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Measurement Level: Secondary 4 / 6 Project aims: The main objectives of this project are to let students: 1. Understand the importance of measurement in our daily life. 2. 3. 4. Explore different methods for measurement. Understand that measurements can never be exact. Compare the accuracy of different measuring methods. Learning objectives: 1. 2. 3. 4. Understand the importance of measurement in science. Gain familiarity with a variety of measuring instruments and methods. Understand that measurements are never exact through error analysis. Recognize the limitation of measuring instruments used. Tasks: In the following project tasks, students are asked to design their own methods for measuring particular object(s), building(s) and distance(s). They have to explain how they make the measurements, identify the possible sources of error of each method, and compare and estimate the accuracies of their measurements. Task 1: Measuring length Measure the length of an object (shorter than one foot long recommended), say a pencil, using a one feet ruler without finer scale. Students may use other tools in their measurement when necessary. However, the tools should not contain any scale or mark, which can directly be used to measure the length of the object. Task 2: Estimating number Estimate the number of small objects inside a container, say, the number of beans in a jar without counting the number directly, or to estimate the number of tiles in a floor which the laboratory is located by measuring the number of tiles per square meter. Task 3: Measuring height Measure the height of a given tall building or structure, such as the Lion Rock or the International Finance Centre in Hong Kong. Task 4: Measuring distance Measure the distance between any two particular places, without the use of any electronic equipment or professional devices Apparatus/Materials: Any appropriate apparatus and materials proposed by students are welcomed. Typical examples include meter ruler (scale covered with paper), identical coins (many), protractor, beaker, balance, measuring tape, tennis ball (as a falling object), stop watch, string / rope with a weight, meter wheel and so on. Pre-knowledge: Measurement is of central importance in our daily life. It is one of the concrete ways we deal with our world. Measurement is particularly important in science. Science is concerned with the description and understanding of nature and measurement is one of its most fundamental tools. One may say that science cannot exist without measurement and we start learning physics by learning how to measure a few physical quantities. Physics attempts to describe nature in an objective way through measurement. However, in every measurement, there is uncertainty associated with it – there is no such thing as a perfectly accurate measurement. There are uncertainties arise from the measuring instrument, the people doing the measurement, the method of making the measurement and the characteristics of the systems being measured. We need to learn to deal with these uncertainties in measurement through error analysis. Questions: 1. 2. Which method of measurement is more accurate? Can you account for this? How could you deal with the difficulty in measurement involving slope and gradient, like staircase? Extension: 1. 2. Measure the thickness of a thin sheet of paper that is too thin to be measured using normal measuring instrument. In room temperature and pressure, one mole of air molecules occupies a volume 3. of 24.5 liter and has a mass of 14.4g. Estimate the mass of air inside a room. Which is the “heaviest”? The people in a room, the chairs and tables, or the air in that room? Let the “cleaning area” of an object be the total area of that object that you wish to clean. (For example, the top surface of a desk and the inner surface of a cabinet but not those outside your visibility.) Clean a small area and record the time needed. Try estimating the time needed to clean the whole cleaning area, the error in estimating the cleaning area and the cleaning time. Raise out the factors we haven’t took into account if this value is outside your expectation. Appendix: Propagation of error in different forms (This appendix is adopted from the draft Error Estimation - 誤差估算 released in October 2005 by Science Education Section of Curriculum Development Institute, EMB.) In many cases, a particular result will be obtained by combining a number of measurements of different quantities in some ways. For instance, when two independent measurements ( A A ) and ( B B ) are combined to give a result X, the associated error terms X will be given by: Combinations of A and B Error of X X A B X A B X A B X AB X A B X A B A X B X kA ( k is a constant) X A or X k A X A X An X A n X A X AB n X A B n X A B The combinations involve addition and multiplication with more variables or more terms can be resolved into combinations listed above. Example 1: Combinations of error involving addition and multiplication A student finds the constant speed of a moving trolley with a stopwatch. The equation used is v s t . The distance s is measured with a meter ruler, the time t is measured with a stopwatch. Their values are s = (4.01 ± 0.01) m; t = (3.2 ± 0.2) s. The speed v = 4.01 / 3.2 ms-1 = 1.253 ms-1 (calculator display) = 1.3 ms-1 This result is corrected to 2 significant figures since the time measured is only known to 2 significant figures. The estimated error in speed, v s t = v t s 0.01 0.2 = 1.253 ms-1 4 . 01 3 . 2 -1 = 0.081437 ms (calculator display) = 0.1 ms-1 Since v has 1 decimal place, the error term ∆v should also has 1 decimal place, i.e. v = 0.1 ms-1. Therefore the speed of the trolley is 1.3 ± 0.1 ms-1. Handling the trigonometric functions of an angle In the tasks, students may measure an angle and apply trigonometric functions of the angle for their analysis. Here shows two examples to deal with the tangent of an angle. Example 2: Handling the trigonometric functions with multiple measurements Students used a protractor to measure the angle of elevation to a building and apply the formula h d tan to calculate the height of the building. If d (8.23 0.005) m and θ = 27º, 29º, 28º, 31º Then the average value of height is h 27 29 28 31 8.23 tan 4 4.515m Since the tangent of an angle is an increasing function, the extreme values of the height are hmax 8.235 tan 31 = 4.948m hmin 8.225 tan 27 = 4.190m hmax h = 4.948 – 4.515 = 0.433m h hmin = 4.515 – 4.190 = 0.325m So, maximum error = 0.433m After rounding off to have the same decimal h (4.52 0.43) m place with d, Example 3: Handling the sine of the angle with single measurement If a student has only one measurement for the angle, and have result d (8.23 0.005) m, (29 0.5) Then the mean and extreme values are: h 8.23 tan 29 = 4.561m hmax 8.235 tan 29.5 = 4.659m hmin 8.225 tan 28.5 = 4.466m hmax h = 4.659 – 4.561 = 0.098m h hmin = 4.561 – 4.466 = 0.095m So, maximum error = 0.098m After rounding off to have the same significant figure with d, h (4.56 0.10) m Further Example (This method is provided for students with strong mathematical background.) For any function f(x), we can find the error by d f x f x x dx f x x f x f x f x f x x x x x x x x x we have f x 1 f x x f x x x 2 x for an infinitesimally small error, that is, x 0 , or, for a small value of error, lim f x df x x dx d f x f x x dx Example 4: Handling error by calculus A student measures the angle of elevation of a building. He finds the angle is 34 0.5 . Find the value of sin and its error. 34 34 180 0.5 0.5 rad 0.593 180 rad 0.00872 sin sin 0.593 0.559 d sin sin cos 0.593 0.00872 0.00723 d sin sin 0.559 0.007 * Notice that we should use radian scale because we are using calculus.