UNIT – 3 Uncertainty Every measuring instrument has some uncertainty. This uncertainty is always added or subtracted from the reading that we receive. It means if the measuring instrument has an uncertainty of 0.1 units, and the reading which is found is 10 units, then the absolute uncertainty is 10±0.1. The length of the an object AB is measured by a measuring instrument which has smallest division of 1cm. The object is placed close to the measuring instrument and parallel to it to minimize experimental error. Maximum Minimum 𝑅! = 4 + 1 𝑅" = 20 + 1 𝑅! = 4 − 1 𝑅" = 20 − 1 There are two types of readings, the maximum reading and minimum reading. 𝐿#$% = 𝑅" – 𝑅! = (20 + 1) − (4 − 1) = 18 𝐿#&' = 𝑅" – 𝑅! = (20 − 1) − (4 − 1) = 14 The smallest division of a measuring instrument is called precision. If a physical quantity is measured by using a measuring instrument, it gives a range of values of measurement rather than a particular magnitude. This range depends on the precision of the measuring instrument. Usually, this range is equal to twice the smallest division of the measuring instrument. A fraction of the smallest division of the measuring instrument cannot be determined using the particular instrument. For a single measurement, uncertainty is equal to its precision. As the number of measurement increases, uncertainty also increases. Uncertainty of measuring instruments can be represented in three different ways: 1. Absolute uncertainty 2. Fractional uncertainty 3. Percentage uncertainty Absolute uncertainty Absolute uncertainty is represented with unit. For example, (𝑥 + 𝛥𝑥) cm or 𝑥 cm ±𝛥𝑥 cm Fractional uncertainty The ratio of uncertainty and the measured value is called fractional uncertainty. For example, 2𝑥 ± ∆% % 3 cm Percentage uncertainty Fractional uncertainty multiplied by 100 gives the percentage uncertainty. Some Common Instruments and Their Precision Measuring Instrument Metre Rule Vernier Caliper Micrometer Screw Gauge Stopwatch Protractor Thermometer Description It is suitable for measuring lengths from 0 to 1m. For large length, measuring tape is used. It is suitable for measuring diameters, and lengths from 0 to 2.5cm. It is suitable for measuring diameters, and lengths from 0 to 2.5cm. It is used to measure time. A large uncertainty is produced due to human reaction. The average human reaction time is 0.25s. It is used to measure angles. It is used to measure temperature. Precision 1 mm 0.1 mm 0.01 mm 0.01 s 10 1 0C Uncertainty of Multiple Instruments/Readings For Length, 𝐿! = 𝑎 ± 𝛥𝑎 𝐿" = 𝑏 ± 𝛥𝑏 Total Length, 𝐿!)" = (𝑎 ± 𝛥𝑎) + (𝑏 ± 𝛥𝑏) 𝐿!)" = (𝑎 + 𝑏) ± (∆𝑎 + ∆𝑏) 𝐿!)" = 𝐿* ± ∆𝐿 Difference between lengths, 𝐿!+" = (𝑎 ± 𝛥𝑎) − (𝑏 ± 𝛥𝑏) 𝐿!+" = (𝑎 − 𝑏) ± (∆𝑎 + ∆𝑏) 𝐿!+" = 𝐿* ± ∆𝐿 For Area, Area, 𝐴 = (𝑥 ± 𝛥𝑥)(𝑦 ± 𝛥𝑦) 𝐴 = 𝑥𝑦 ± 𝑥𝛥𝑦 ± 𝑦𝛥𝑥 ± 𝛥𝑥𝛥𝑦 𝐴 = 𝑥𝑦 ± (𝑥𝛥𝑦 + 𝑦𝛥𝑥 + 𝛥𝑥𝛥𝑦) 𝐴 = 𝐴 ± 𝛥𝐴 Errors Experimental Errors Reliability of an experiment decreases due to experimental errors. There are two types of experimental error. Random Error: These errors are produced due to variation in performance of instruments or the operator of the experiment. During this error, readings are shifted along both side of the actual value. Some sources of random error are: • • • Variations of temperature Readings taken under lack of concentration Parallax error Random errors can be minimized by: • • Taking large number of measurements. Producing a best fit curve with graphical method. Systematic Error: These errors are produced due to measuring apparatuses. For this error, the reading may be shifted to a particular side only. The reading will either be increased or decreased from the actual value. This error cannot be minimized by taking several readings and calculating average. It can only be reduced by using more precise instruments. Thus, systematic error is also known as instrumental error. Some sources of systematic errors are: • • • Zero error of instrument Poor calibration of instrument Instrumental parallax error Some Common Techniques to Improve an Experiment • • • • • • • • • • Check zero error of instrument. Repeat the measurement at appropriate position. Different method of measurement should be used. Readings should be taken from eye level to avoid parallax error. Use set square to ensure vertical position where necessary. Using trigonometric calculations to measure angle. All readings must have same precision. Sufficient number of readings should be taken for graph plotting purposes (at least 6 for a straight line and 9 for a curve). Each reading should be repeated and the average should be taken. There must be consistent gap between readings.
