Projects Now for some choice! The experiments in the Projects segment have been collected to demonstrate a variety of physical principles and make use of a large suite of experimental measuring techniques and analytical tools. No continuous theme connects the experiments; rather, they are self-contained. A simple, but important physical principle or procedure forms the basis of each. You will find that many of the projects are more open-ended than those in other segments. This has been a conscious decision by those developing the laboratory course, with the aim of developing your skills in experimental design and evaluation. Don’t feel as though you are expected to meet the challenge without help - your demonstrators have been chosen for their ability to encourage you in this process. Organisation In this laboratory you and your partner will make a selection from the available experiments. The equipment is randomly arranged throughout the room so you will move from table to table during the segment. Upon your arrival in the laboratory, inspect all the apparatus supplied for your particular experiment. See how you can optimally manage it to reduce errors during the data collection. Rearrangement of the relevant equations (given as part of some experiments) invariably suggests the most sensitive way of calculating the final quantity required. Both demonstrators in the laboratory are available to help you improve your data collection and analysis, but only one will be concerned with the marking of your book. Both partners should be involved in the planning, execution and calculation phases of all experiments, but particularly so in this laboratory. As in the other laboratories, your logbooks must be written independently. Before you leave the laboratory each week, select a number of experiments (2 or 3) that you would want to do the following week. Inform your demonstrator of your preferred option and s/he will make a record of it in the Projects Booking Book. Make sure you remember your selection (write it in your prac-manual, maybe) as this is the experiment you will be doing in the next week. Where class sizes permit, a maximum of two students may do any one experiment in a given week. Preparation The notes provided in this manual are not the pracs themselves—these will be provided to you in the laboratory. They consist instead of a brief synopsis of the experiment and the prelab exercises that you should complete before you arrive. These prelab exercises are designed to assist you in understanding the experiment. In addition to the assessment contribution, students attempting the exercises will be able to walk into the laboratory with some pre-prepared questions for the demonstrator and will thus find themselves with more time available to complete the prac. Remember, you will be give credit for an honest attempt at the prelab exercises. They are designed to start your thinking. If you require a copy of the experiments that you have completed you should inform your demonstrator at the end of the laboratory session and s/he will make one available to you. Physics 121/2 and 141/2 Laboratory Manual P-1 P-2 Physics 121/2 and 141/2 Laboratory Manual Experiment 1 Physics of the guitar Note that you may not choose this experiment and Experiment 13, The Sonometer. References 121/2: Section 17.2. 141/2: Sections 17.11 to 17.12; 18.6; 31.4 (Electric Guitars). In this experiment you will investigate some of the physical and musical principles behind the functioning and construction of the guitar. In order to do this you will need to learn some basic music theory from the lab notes provided to you in the class. In these notes we will pre-empt the experiment by examining the vibrations of a guitar string from a purely geometrical viewpoint. When a guitar string is plucked it vibrates in all of the modes available to it. Some of these modes are prohibited by physical constraints and die out almost instantly. Other modes are strongly favoured by the geometry etc and are sustained for a long time. These are the notes we hear when a guitar string is plucked, and it turns out that these are the standing waves of the string. The first (or fundamental) standing wave may be represented as: n=1 L = 2L You can see this as the amplitude envelope of the string as it vibrates: note that the ends must have zero amplitude as they correspond to the fixed ends of the string. Also note that the whole of the string oscillates in phase: it is not a travelling wave. We can thus identify by inspection the wavelength of the fundamental mode as twice the length of the string, ie =2L. Similarly, the second harmonic can be represented by: n=2 =L L Prelab Exercise: Draw the first four standing waves on a string of length L with n = 1,2,3,4. Write down their wavelengths in terms of n and L. Hence derive a general expression for the wavelength of the nth harmonic. = = = = In general,= Physics 121/2 and 141/2 Laboratory Manual P-3 Experiment 2 The speed of sound In this experiment you will measure the speed of sound using time of flight techniques. That is, you will directly measure the time of propagation of a sound wave over a variety of distances to determine the rate at which the sound wave travels. A spark generator will be used to produce a burst of sound which will be detected by the microphone at some position S at some time later. The amount of time later that the sound is detected will allow a measurement of the speed of propagation of the sound wave. Schematically, this will occur as in the diagram below: travelling sound wave spark microphone s? s? As you can see from the picture, there is some difficulty in determining S accurately. Since the time we measure will indicate when the sound is detected by the microphone, it appears that we need to know exactly where the sound is detected. Apart from the fact that this is a tricky technical question, requiring much knowledge about the construction of this particular microphone, it is also quite Zen: where is the microphone? Is it at the front or the rear of the piezo crystal which creates the voltage output? These objections may seem to be overstated, but they highlight a fundamental measuring problem: sometimes it is technically impossible to measure a quantity precisely. In this case we have an (unremovable) systematic error (look up Notes on Confidence Limits in Experimental Physics if you are unsure what a systematic error is.) Nevertheless, it may be possible to obtain an extremely good result by being clever about the experiment. In order to see how this might be done it is necessary to look at the analysis of the experimental data. In the above example we were going to find the speed of sound using v = S / t. We can write the S that we measure, SM, as a combination of the ‘true’ S, ST, (which we mortals cannot know) plus some extra amount S which represents the amount by which we were wrong. That is, SM = ST + S. Then the result we would get from the above calculation would be vM = vT + S / t. Thus our result would be fundamentally wrong. (Note that this result becomes more accurate as we let S (and thus t) become large.) The challenge facing us is to wonder whether we might find the true v, provided we do not have to do too much work to get it. Part of the clue to doing this may be found in the above expression, SM = ST + S. Notice that if we move the microphone to a second position, S2M will relate to S2T with the same S. That is, S2M = S2T +S. From this we might notice that the difference between these two measurements, S, is independent of S. That is; S = S2M – S1M = (S2T +S) – (S1T +S) = S2T – S1T So, while our measurements of the distances are clearly incorrect, the difference between them remains absolutely correct. Thus, if our analysis were to rely only on the change of distance, we might expect it to give the correct result. This can be done by rethinking our analysis of the dS velocity of sound: let us now use v . dt Of course, this velocity is the gradient of an S vs t graph. Let us look at our graph, to see what is going on… P-4 Physics 121/2 and 141/2 Laboratory Manual SM = ST + S S ( t , SM ) ST S t It can be seen from this view of the data that an analysis based on the gradient will be independent of the systematic error S. In fact, there is no longer any good reason to minimise the systematic error, S: the value of v obtained will be ‘good’ however large S. Returning to the ‘old’ way of getting v, ie v = S / t, we can interpret this calculation as the gradient of a line from the origin to any point on the SM line. Thus we can see that this result has little to do with the speed of sound but that it will return a more accurate value of v if the measured data point is at (large S, large t). Prelab Question: Will the result be similarly affected by a systematic error in t? (In order to answer this conclusively you may need to go through the algebra, as above.) Experiment 3 Microwave optics References 121/2: Section 39.1, 40.1 and 37.4. 141/2: Section 17.11, 37.3 to 37.4 and 34.6. The electromagnetic spectrum consists of all possible electromagnetic waves satisfying v = c = f . Thus, visible light is an electromagnetic wave with properties that are reasonably familiar to us: reflection, refraction, diffraction etc. The microwaves you will use have a frequency of approximately 1010 Hz, which is about 1/100,000th that of visible light. Prelab Question: What is the approximate wavelength of the microwaves that you will use? Physics 121/2 and 141/2 Laboratory Manual P-5 As a consequence of the macroscopic wavelength of these microwaves you will be able to directly measure the electric field component of the microwaves, and you will be able to investigate and interpret light-like phenomena on a macroscopic scale. In particular, you will directly measure and interpret the polarisation of electromagnetic radiation (please refer to your textbook if you are not familiar with polarisation). You will also set up a standing microwave field to measure the wavelength of the microwaves. Further experiments examining the diffraction of microwaves and the reflection, transmission and absorption of microwaves by various materials will also be available. Experiment 4 Equipotentials and field lines References 121/2: Sections 23.3 and 25.4. 141/2: Sections 23.3 and 25.3. In this experiment you will study the field which surrounds an electric charge. This electric field, E, has very similar properties to that of the gravitational field, g, differing only in that: electric charges, q, can be either positive or negative, whereas gravitational ‘charges’, m, are always positive opposite electric charges attract and like gravitational charges attract. The ‘strength’ of the electric field is far greater than that of the gravitational field. These differences, however small, produce the world we see. This world is generally populated by electrically neutral objects (down to the atomic scale) due to the attraction of unlike charges. The microscopic size of this scale is testament to the greater inherent strength of the electric force. Gravitationally we see the existence of large clumps of matter (planets, galaxies, etc). If the gravitational force were comparatively as strong as the electric force we would feel significant gravitational attraction to small objects, which we do not. The upshot of these properties, comparatively speaking, is that we can, with the aid of a voltage supply, observe the local (near the field source) and global (far away from the field source) properties of the electric field with a relatively small experiment. To understand the physics of the electric field it can be useful to exploit the similarity with the gravitational field. As a preparatory exercise, consider the gravitational field: Prelab Question: If U is the gravitational potential energy on a ‘flat’ earth (U = m g h) show that the field lines (g) are at right-angles to the equipotential surfaces. Hint: P-6 Write out the condition for an arbitrary vector lying on an equipotential surface, and show that this vector is always at right-angles to g. (recall that an equipotential surface is the collection of points with U = 0) Physics 121/2 and 141/2 Laboratory Manual Experiment 5 The speed of light References 121/2: Section 34.3. 141/2: Sections 33.5 to 33.7. Using Maxwell’s equations it can be shown that light is an electromagnetic wave with a speed of propagation given by: 1 c 0 0 where 0 and 0 are the permittivity and permeability of free space. These quantities provide a measure of the response of space to B and E fields. Inasmuch as this is so, it should not be surprising to see that light, as an electromagnetic wave, has a speed which is dependent only on these quantities. In this experiment you will use a resonant inductor-capacitor ( L-C ) circuit to measure the product ., from which you can obtain a measurement of the speed of light. In these notes we hope to give you some understanding of the functioning of the L-C circuit shown below. + + Signal Generator L C (Inductor) - C.R.O. - In this circuit the inductor (L) and capacitor (C) are in parallel. This means that their effect on the circuit is similar to that of resistors in parallel, i.e.(for our purposes): 1 RTOTAL 1 RCAPACITOR 1 RINDUCTOR So, if we know how the C and L act in the presence of oscillating voltages, we will know how this circuit behaves. Frequency dependence of the ‘resistance’ of the capacitor In the electronics laboratory you will have seen that a capacitor is an open circuit (break in the wire) which provides no resistance to rapidly varying voltages. Recall that this is because the plates of the capacitor are charged up by the applied voltage, and that if this potential changes rapidly enough the capacitor does not get to be ‘fully charged’, and thus does not become significantly resistive. In fact, the characteristic of the capacitor is such that its effective ‘resistance’ is inversely proportional to the frequency of the applied voltage. That is: RCAPACITOR 1 2fC Frequency dependence of the ‘resistance’ of the inductor The inductor is simply a piece of wire wrapped into a large coil. When a slowly varying (approximately DC) voltage is applied to it, it acts as a piece of wire. When the applied voltage is rapidly varying, however, the current sets up a magnetic field in the coil which acts to oppose a change in the applied voltage. Thus the inductor becomes ‘resistive’ at high frequencies. (Note Physics 121/2 and 141/2 Laboratory Manual P-7 that this occurs for the slowly varying voltage also, but that the decay time of the B field is much quicker than the rate of change of the applied voltage, and thus the effect of the decaying B field is minimal.) Thus the characteristic of the inductor is such that its effective ‘resistance’ is proportional to the frequency of the applied voltage. That is: RINDUCTOR 2fL Graphically these become (note the logarithmic scale chosen for the frequency axis) 80 60 R capacitor (effective) C = 3.2 microF Effective Resistance () R inductor (effective) L = 6.4 mH 40 R total (effective) 20 0 100 Frequency (Hz) 1000 10000 The parallel resistance formula stated earlier really just means that for parallel resistors it is the lower of the two resistors which passes the most current and thus dominates the overall conductivity of the circuit. Thus we can see that at high f the capacitor becomes unresistive and for low f the inductor is unresistive. Somewhere in the middle the total R is a maximum. Prelab Exercise: If the effective resistance of the circuit is given by: R EFFECTIVE 1 2fC 2fL 1 Show that the resistance of the circuit will be a maximum when the frequency satisfies: f 1 2 LC From this it can be seen that, if L and C can be related to and , then a measurement of the speed of light might be obtained by measuring the resonant frequency of the LC circuit. A similar argument applies to the series LC circuit which you will use to determine the speed of light. P-8 Physics 121/2 and 141/2 Laboratory Manual Experiment 6 The saw tooth wave generator References 121/2: None - see notes below. 141/2: Sections 28.8 and 28.7. In this experiment you will investigate a circuit which can be used to provide the time base signal for the CRO. In the process of doing this it is hoped that your understanding of the operation of the CRO will be extended. In addition you will investigate the properties of an electronic device known as a neon diode. Prelab Exercise: Before you attempt to build this circuit it is useful to appreciate exactly what the time base circuit does. Therefore it is suggested that you read the introductory notes for the projects experiment The Cathode Ray Oscilloscope (Experiment 15). Briefly summarise the role of the time-base circuit here, and then complete the prelab exercises for Experiment 15.. Experiment 7 The pendulum References 121/2: Sections 5.2 - 5.3, 6.2; Chapter 15 (introduction), Section 15.1; 15.3 and 15.4. 141/2: Sections 5.5, 5.6 (Weight and Tension); 16.1, 16.2; 16.4 and16.6. In this experiment you will use a pendulum to measure g, the acceleration due to gravity. NOTE: the following (full) derivation is provided for the interested student only. You are not required to memorise the details, but must follow its overall structure. m g cos m g sin mg Physics 121/2 and 141/2 Laboratory Manual P-9 Derivation: the period of a pendulum for 1 From the above diagram it can be seen that the restoring force on the pendulum has the form: F ( ) mg sin Let us suppose for now that the approximation sin is valid. Then we have: F ( ) mg Now, we also have from first principles that: F ma m so that: g Prelab Exercise: Confirm by substitution that: ( t) sin t g is a possible solution to this equation. This equation describes SHM with a period T given by T 2 . g Using this equation you will try to find a value of g by measuring T for a number of pendulum lengths, . In obtaining this result you will hope that the approximation sin is a valid in this case. For an approximation to be experimentally valid it must not induce an error in the result which is larger than the experimental error associated with the apparatus. Prelab Exercise: P-10 Sketch a graph of y = x and y = sin x (x in radians) on the same axes to show that the approximation x sin x may be valid for a sufficiently small range of x. Physics 121/2 and 141/2 Laboratory Manual To check that the approximation sin is valid it is necessary to look at the implications of the approximation explicitly. Suppose that the length of the pendulum can be measured to 2 parts in 1000 accuracy (i.e. 2 mm in 1 m). Roughly what do you think will be the maximum that you can swing the pendulum at while still obtaining an accurate value of g ? Experiment 8 The acceleration due to gravity References 121/2: Chapter 2 (introduction); Sections 2.1 - 2.6. 141/2: Sections 2.1 - 2.8 (2.7 and 2.8 in particular). In this experiment you will use time-of-flight techniques and some interesting analysis to determine the acceleration due to gravity. The experiment consists of timing the flight of a bob as it falls through a number of distances. The bob will not ‘start’ from rest, but will have the same initial velocity for each drop. Conceptually, the ball will follow the path shown in the diagram below: Ball Released: t<0 s<0 v=0 Ball passes first sensor: t=0 s=0 v0 0 Ball passes second sensor: t=t s=s Physics 121/2 and 141/2 Laboratory Manual P-11 Prelab Exercise: Starting from: a d2s dt 2 g ( constant ) for a mass falling in a constant gravitational field, derive the equation of motion describing the position of a particle at any time t. In the course of this experiment you will measure (s, t) data (as shown above) for the mass. Using your equation of motion, consider how you might create a linear graph whose gradient is proportional to g. Describe your strategy here. Consider the effect of a systematic error in s on this graph, and think about how you might analyse your data so that your result is independent of this error. (Look up Notes on Confidence Limits in Experimental Physics if you are unsure what a systematic error is.) Experiment 9 Mechanical resonance References 121/2: Chapter 15 (introduction); Sections 15.1 - 15.2, 15.6. 141/2: Sections 16.1, 16.8, 16.9. All objects have a natural frequency of (mechanical) resonance which depends largely on their shape, composition and configuration. A swing, for example, has a natural frequency dependent only on its length. If the swing is pushed at a frequency which is not (an integer fraction of) its natural frequency it will not swing very highly. If, however, the swing is pushed at the ‘right’ rate it will attain a large amplitude of oscillation (and ‘fun’ will be had). In this experiment you will investigate the resonant behaviour of a spring - mass system. You will relate this behaviour to properties of the spring (k) and the mass (m). This will be done by measuring k as the gradient of an F vs x graph and by using a mass bar driver system to drive (= push) the mass at resonant and off-resonant frequencies. P-12 Physics 121/2 and 141/2 Laboratory Manual Prelab Exercise: Show that: k x(t ) A sin t m F mx m d2x dt 2 is a solution to the spring equation: kx and thus that the period of the oscillation is given by: T 2 m . k Experiment 10 Measuring the wavelength of light with a ruler References Diffraction gratings are not discussed in detail in either text book, however, the following references should help with the principles of interference and diffraction. 121/2: Chapter 39 (introduction); Sections 39.4 - 39.5; Chapter 40 (introduction); Section 40.1. 141/2: Sections 36.1, 36.4, 37.1 - 37.4, 37.7. In this experiment you will measure the wavelength of red laser light (approximately 6 10-7m.) with a ½ mm graduated ruler to an accuracy as high as 0.1 % ! This can be achieved by using the ruler as a reflection diffraction grating (shown to the right) and observing the resultant diffraction pattern. Xn n n Xc Metal ruler i laser L Physics 121/2 and 141/2 Laboratory Manual P-13 The light incident on the steel ruler is scattered by the engravings, each of which becomes a source of spherical (Huygen) wavefronts. This scattering occurs at many sites along the ruler and the light is allowed to recombine at the screen. As the light from neighbouring scratches on the ruler travels different distances there will be a phase difference between the scattered wavefronts and there will thus be interference. Let us consider the light incident on the ruler in more detail: We can use an approximation here that the outgoing (n) waves are parallel because the screen is far away, i.e. L>>d.) Prelab Questions: If the incident light is coherent (in phase), what must be the path difference - in terms of the wavelength - for the light arriving at the screen to be in phase also? Is this the condition for constructive or destructive interference? Using the geometry of the experiment, write an expression for the path difference in terms of i, n and d. When you do this experiment you will find that the central maximum (n = 0) is readily distinguished from the other maxima. Given that you will be measuring only n and n, can you work out how you will eventually determine n ? (Note that i is fixed but that you cannot directly measure it with great accuracy. Can you calculate i, from your knowledge of 0 ? P-14 Physics 121/2 and 141/2 Laboratory Manual Experiment 11 X-ray diffraction: (lattice spacing of NaCl) References Prelab exercise: Read appendices F and G for background on x-ray production and mounting crystals in the 580 Tel-X-ometer. 121/2: Page I3 (not very useful however - use the reference below). 141/2: Section 37.9. In this experiment you will use x-rays – very short wavelength, high energy electromagnetic radiation – to determine the separation of neighbouring atoms in a crystal of NaCl. In order to do this you will need to understand the processes of Bragg diffraction and x-ray production. The following notes will help you to teach yourself about Bragg diffraction… Bragg diffraction A NaCl crystal has a regular (face centred) cubic structure, as shown here. For the sake of simplicity it is useful to confine our treatment of the physics involved so that it lies entirely in one plane. This is achieved by orienting the NaCl crystal so that its crystal planes coincide with the plane of the x-ray source and detector. If we do this then the crystal looks like (from ‘above’): d Each of the Na and Cl atoms will scatter the x-rays in all directions, but because of the regular spacing of the atoms in the lattice, there will be interference of these scattered x-rays. This occurs because the x-rays interact with all of the atoms in the crystal, and thus there are many scattered beams. This process involves exactly the same physics as other wave interference phenomena you are familiar with: when the path difference between the scattered beams is an integer number of wavelengths, constructive interference will result and (if the radiation is in the visible region) a bright spot will appear. In practice, there will be many combinations of and having a maximum intensity of x-rays. Measurement of the lattice spacing is achieved by further restricting the geometry of the experiment. In order to find out how we may do this, let us think through the process of Bragg diffraction. Physics 121/2 and 141/2 Laboratory Manual P-15 Consider the crystal to be a collection of rows of atoms. By doing this we can break our analysis into two parts: we can look at the diffraction from one row of atoms and then we can consider the diffraction from a collection of these rows. Considering the x-rays incident on one of these rows: 2 1 B D A Note that in general because the x-rays are scattered. C d Note that path 1 travels a distance AD further and a distance BC shorter than path 2. We can see that there will be a constructive interference for any , where this path difference (AD-BC) is an integer number of wavelengths. Prelab exercise Draw an expanded version of the diagram above and use it to show that the path difference is given by: P.D. = AD–BC = d (cos – cos ) = n (nI) In principle, this argument may be used to determine the structure of the crystal. As we have discussed earlier, however, successive layers of the crystal will also diffract and may cause a destructive interference at the same , . If this were to happen, the interference peak would be missed and the results incorrect. In order to avoid this, we can force = so that we always have a constructive interference from this process: by doing this we will always be observing the zeroth order diffraction, of reflection. If this is done we will be able to observe the interference from neighbouring layers of the crystal without having to worry about how these two effects compete. So, let us consider diffraction from the collection of rows that make up the crystal: d A C reflecting ‘layers’ of crystal B P-16 Physics 121/2 and 141/2 Laboratory Manual As before, the x-rays scattered from adjacent rows will be in phase only if the path difference is an integer number of wavelengths. i.e.: P.D. = ABC = 2 AB = 2 d sin = n (nI) This is the Bragg condition we will use in this experiment: by holding = we can look at the interference of x-rays scattered from neighbouring planes within the crystal and determine the lattice spacing, d. It is possibly instructive to note that, by treating the crystal in this way we have transformed it from a single three dimensional diffracting volume into a diffraction-grating like part (a row of scattering objects) and a thin-film like part (where the path difference is due to the separation of the crystal planes). If you do not understand how this analysis works, it is recommended that you read these notes again. Ask your demonstrator a little while after the practical begins if you have any questions. Experiment 12 Single-slit diffraction References 121/2: Chapter 40 (introduction); Section 40.1. 141/2: Sections 37.1 - 37.4. In this experiment you will use a laser and a photo-detector to determine the width of a narrow slit by examining the diffraction pattern produced by the slit. After doing this you will be able to measure the width of your hair by considering it to be an inverse slit, which (interestingly) produces the same diffraction pattern as a slit. To detect the maxima and minima in the diffraction pattern, you will use a photosensitive resistor which has a resistance that is approximately inversely proportional to the light intensity illuminating it. The photoresistor is enclosed in a lightproof container, with a small circular aperture, as shown in fig. 1. Only light entering from (nearly) perpendicular to the detector face will reach the photo-resistor: thus, stray light from the room should not affect your results. Figure 1 : the construction of the photo-sensitive resistor ohm meter LASER slide photo-sensitive resistor Physics 121/2 and 141/2 Laboratory Manual P-17 Prelab Question: It may be apparent that the diameter of the aperture will affect your ability to locate the exact positions of the maxima and minima of the diffraction pattern. With reference to the diagram, discuss the relationship between the relative sizes of the aperture/diffraction pattern and your ability to accurately measure the position of the features of the diffraction pattern. (It may be useful to consider the limiting physics by increasing / decreasing and examining the problems first encountered in each of these limits.) I DETECTOR Figure 2: Single slit diffraction pattern and the detector with collecting aperture . Experiment 13 Note: The sonometer You may not chose this experiment and Experiment 1, Physics of the Guitar. References 121/2: Sections 16.3 and 16.6. 141/2: Sections 17.6 and 17.11 to 17.12. In this experiment you will investigate the vibrations of a wire and you will try to determine how the physical variables constraining the wire, such as length, mass density and tension affect the resonant frequency of the wire. As an introduction to these concepts it is useful to examine the vibrations of a string as determined by the geometry of the situation. When a string is plucked it vibrates in all of the modes available to it. Some of these modes are prohibited by physical constraints and die out relatively quickly. Other modes are strongly favoured by the geometry etc and are sustained. These are the notes we hear when a guitar string is plucked, and it turns out that these are the standing waves of the string. The first, or fundamental, standing wave may be represented as: P-18 Physics 121/2 and 141/2 Laboratory Manual n=1 L = 2L You can see this as the amplitude envelope of the string as it vibrates: note that the ends are stationary, corresponding to the fixed ends of the string, and also that the string oscillates in phase: it is not a travelling wave. We can thus identify the wavelength of the fundamental mode as twice the length of the string, ie = 2L. Similarly, the second harmonic can be represented by: n=2 =L L Prelab Exercise: Draw the first four standing waves on a string of length L with n = 1,2,3,4. Write down their wavelengths in terms of n and L. Hence derive a general expression for the wavelength of the nth harmonic. = = = = In general,= Experiment 14 Mechanical measurement of the velocity of light It is not immediately obvious from observation that the speed of light is finite. In 1675 Olaf Röemer measured the speed of light by observing that the period of Jupiter’s moons varied. As the moon orbits Jupiter, it is eclipsed by the planet for a time. Röemer noticed that the duration of these eclipses was shorter when the Earth was moving towards Jupiter than when the Earth was moving away. This time difference was attributed to the distance that the earth had travelled while Jupiter’s moon was eclipsed, and to the finite velocity of light. Thus the velocity of light was first measured to be 2 108 m/s. This value is too slow by a factor of 1/3 primarily because of inaccurate estimates of the planetary distances involved. Physics 121/2 and 141/2 Laboratory Manual P-19 Earth orbiting the sun with (relatively) constant speed SOL moon of Jupiter having constant period position of Earth when moon reemerges position of Earth when moon eclipsed Jupiter time measured for moons occlusion is shortened by a time t = x / c. In 1862 Foucault devised an experiment which employed a rapidly rotating mirror and some accurate distances to measure the velocity of light, which Michelson then modified to measure the velocity of light in two orthogonal directions simultaneously. In this experiment you will measure the speed of light (wholly within the laboratory!) using apparatus similar to that used by Foucault in his experiment. In this introduction we will consider a simplified version of the Foucault apparatus which demonstrates the physics of the measurement of the speed of light. The apparatus is configured as in the diagram below. Rotating Mirror (RM) LASER Fixed Mirror (FM) In this arrangement the laser is directed at the rotating mirror (RM) which, as it rotates, sends the beam spinning around the room. In one position the beam will be reflected towards the fixed mirror (FM) which has been adjusted so that it will reflect the light directly back to the RM. By the time the beam has travelled back to the RM it will have rotated a bit, and thus the light reflected from it will not go directly back to the laser. Rather, it will make an angle with the incident beam which depends directly on the speed of light. In order to measure the speed of light we need to relate the displacement of the return beam to the geometry of the experiment. screen R LASER (RM) x D (FM) P-20 Physics 121/2 and 141/2 Laboratory Manual Prelab Exercise: Using the above diagram to define your variables, calculate the following quantities: How long will the light take to travel from the RM to the FM and back? Through what angle will the mirror rotate in this time? Note that (2 t) is independent of In order to work out the angle that the return beam makes with the incident beam one must note that the beam is deflected by twice the angle that the mirror has rotated through, as in the diagram to the right. 2t incoming beam –t –t return beam Prelab Exercise: Continue your analysis of the apparatus by finding: the angle that the light is reflected from the mirror (relative to the incident light beam); Where will the beam be found on a screen placed at some distance R from the RM? After this calculation you should have the following expression: x 4DR c Evaluate the expected size of the displacement x for the (representative) values D = 1 m, L = 1 m, = 2 f = 6000 rad/sec, c = 3108 m/sec. While this apparatus forms the basis of the equipment on your bench, it turns out that some extra components need to be added in order to use it to make real measurements. This is because the displacement x that you have calculated above is very small: so small that the return beam does not fully emerge from the incident beam! When you arrive in the laboratory you will find a full explanation of the modified apparatus in the lab notes. Physics 121/2 and 141/2 Laboratory Manual P-21 Experiment 15 The Cathode Ray Oscilloscope The CRO is a complicated-looking instrument that does a very simple job. Fundamentally, the CRO performs the same job as the needle-pointer voltmeters that you have already used. What makes the CRO a powerful instrument is the extra functions that it makes available to the user. However, in order to understand these features it is useful to understand how the CRO functions as a voltmeter first. In order to explain the operation of the CRO we will attempt to construct it bit by bit. Inside the CRO there is an electron gun which aims a thin beam of electrons at a phosphorescent screen. When this beam hits the screen, the screen glows. In between the electron gun and the screen there is a pair of deflection plates. These are two metal plates, one on each side of the beam, which can have a potential difference applied to them. If we recall that electrons carry a negative charge, we can see that these deflection plates can be used to deflect the electron beam so that it hits the screen in a different place. electron beam deflection of the beam is proportional to the voltage on the plates +V electron gun –V deflection plates path of the undeflected electron beam (when there is zero voltage on deflection plates) phosphorescent CRO screen Fortunately, the deflection of the electron beam is proportional to the voltage on the plates. Thus if we can put the voltage we wish to measure across the deflection plates then our CRO will represent this voltage as a displacement of the spot on the screen. Thus the CRO can be used as a voltmeter, although as a voltmeter our CRO suffers from one problem: it does not have an adjustment to allow for the measurement of other scales of voltages, as a standard voltmeter does. This problem is overcome in the CRO by inserting an amplifier into the circuit. P-22 Physics 121/2 and 141/2 Laboratory Manual deflection of the beam is proportional to the gain on the amplifier electron gun AMPLIFIER with gain control input signal (voltage to be measured) It can be seen from this picture that if the gain on the amplifier is turned up, the spot will be deflected further up the screen, and thus the input signal can be magnified. This is the same as choosing a different scale on the voltmeter. There is a small difference here that is often a source of confusion when using the CRO: when you choose a different scale on a voltmeter you also commit yourself to reading one of the scales printed on the voltmeter, and do so accordingly. The CRO, however, does not have scales printed on the screen; instead, the amplifier gain control has the scale printed on it, where the scale is quoted in volts per division (often 1division = 1 centimetre). This means that , for a given gain chosen, one division on the screen represents a certain (stated) number of volts. Obviously (think about it!) the further you turn up the gain the lesser the voltage input per division of deflection on the screen. Our CRO now functions as a perfectly serviceable voltmeter! Now lets try to jazz it up a bit… Firstly, let us add to the CRO another, independent set of deflection plates. These plates will control the X-position of the spot independently from the other deflection plates which influence only the Y-position of the spot. Our CRO then becomes: Vertical (Y) deflection electron gun Y AMP X AMP Y inputs X inputs Horizontal (X) deflection It may be useful to think of this CRO as being two voltmeters whose outputs are represented at right angles. As stated at the start of this discussion, the CRO is just a voltmeter, albeit quite a jazzy one. You may be aware that a standard voltmeter is often capable of measuring two types of voltages: DC (ie, constant) and RMS (for where AC signals are concerned). These two cases are a bit limiting: what to do if we wish to look at the structure of the voltage in detail, rather than just the average DC or RMS level? It is possible to do this using our CRO! In order to represent a time varying signal on our CRO, using the X axis as the time axis, we will need to input a signal to the X axis which can pull the spot across the screen at a constant rate. The Y-input would then alter only the vertical position of the spot as the X-input drew the spot across the screen. Physics 121/2 and 141/2 Laboratory Manual P-23 Prelab Exercise: Sketch the X-input required to draw the dot across the screen at a constant rate. Time Base Fortunately the CRO has an in-built circuit to perform this function. The sweep function is called a time base circuit. The time base controls on the CRO are labelled SWEEP TIME / DIV (read as “sweep time per division”). The rapidity with which the spot is drawn across the screen can be adjusted by altering the gradient (but not the amplitude!) of the signal you worked out for the above exercise. Note that, once the spot has traced across the screen at a constant rate, representing the Y-input voltage as a Y-deflection, you ideally want it to return to the ‘beginning’ of the screen to start again and to re-trace the signal. (If it did not do this you might miss your signal: for example, the AC mains run at 50 Hz. If you wanted to view the AC mains voltage in detail, you would need to look at a signal which lasted for only 1/50th of a second. Can you see something lasting for only 1/50th of a second?) Prelab Exercise: Sketch the X-input required to draw the dot across the screen and back again three times at twice the rate of your previous answer. P-24 Physics 121/2 and 141/2 Laboratory Manual