Comment The fractional error in one mass of Br is about a tenth of the fractional error in the other mass of AI. The result is that the fractional error in n is very nearly equal to the fractional error in the mass of AI. When two fractional errors are combined the larger one is often dominant because they are squared in equations 4, 8 and 9. So if you notice that one fractional error is about a third or less of another, you can ignore the smaller one. The diameter of a sphere is measured to be 7.
Using equation 10 in Box 1, 3 xO. Because errors increase so rapidly when powers are taken, you should always take particular care to reduce errors when measuring quantities that will be raised to some power.
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Errors, by their very nature, cannot be precisely quantified. As a general rule : Errors should usually be quoted only to one significant figure; two significant figures are sometimes justified, particularly if the first figure is a 1. You should bear this in. Don't be concerned about 30 per cent errors in your errors- they really don't matter.
Neglecting small errors As you have seen in Example 4, the total error in a result may be a combination of several contributing errors, and these contributing errors may have widely varying sizes. Concentrate on reducing the dominant errors 30 As we have just shown, the largest errors will dominate the error in the final result, and small errors can often be neglected. It is therefore, very important when doing experiments not to waste a lot of time reducing small errors when much larger errors are also present. As a general rule : Find out as early as possible in an experiment what the dominant errors are, and then concentrate your time and effort on reducing them if the precision of your experiment is not sufficient for your purpose.
This is a 50 per cent error compared with only about 4 per cent in the individual measurements! The fractional error in the volume is three times greater than in the length measurement. As a general rule: If calculating the result of an experiment involves taking the difference between two nearly equal measured quantities, or taking the power of a measured quantity, then pay particular attention to reducing the errors in those quantities.
Graphs are frequently used to represent the results of experiments, and some of the reasons for their use should become clear if you compare Table 1 repeated from Section 1 with Figure 5. The Table and the graph both summarize the results obtained in an experiment in which the extension of a copper wire is measured as a function of the load suspended from it. In fact, if the wire is extended more than 2. The straight line drawn averages out experimental errors in individual measurements.
In this case the straight line on the graph is used to interpolate between measured values: g The gradient of the line relating extension e to mass m is 2. However, it would take a long time to arrive at a - g from the tabulated information, whereas they can all be very rapidly deduced from the graph. This is the great advantage of graphs as visual aids. The form of the relationship between measured quantities, the typical errors in measurements and the presence of anomalous measurements are readily apparent, and graphs allow straightforward averaging of experimental measurements, interpolation between measurements and in simple cases determination of the equation relating measured quantities.
The doubtful point has coordinates 1. Do not make your life difficult by making 10 small divisions equal 3 or 7; you would take much longer to plot the graph, and the chances of misplotting points would be very much higher. In some cases this may mean excluding zero from the axis : for example, if lengths measured in an experiment varied between 5.
The mass is the independent variable and is plotted horizontally, the extension is the dependent variable and is plotted vertically, and Figure 5 shows how the extension of the wire depends on the mass hung from it. Either crosses X or dots with circles round them 0 are preferable. Figure 7a is poor, Figure 7b is better. You will also be able to 36 check whether the points are reasonably evenly spaced, or whether you need some more closely spaced points e.
In most cases, graphs represent some smooth variation of one quantity with another, and so a smooth curve is usually appropriate. Thus the continuous line in Figure 9 is usually preferable to the broken line. SAQ 2 Table 3 shows the percentage of men surviving until various ages as ascertained in a particular survey. SAQ 3 Having noted the speed of his car at various times Table 4 , a naive graphplotter presents his measurements in the way shown in Figure His graph can be criticized on at least five counts. Point out the shortcomings, and correct them by replotting the results in Table 4.
Showing likely errors on graphs In any quantitative scientific experiment, it is essential to indicate the likely errors, or confidence limits, in any measured quantity. The most difficult part of this procedure is actually assessing how large the error in a measurement could be, and this problem was discussed in Section 2. However, once an error has been estimated, it is straightforward to represent it on a graph. The error bars extend 0. This uncertainty would be represented by a horizontal error bar, which extends 0. Generally both horizontal and vertical error bars should be shown, and either of them omitted only if the associated error bar is too small to plot on the graph.
The absence of horizontal error bars implies that the mass is known much more accurately. Plotting error bars is slightly more complicated if you are not plotting a measured quantity x directly, but are plotting x 2 , x 3 , sin x, etc. We shall illustrate the procedure by taking as an example an experiment in which the speed v of a falling ball is measured after it has dropped through various distances s. Suppose that we have already discovered, by plotting various graphs, that v2 is proportional to s.
Using the data in columns A and C of Table 5, we get the plot shown by the circled points in Figure Note that even though the size of the errors in v are all the same, the errors in v2 get larger as v gets larger. The presence of error bars on a graph serves a number of useful purposes. If the graph is one which we would expect to be a smooth curve then we would expect the results to be scattered around that curve by amounts ranging up to the size of the error bar. Should the results deviate from a smooth curve by much more than the error bars, as shown in Figure 15, then either we have underestimated the errors, or the assumption that a smooth curve should describe the results is not valid.
On the other hand, if all of the results deviate from the expected curve by much less than the error bars, as in Figure 16, then we might well have overestimated the likely errors. An alternative explanation in this latter case would be that the dominant contribution to the error bar was from a systematic error that would shift all results in the same direction by as much as the error bar.
Error bars are also very helpful in identifying mistakes in measurements or plotting of points. All results lie on the broken line except for one, which deviates from the straight line by about three times the magnitude of the error bar. This immediately suggests a mistake. In cases like this, the plotting of the point on the graph should be checked first of all, and any calculations made to get the numbers plotted should be checked as 42 well. If these don't show up a mistake, then the measurements that gave rise to the suspect point should be repeated.
No hard and fast rule can be made about this, but you should always be aware that what appears to be an anomalous measurement may indicate a real and possibly as yet undiscovered effect. In the case of gadolinium, more detailed measurements show that the real behaviour is as shown by the continuous line in Figure By ignoring the apparently anomalous result, we would have overlooked a real anomaly which is, in fact.
An important piece of advice follows from this: if at all possible, plot a graph while you are taking measurements. If you do this, you can immediately check for errors and investigate any anomalies that appear. Plotting as you go along also helps to ensure that the results are reasonably distributed on the graph, and this usually means having them equally spaced.
Remember also that the slope of the line is constant, and does not depend on which pair of points on the line are used to calculate it. However, we can only read off the coordinates of the points with limited accuracy, and so, in practice, different pairs of points will produce slightly different values of the slope.
Note that the best practice is to choose two points that are as widely separated as possible, since then the errors in reading the coordinates will be a much smaller fraction of the difference between the coordinates than if the points were close together. The straight line in Figure 19 is the one we think is best fitted to the results plotted. Displacements of some points above the line are balanced by displacements of other points below the line.
However, the error bars indicate the uncertainty in the experimental results, and other lines with different slopes could be drawn to pass through all of the error bars. So in order to estimate the uncertainty in the slope, we draw lines that pass through all of the error bars with the maximum and minimum possible slopes. These lines are shown in Figure 20, and their slopes are 6. These values differ from the slope of the 'best' straight line by 0. We therefore quote the value of the slope as 5. The other quantity that is normally used to specify the position of a straight line on a graph is its intercept with the vertical axis.
Figure Note that again we must quote the appropriate units for the intercept. The maximum and minimum likely values of the intercepts are found by drawing other lines through the error bars, and in this case they are the intercepts of the lines with maximum and minimum slopes shown in Figure Converting curves to straight lines In many cases, if you plot the dependence of one variable in an experiment against another, the sequence of experimental points will be curved in some way, and cannot be fitted by a straight line.
In such cases, it is much more difficult to draw a line on the graph to represent the average behaviour, and even more difficult to deduce what equation might represent the results. However, it is often possible to plot such results in an alternative way so that a straight line is obtained, and an equation can then readily be deduced.
Of course, the real problem then is deciding what to plot along each axis in order to get a straight line. If we know a priori the form of the equation relating the measured quantities, then the decision is fairly easy. Take the example of the dependence of speed v of a ball against distance fallen s, that we discussed in another context in Section 3.
If we had plotted v against s, we would have obtained the graph shown in Figure 22certainly not a straight line. Don't worry about how this equation was derived. So what advantages, or additional information, do we get by plotting graphs in such a way as to get a straight line? First, we can confirm or disprove!
Looking at Figure 22, it is difficult to say whether v2 is proportional to s : the graph is curved, as one would expect, but is it curved in precisely the way predicted by v2 ex: s? By plotting v2 against s, as in Figure 14, we can readily see whether a straight line is consistent with the experimental results and, since it is, we can conclude that the results are consistent with the prediction v2 ex: s.
Second, by determining the slope and intercept of the straight line, we can actually deduce numerical values of constants in the equation relating measured quantities. Thus, from the slopes of straight-line plots, we readily deduce values of k and n in the two examples given above. Getting these values from curves would be extremely difficult. Another important advantage of a straight-line graph is that it is extremely easy to extrapolate beyond the range of measured values : you only need a ruler.
Extrapolating a curve is far more difficult and is subject to much more uncertainty. Numbers of adult moths surviving were counted on six occasions in the last five months of the study period. All other numbers are of adult moths. When this data is plotted, the graph shown in Figure 23 is obtained. This is not a very convenient way of plotting data. Six of the seven experimental points are clustered together near the bottom right-hand corner of the graph, lea ving a large gap between 16 survivors and the original eggs.
The shape of the curve we have drawn is largely a matter of guesswork. When you do this, you sometimes find that the result is a straight line. There are many logarithmic relationships in science. The data that we need to plot are in Table 7, and log N is plotted against time in Figure You can see from Figure 24 that we obtain a graph in which the points are more evenly spaced and we can more easily draw a line through the points. Admittedly, our line doesn't go through all the points, but this is hardly surprising when the difference of one moth makes such a large change in the position of a point on the right of our graph.
Working out the log values for such a plot is a trifle tedious even with your calculator! The way around this is to use graph paper where the lines on one axis have been drawn in a logarithmic fashion. On the paper in Figure 25 the horizontal scale is an ordinary one, in which the large divisions are divided into tenths and each division has the same size. The vertical scale is however a logarithmic scale, in which each power oJten or decqde corresponds to the same length oJscale.
Figure 25 shows the moth data plotted on this semi-logarithmic paper. Notice that, in each decade, the divisions become progressively compressed towards the upper end, In the same way as the logarithms of numbers increase more slowly than the numbers themselves. Table 8 compares the logarithmic scale and the linear scale.
Figures 24 and 25 show this clearly. What is the slope of the line in Figure 25? If we replace log N by y as the dependent variable, i. This graph has a constant slope which we shall call k and it has a y-intercept which we shall call b. To do this we need to recall the definition of a logarithm see Appendix 1. It shows a population that is decreasing in such a way as to be halved every 1. Thus, starting at , it drops to at 1. This population is said to be decreasing logarithmically, with a half-life of 1.
This mode of decrease, or 'decay', which is also called 'exponential decay', is characteristic of many natural phenomena, from populations subject to predation, to radioactive elements subject to instability. The equation for exponential decay may be written in another form, in which the half-life appears explicitly. Furthermore, as you can see from Figure 25, the half-life can be read simply and directly off the graph.
The type of graph paper illustrated in Figure 25 is called log-linear or semilogarithmic paper, because one axis is marked off on a logarithmic scale and the other on an ordinary linear scale. It is useful, as you have seen, for plotting a quantity that varies by several orders of magnitude against another quantity that does not. In the example we have just considered, N varies from to 2, i. Sometimes we find that we wish to plot a graph where both variables range over several powers of ten.
For example, consider the data in Table 9 which show how one quantity T depends on another quantity R. It would thus be convenient to plot both T and R on a logarithmic scale. For this you need 'log-log' graph paper. In Figure 26, we have plotted T against R on log-log graph paper p. The points lie upon a straight line. To find the slope of a straight line plotted on log-log paper, you simply measure off a convenient baseline, such as the line AB, which we have made 8.
This procedure is, of course, valid only if the logarithmic scales on both axes are the same, that is, each step in the powers of 10 has the same length on each axis. In such a case the slope is also the proportionality constant between log T and log R. SAQ 4 Table 10 gives data for the mean growth in length in mm of a sample of moth caterpillars from the time they emerge from the egg to when they turn into a pupa. What type of graph linear, log-linear or log-log is most suitable to illustrate this data graphically.
Note that when finding the slope of a straight line, the use of of a Jarge triangle ABC is preferable to A'B 'C '; given the same ruler, fractional errors are reduced. In this Section we introduce ways of expressing the range of values in a set of measurements and how the values are distributed about the mean value. The kind of expressions we introduce here are important in quoting data on populations encountered in Biology and Earth sciences, as well as in expressing the reliability of a value that is the average of several measurements.
She has your data from the water-flow experiment in Section 2. She also has a value of her own : She wants the best possible value. She cannot choose one and ignore the other, for she has no grounds for making such a choice. Does she take the mean of the two results : If she had taken only one reading, viz If the single reading taken by the second experimenter had the same reliability as anyone of the readings you took, what would be the best estimate in these circu mstances?
If the second experimenter had taken 20 readings of the same reliability as yours, what would be the best estimate? There is no need to know what the 20 individual values were. If the average of 20 readings was And if the average of your 5 read- 57 ings was So the sum of all 25 readings is 20 x Clearly the number of readings taken is one important factor in determining the precision of a result. Is it the only factor? Take a look at Figure In Figure 27a are displayed the five readings that gave a mean value of If the number of readings were the sole criterion for judging the precision of a result, the two values- Does it look from Figure 27 as though they ought to?
It would seem not. The measurements in the second set agree with each other more closely. Perhaps the second experimenter had a better way than you had of cutting off the water flow at the end of the given time period. It is only right that she should be given credit for her extra care. The extent to which the readings are spread about the mean position has to be taken into account. II In the example we have been considering, the mean of the first set of measurements was The method is evidently subjective as well as rough and ready.
If it is impossible to make more than one or two measurements of some quantity, this simple 'difference method' of estimating the error will obviously not work at all. We shall return to the question of what to do in such circumstances. An objective method of reporting the spread in a set of measurements which has significance for statisticians is to calculate a quantity called the standard deviation s. If you quote the spread of your results in terms of the standard deviation, other people will know exactly what you mean. The method of calculating the standard deviation is illustrated in the bottom part of Table Then calculate the differences, by subtracting the mean from each individual reading.
We have called these differences d in Table Now calculate the square of each of these differences. We have done this in the fourth column of the Table. Note that d 2 is always a positive quantity. Now add up all the values of d 2 and divide this sum of the number of measurements five in this example. Now take the square root of S2 to get s. With more than just a few readings, the standard deviation would give a more reliable and less pessimistic measure of the spread than the roughand-ready method would. X4, X Unless otherwise stated, it is understood throughout HED that we are taking the positive square root, i.
If s has been determined for a sample consisting of a great many readings, it gives a measure of how far individual readings are likely to be from the true value. If there are many readings of the same quantity, it is difficult to make a quick assessment of the precision of an experiment and the likely mean value by examining a long list of numbers displayed in a table. It is more convenient to plot the readings in the form of a histogram Figure The range of measured values is divided into equal intervals and a note made of the number of readings falling within each interval.
For instance, as illustrated in Figure 28, the intervals may be from zero up to, but not including 1 s; from 1 sup to, but not including, 2 s; from 2 s up to, but not including, 3 s; and so on. For instance, in the histogram shown in Figure 28, the number of readings between 6 and 7 seconds is With ihis histogram you can see at a glance that the mean value is about 6. Using the 'rough-and-ready' rule, you might quote the result as 6. As expected, the rough estimate is on the pessimistic side. It gives too much weight to the highest and lowest readings.
As the number of readings is increased, so one can take smaller and smaller intervals and still have a reasonable number of readings in each. Eventually, the 'step-like' character of the histogram is no longer noticeable and one is left with a smooth curve as in Figure So much for the question of how far an individual reading is likely to be from the true value. This, however, is not our main concern, because normally you will take several readings and find the mean. We are more interested in how far the mean of the reading x is from the true value.
Of course, it is not possible to say exactly how far away an individual reading is from the true value. We must emphasize that we have no way of telling what the true value is. However, it is desirable to be able to assign a probability to the mean lying within a certain range of the true value.
This range will depend not only on the spread of the individual readings but also on n, the number of such readings. Therefore, Sm is a 61 measure of how close the mean value of the given sample x is to the unknown true value. Once again we must stress that, unless you are prepared to take a course in statistics, this is an equation you must simply accept and use as a tool.
But note the difference between the standard error on the mean Sm and the standard deviation of the individual measurements s. In the experiment we have been considering, x is the mean of the five readings and has a value of Therefore we have included both these numbers in the brackets.
One important use of the standard error on the mean is in deciding about the significance of an experimental result. Suppose that a new theory suggests that a particular physical quantity should have the value units. An experiment is done in which a large number of measurements are made of this quantity. Does this experiment confirm the theory, or refute it? Thus the probability that the experiment confirms the theory is certainly no higher than 68 per cent. In this case, the 'true value', assuming it is units; differs from the mean by more than twice the standard error.
It follows that the chances that the theory is wrong are greater than 95 per cent. However, it does so only slowly because n - 1 appears as a square root. Thus a tenfold increase in the number of readings improves the precision by a factor of 3. A large number of sloppy readings, therefore, is no substitute for taking care and keeping the spread of the individual readings, and hence s, as small as possible.
You should now be able to calculate the mean value, the standard deviation and the standard error on the mean value of a set of measurements subject to random error, using one or other of the formulae which we have collected together for your convenience in Table The masses in grams g ofthe 20 individuals at days after birth were as follows: , , , , , , , , , , , , , , , , , , , SAQ 6 An experiment was carried out to measure the period of oscillation of a pendulum. The mean value of a hundred measurements was The theoretically predicted period of oscillation is If the theory is correct, what is the chance of obtaining the above value in a properly conducted experiment?
What could such a result indicate? It is more usual to find that the mean value is not the most 'popular' value in the distribution. For example, if you were to stand on a motorway bridge and make a survey of the lengths of the cars passing beneath you, then you would be very likely to obtain a distribution like that in Figure 30, because the most popular cars are of less than frequency mode mean CORTINAS increasing car length FIGURE 30 A positively skewed distribution of car lengths on a motorway.
There is also a larger range of sizes among the large cars on the motorway, and this gives the distribution a 'tail'. By definition, Figure 30 shows a positively skewed distribution. On the other hand, if you were to include all the vehicles on the motorway in your survey at a time when the number of lorries exceeded the number of cars, then the range of car sizes would form the 'tail' and the distribution might be negatively skewed Figure How can we make a quantitative measure of skewness?
We know from Figure 29 that 68 per cent of the values in a symmetrical frequency distribution lie within 64 one standard deviation of the mean. In all examples, we shall assume that 50 per cent of the observations lie to either side of the mean value ,x is defined strictly as the sum of the x values of all the observations divided by the number of observations- equation 21 - and is close to the 50 per cent mark of most distributions.
The 16, 50 and 84 percentiles have been marked, and each of the shaded areas includes 16 per cent of the observations. In the symmetrical distribution, these 16 and 84 percentiles, as they are known, are equidistant from the mean or 50 percentile. But in a skewed distribution this is no longer the case. In Figure 29, skewness is zero and this defines the symmetrical distribution. The importance of skewed distributions is demonstrated in the interpretation of grain sizes in different samples of sand.
Although accumulations of sand on beaches, on sand dunes and in the bed of a river may appear to be all the same, we find that the ranges of grain sizes present vary tremendously depending upon whether the sand was blown by the wind, carried by water in a river or winnowed gently by waves. The populations of sand grains are often positively or negatively skewed and geologists can use this property of skewness to help them decide how an ancient sand deposit, now formed into a solid rock, was originally laid down.
It was as though they considered the results of their experiments a private matter; fodder for their own curiosity. However, with the development of technology capable of exploiting new scientific discoveries, this attitude changed. It became clear that scientists who did not publicize their findings could needlessly delay the pace of technological advance, and possibly forfeit the credit for their discoveries, losing not only fame, but possible material rewards as well.
For example, an eighteenth-century British scientist, Henry Cavendish, came, through his experiments, to a deep understanding of electrical phenomena.
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He had, however, one serious fault- he rarely took the trouble to write up his findings. The result was that for many years people continued working on problems he had solved long ago. For example, he discovered Ohm's law of electrical resistance fifty years before Ohm did. Yet the law is quite rightly ascribed to Ohm, because it was the German schoolmaster who made the information available to others.
Moral: you may get personal satisfaction from doing good experimental work but, if you wish to get credit for it and to benefit the community at large, you have to develop an ability to communicate your findings to others. Such an ability is particularly important when results challenge orthodox thinking. The chemical world had to wait 50 years for the insight into chemical bonding which derived from the work of Avogadro in the early nineteenth century. His work was ignored during his lifetime, but brilliantly publicized by his countryman Cannizaro at one of the first scientific conferences, a gathering of chemists at Karlsruhe in It is noteworthy that it was not verbal argument which convinced most of the chemists present, but the distribution of a pamphlet in which Avogadro's results and arguments were set out clearly and forcibly.
So, skill in the presentation of experimental results and conclusions derived from them is something that is in the interests of every scientist to acquire. It does not matter how skilful your experimental technique is or how careful your observations are, if you fail to communicate clearly both the results and their significance to your reader, this skill and care will have been pointless. The main requirement of the notebook is that it should contain all necessary information in a form that is accessible to you. The final report must still contain all necessary information, but organized in a form accessible to the average reader, and retrievable in as short a time as possible.
Clarity, concise- 66 ness, and presentation are all at a premium ; any conclusions you reach should be stated simply, with a clear indication of their limitations. There are a few basic rules for the presentation of experimental reports, which you should find useful.
It may not be the first thing that you write, but nonetheless it should appear at the beginning because it helps your reader 'see the wood for the trees' while working through the necessarily detailed treatment of experimental results. It is worth taking some trouble to organize data into an easily digestible form.
Where possible even if the data are qualitative use a table. Allow plenty of space for this so that the results are not cramped.
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Graphs should always be supported by appropriately labelled tables of raw data. It is important to remember to include all the original data in your report; it is impossible for someone else to judge your work if the data are incomplete. You could find it difficult enough yourself to make sense of an incomplete report in a couple of months time! Careful recording of data will prevent the need to repeat the experiment to supply missing data.
When drawing diagrams or graphs you should initially use a pencil rather than a pen so as to be able to make corrections if necessary. The drawings should be made large; do not think that by using a thick pencil and cramming the drawing into a small space you can avoid being careful over details.
It is a good idea to make suggestions at this stage about how experimental design could be improved by reducing the dominant error, and about how the phenomenon being studied could be investigated more thoroughly and extensively. Make sure to include a discussion of any unexpected behaviour, and a comparison of your result with the predictions of theory, if this is appropriate. It is better to use simple terms which you fully understand than more complex jargon which you may misuse.
Throughout the discussion try to cut down on unnecessary words and information. Although in presenting data you can hardly be too generous with information, the ideal to aim for in presenting arguments is a concise style, avoiding irrelevancies and repetition.
It is a good plan to reread your report critically after a lapse of a few hours or even a day to make sure that it is easy to read, and that it says exactly what you mean to say. The changes that you will almost certainly want to make at this stage are bound to be improvements! It may take the form of a verbal generalization, or it may be a numerical result, which should never be separated from the estimate of its error limits.
If you are studying the transformations involved in a chemical reaction, you should express your conclusions in the form of a chemical equation; if you are studying the relationship between measured quantities, such as, for example, distance and time you should include an equation relating the quantities under study. If, for example, during the experiment, you took certain precautions and made certain checks, but subsequently omitted to point these out in your account, it will be assumed that you did not do them and, moreover, that it did not even occur to you that you ought to have done them.
Keep referring to them until, with experience, they become second nature to you. Planning an experiment If possible perform a quick preliminary experiment as a rehearsal, in order to allow you to plan the way you will do the experiment. Keeping a laboratory notebook 2 Do not use scraps of paper ; put readings and comments into a special notebook kept for the purpose. Hints on calculations 10 Average raw readings rather than processed data in order to save the labour of unnecessary multiplications.
The final result of a multiplication or division can have no more significant figures than those possessed by the factor with the fewest. Check that inaccuracies introduced by rounding-ofT are negligible compared with the experimental error. Errors 17 Always estimate errors in experimental measurements and calculated results.
These are listed for easy reference on p. Hints on how to plot graphs 23 Plot the dependent variable i. Drawings of specimens should be done initially at least in pencil, and should be large. Where necessary, you should indicate the scale. If you follow the advice contained in this summary, and apply it in your practical work, you should acquire an ability not only to present data, but also to detect errors or deficiencies in somebody else's presentation.
You may like to check your ability to do this by trying SAQ 7. SAQ 7 Read the following account of an experiment in which the extension of an elastic rubber band is measured as a function of its load. Imagine you are a tutor assessing the work. Criticize the design of the experiment and comment on the adequacy of reporting. To which deficiencies of presentation would you draw most attention? The rubber band was attached by means of a hook to a fixed point the underside of a cupboard and a mark made on it by means of a felt pen.
Each weight was tied in turn to the end of the rubber band, using a light but strong thread, and the position of the mark for each load measured with a ruler. The Figure shows the apparatus. The results are as I, listed in the table. An extension of The reason that this extension is possible is that the' band is made of a polymer which can exist both in a coiled and an uncoiled form. In the coiled form, weak interactions between polar side chains hold the coils together; however, these weak interactions are easily overcome, allowing the coils to straighten out under the extending force.
This is an abbreviation for Systeme Internationale d'Unites. It was formally approved in by the General Conference of Weights and Measures. In the SI system there are seven basic units Table TABLE 13 Physical quantity Name of unit Symbol for unit length mass time electric current thermodynamic temperature luminous intensity amount of substance metre kilogram second ampere kelvin candela mole m kg s A K cd mol Symbols for units do not take a plural form e. Notice that these are now defined in terms of SI units although originally they were not.
We can, of course, always get over the problem of writing many zeros before or after the decimal point by using powers of 10, for example, 10 - 6 kg instead of 0. It is convenient, however, to have fractional or mUltiple units, and these are denoted by placing a prefix before the symbol of the unit. It is normal to restrict the multiples and fractions of a unit to powers of hence the centimetre, which is 2 m, is not strictly in accord with SI. The allowed prefixes are shown in Table TABLE 16 Fraction Prefix Symbol Multiple Prefix Symbol 10 - 1 10 - 2 10 - 3 10 - 6 10 - 9 10 - 12 10 - 15 10 - 18 deci centi milli micro nano pico femto alto d c m 10 10 2 10 3 10 6 10 9 deka hecto kilo mega giga tera da h k M G T Il n P f a Compound prefixes should not be used, e.
Also note that until such time as a new name may be adopted for the kilogram the basic unit of mass the gram will often be used, both as an elementary unit to avoid the absurdity of mkg and in association with numerical prefixes, e. The values are given in scientific notation to five significant figures. In scientific notation, the quantity is expressed with one figure before the decimal point and with the appropriate power-of-ten multiplier.
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To test your facility with units and their fractions and multiples, you should try SAQs 8 and 9 preferably without referring back to the lists in this Section. Fill in the blank in each row of the Table. So if a mass of I kg is accelerated by 1m s- 2 we have, by definition, a force of I N, i. By way of an exercise, see if you can write the SI symbols for the quantities named and defined in Table Area, for example, obtained by mUltiplying a length by another length, has the dimensions of length squared.
No matter what units the area is expressed in e. A speed, which is obtained by dividing a distance by a time, has the dimensions of distance per time or distance x time - 1 : i. As a simple exercise, enter the appropriate dimensions in the fourth column of Table Compare your list with the one in Table 20 at the end of Section 7.
Note that some quantities have units but are dimensionless. For example, angles are measured in units of degrees or in units of radians. In both cases, these units are defined as a ratio of one length to another. A valid physical equation applies both in respect of number and of units for each term.
The consequence of this is that a change from one system of units to another does not invalidate the equation. Furthermore, each term in a correct physical equation must have the same dimensions. An equation which is not 'balanced' in terms of dimensions is certain to be wrong. To take an example, you might decide that an expression of the form: 78 describes how the weight of a body alters when it is in a fluid, where : Wa is the weight in air the force of gravity on the body , Wr the apparent weight registered by the force on the balance when the body is immersed in fluid, p is the density of the fluid, gE the acceleration due to gravity, and V the volume of the body.
You should check whether your logic in deciding on this expression was correct, by examining its dimensional balance. The dimensions of the term on the lefthand side of the expression are simply those of force; i. This analysis shows at once that the original expression was wrong ; further, it indicates the nature of the error. The discrepancy involves dimensions of length, which are much too small in the second term. The dimensions of the second term on the right-hand side of this equation are: This equation is dimensionally balanced and, therefore, may be correct.
To take a somewhat more complex example, suppose you want to calculate the speed of propagation of a seismic wave through a part of the Earth's interior. You believe that the speed should depend on the ratio of the elastic modulus to the density, but you are not sure whether it is just proportional to this ratio, or to its square or square root, or to some other power of it. It is meant to be used for reference, particularly in conjunction with experimental work.
If you use HED in this way, it should help you to acquire certain skills and abilities. SAQs 2, 3 and 7 4 a Specify the SI units for the following physical quantities: length, mass, time, electric current, amount of substance, angle, frequency, force, acceleration, momentum, energy, power, stress, volume, density, electric charge, electric potential difference. SAQ 10 6 Report an experiment correctly using : a tabulated data b graphical presentation of data c a concise description d a summary e estimates of errors f a dimensional check of results SAQs 1 and 4 82 Appendix 1 Logarithms Logarithms are closely related to powers of numbers.
Take for instance, the number If you ask: 'by what power must ten be raised to produce a hundred', what should the answer be? Clearly, the answer is 2. For example, The word 'logarithm' means 'a reckoning number', the name indicating the purpose for which logarithms were used when they were invented in the seventeenth century. The idea was that much tedious arithmetic could be avoided if multiplication and division could be replaced by addition and subtraction.
However, the advent of cheap electronic calculators has made the tables redundant as a computational aid, so you will not be using logarithms for this purpose. They are still used in science, however, to express the values of quantities that can vary over a wide range, so you should be familiar with their evaluation and manipulation.
The most commonly used logarithms are those to the base 10, and these are usually represented simply by log, i. Before you do this try to make an estimate of the range i. The definition of a logarithm to the base 10 may be expressed more generally. Answers to Exercises in Appendix 1 Exercise 1 a 3, 6, 9.
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Numbers between 1 and 10 have logs between 0 and 1, as you should expect. As lies between and i. This is not possible, however, when the logs are not integral, although you can see that the number whose log is 0. Moreover, because of the properties of indices the number whose log is 1. Hence, n can be evaluated as 5. SAQ 2 The graph is shown in Figure Notice how the axes are chosen- the independent variable is age.
Notice too that the axes are labelled so that the numbers written on them are dimensionless. The choice of scale makes best use of the paper available. The points are plotted as dots within circles and they are joined by a smooth curve. SAQ 4 Since the mean length varies by more than two orders of magnitude, a logarithmic plot is most suitable to illustrate the data.
If you tried to plot the data on linear graph paper, you would find as Figure 34 shows that the points are grouped at the bottom of the graph and that the plot is curved. The most suitable plot is a log-linear one with the mean length plotted on the log scale, as in Figure Notice that this Figure uses three-cycle semi-log log paper because I varies by more than two orders of magnitude.
The largest reading is g which is 11 g above the mean, and the lowest reading is which is 16g below the mean. The measured mean value of The disagreement indicates a 95 per cent probability of some systematic error in the measurements. This Physics Factsheet aims to look at representing experimental data correctly in results tables, determining the significant figures in both results tables and calculations and how this relates to the precision of raw data.
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How gravitational attraction provides This Physics Factsheet aims to cover basic graph skills including plotting scatter graphs and This Physics Factsheet looks at examples of the way charges flow, including electrons flowing Electricity can seem very abstract and difficult to understand.