. Period T B. T C. T 2 T 2 D. Total energy E 2 E 4 E 2 E 4 (1) © Oxford University Press 2015 www.oxfordsecondary.co.uk/acknowledgements 28 2 IB Physics Assessment paper: 9 Wave phenomena (AHL) 7. Two lamps producing light of the same colour are placed close to one another. A two source interference pattern is not observed because A. the lamps do not emit light of a single frequency. B. the phase difference between the light from the lamps is continually changing. C. the intensity of the light emitted by the lamps is not the same. D. the two lamps are not exact point sources. (1) 8. Light from a double slit arrangement produces bright and dark fringes on a screen in the region near point P, as indicated in the diagram below. The light from the two slits has equal amplitudes on reaching point P. Which one of the following gives the change, if any, in the appearance of the bright and the dark fringes when the amplitude of the light wave from one slit is reduced? A. B. C. D. Bright fringes Remains the same Becomes less bright Becomes less bright Remains the same Dark fringes Remains the same Remains the same Becomes more bright Becomes more bright (1) 9. Waterwaves of wavelength 2.0 m are produced by two sources S1 and S2. The sources vibrate in phase. Point P is 1 m from S1 and 3 m from S2. S1 alone and S2 alone each produce a wave of amplitude a at P. Which one of the following is the amplitude of the resultant wave at point P when S1 and S2 are both emitting waves? A. 2a B. a C. 1 a 2 D. Zero (1) © Oxford University Press 2015 www.oxfordsecondary.co.uk/acknowledgements 29 3 IB Physics Assessment paper: 9 Wave phenomena (AHL) 10. In a double-slit experiment using coherent light of wavelength λ, the central bright fringe is observed on a screen at point O, as shown below. At point P, the path difference between light arriving at P from the two slits is 4λ. Which of the following correctly describes the observed fringe pattern? nature of fringe at P A. B. C. D. bright bright dark dark number of dark fringes between O and P 3 4 3 4 (1) © Oxford University Press 2015 www.oxfordsecondary.co.uk/acknowledgements 30 4
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