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READINGS AND MEASUREMENTS EXPLAINED BY MR. AKINS MEASUREMENTS

Making Measurements

A measurement should always be regarded as an estimate. The precision of the final result of an experiment cannot be better than the precision of the measurements made during the experiment, so the aim of the experimenter should be to make the estimates as good as possible. There are many factors which contribute to the accuracy of a measurement. Perhaps the most obvious of these is the level of attention paid by the person making the measurements: a careless experimenter gets bad results! However, if the experiment is well designed, one careless measurement will usually be obvious and can therefore be ignored in the final analysis. In the following discussion of errors and level of precision we assume that the experiment is being performed by a careful person who is making the best use of the apparatus available. Systematic Errors If a voltmeter is not connected to anything else it should, of course, read zero. If it does not, the "zero error" is said to be a systematic error: all the readings of this meter are too high or too low. The same problem can occur with stop-watches, thermometers etc. Even if the instrument can not easily be reset to zero, we can usually take the zero error into account by simply adding it to or subtracting it from all the readings. (It should be noted however that other types of systematic error might be less easy to deal with.) Similarly, if 10 ammeters are connected in series with each other they should all give exactly the same reading. In practice they probably will not. Each ammeter could have a small constant error. Again this will give results having systematic errors. For this reason, note that a precise reading is not necessarily an accurate reading. A precise reading taken from an instrument with a systematic error will give an inaccurate result. Random Errors Try asking 10 people to read the level of liquid in the same measuring cylinder. There will almost certainly be small differences in their estimates of the level. Connect a voltmeter into a circuit, take a reading, disconnect the meter, reconnect it and measure the same voltage again. There might be a slight difference between the readings. These are random (unpredictable) errors. Random errors can never be eliminated completely but we can usually be sure that the correct reading lies within certain limits. To indicate this to the reader of the experiment report, the results of measurements should be written as Result ±Uncertainty For example, suppose we measure a length, L to be 25cm with an uncertainty of 0.1cm. We write the result as L = 25±0.1cm By this, we mean that all we are sure about is that L is somewhere in the range 24.9cm to 25.1cm. Quantifying the Uncertainty in a Measurement The number we write as the uncertainty tells the reader about the instrument used to make the measurement. We assume that the instrument has been used correctly. Consider the following examples. Example 1: Using a Ruler The length of the object being measured seems to be somewhere near 4.3cm (but it is certainly not exactly 4.3cm). The result could therefore be stated as 4.3cm ± half the smallest division on the ruler In choosing an uncertainty equal to half the smallest division on the ruler, we are accepting a range of possible results equal to the size of the smallest division on the ruler. However, do you notice something which has not yet been taken into account? In situations like this one has a tendency to concentrate on the right hand end of the ruler. A measurement of length is, in fact, a measurement of two positions and then a subtraction. Was the end of the object exactly opposite the zero of the ruler? This becomes more obvious if we consider the measurement again, as shown here. We now notice that the left-hand end of the object is not exactly opposite the2cm mark of the ruler. It is nearer to 2cm than to 2.1cm, but this measurement is subject to the same level of uncertainty. Therefore the length of the object is (6.3 ± 0.05)cm - (2.0 ± 0.05)cm so, the length can be between (6.3 + 0.05) - (2.0 - 0.05) and (6.3 - 0.05) - (2.0 + 0.05)cm that is, between 4.4cm and 4.2cm We now see that the range of possible results is 0.2cm, so we write length = 4.3cm ± 0.1cm so, in general, we state the result of a measurement as reading ± the smallest division on the measuring instrument Example 2: Using a Stop-Watch Consider using a stop-watch which measures to 1/100 of a second to find the time for a pendulum to oscillate once. Suppose that this time is about 1s. This means that the smallest division on the watch is only about 1% of the time being measured. We could therefore write the result as T = 1s ± 0.01s which is equivalent to saying that the time T is between 0.99s and 1.01s This sounds quite good until you remember that the reaction-time of the person using the watch might be about 0.1s. Let us be pessimistic and say that the person's reaction-time is 0.15s. Now considering the measurement again, with a possible 0.15s at the starting and stopping time of the watch, we should now state the result as T = 1s ± (0.01+ 0.3)s In other words, T is between about 0.7s and 1.3s We could probably have guessed the answer to this degree of precision even without a stop-watch! Conclusions from the preceding discussion If we accept that an uncertainty (sometimes called an indeterminacy) of about 1% of the measurement being made is reasonable, then a) a ruler, marked in mm, is useful for making measurements of distances of about 10cm or greater. b) a manually operated stop-watch is useful for measuring times of about 30s or more (for precise measurements of shorter times, an electronically operated watch must be used) How Many Decimal Places? Now let me address the question of how many decimal places should a reading based on precision must be written Suppose you have a timer which measures to a precision of 0.01s and it gives a reading of 4.58s. The actual time being measured could have been 4.576s or 4.585s etc. However, we have no way of knowing this, so we write t = 4.58 ±0.01s If we now repeat the experiment using a better timer which measures to a precision of 0.0001s. The timer might still give us a time of 4.58s but now we would indicate the greater precision of the instrument being used by stating the result as t = 4.5800 ±0.0001s So, as a general rule, look at the precision of the instrument being used and state the result to that number of decimal places. Another point to remember is that very often we will be using our results to plot a graph. On most graph paper you can represent a result to a precision of 3 significant figures (3 sig. fig.). So, assuming that your measurements allow for this level of precision, convert your table of results to lists of numbers in standard form and give them to two decimal places. By "standard form" we mean a number between 0.00 and 9.99 multiplied by the appropriate power of 10 How Does an Uncertainty in a Measurement Affect the Final Result? When considering how an uncertainty in a measurement will affect the final result, it is important to remember that what really matters is that the uncertainty in a given measurement should be a small fraction of the measurement itself. For example, if you write, "I measured the time to a precision of 0.01s", it sounds good, unless you then inform your reader that the time measured was 0.02s!  The uncertainty is 50% of the measured time so, in reality, the measurement is useless. We will define the quantity relative uncertainty as follows: and to emphasize the difference, we use the term "absolute uncertainty" where we simply said "uncertainty". The most common way to express a relative uncertainty is as a percentage, in which case the fraction above is simply multiplied by 100. Thus, the result of a measurement of, say, length would be stated as L = 15.2mm ±1% which would be the equivalent of saying L = 15.2mm ±0.152mm Therefore we can write L = 15.200mm ±0.152mm We will now see how to answer the question in the title of this page. It is always possible, in simple situations, to find the effect on the final result by straightforward calculations but the following rules can help to reduce the number of calculations needed in more complicated situations. Rule 1 If a measured quantity is multiplied or divided by a constant then the absolute uncertainty is multiplied or divided by the same constant. (In other words the relative uncertainty stays the same.) Rule 2 If two measured quantities are added or subtracted then their absolute uncertainties are added. Example Rule 3 If two (or more) measured quantities are multiplied or divided then their relative uncertainties are added. Example Rule 4 If a measured quantity is raised to a power then the relative uncertainty is multiplied by that power. (If you think about this rule, you will realize that it is just a special case of rule 3.) A few simple examples might help to illustrate the use of these rules.  Example to Illustrate Rule 1 Suppose that you want to find the average thickness of a page of a book. We might find that 100 pages of the book have a total thickness of 9mm. If this measurement is made using an instrument having a precision of 0.1mm, we can write thickness of 100 pages, T = 9.0mm ± 0.1mm and, the average thickness of one page, t, is obviously given by t = T/100 but the uncertainty is also divided by 100 so our measurement is t = 9/100mm ± 0.1/100mm, or t = 0.090±0.001mm Example to Illustrate Rule 2 To find a change in temperature, ΔT, we find an initial temperature, T1, a final temperature, T2 and then use ΔT = T2 - T1 If T1 is found to be 20°C and if T2 is found to be 40°C then ΔT= 20°C. But if the temperatures were measured to a precision of ±1°C then we must remember that 19°C < T1 < 21°C    and    39°C < T2 < 41°C The smallest difference between the two temperatures is therefore (39 - 21) = 18°C and the biggest difference between them is (41 - 19) = 22°C This means that 18°C < ΔT < 22°C Therefore, ΔT = 20°C±2°C Example to Illustrate Rule 3 To measure a surface area, S, we measure two dimensions, say, x and y, and then use S = xy Using a ruler marked in mm, we measure x = 50mm ± 1mm and y = 80mm ± 1mm This means that the area could be anywhere between (49 × 79)mm2    and    (51 × 81)mm2 that is 3871mm2 < S < 4131mm2 To state our answer we now choose the number half-way between these two extremes and for the indeterminacy we take half of the difference between them. Therefore we have so, S = 4000mm2 ±130mm2 Example to Illustrate Rule 4 To find the volume of a sphere, we first find its radius, r, (usually by measuring its diameter). We then use the formula Suppose that the diameter of a sphere is measured (using an instrument having a precision of ±0.1mm) and found to be 50mm. Diameter = 50.0mm ± 0.1mm so  r = 25.00mm ± 0.05mm This means that V could be between (4/3)(24.95)3   and   (4/3)(25.05)3 As in the previous example we now state the final result as which gives, V = 65451mm3 ± 393mm3 Check: Relative uncertainty in r is 0.05/25 = 0.002 Relative uncertainty in V is 393/65451 = 0.006 so, again the theory is verified For  questions and contributions: call  08160001151 or  email me at akinstutor@yahoo.com