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| Arithmetic Operations With Significant Figures |
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| The accuracy of a result is taken to be equal to the least accurate among the numbers, when 2 or more numbers are used to add, subtract, multiply or divide. The number of significant figures in the result, is equal to the number of significant digits in the least accurate one among them. |
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| When we make use of various measuring instruments, we encounter various types of errors. They are: |
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| Constant errors |
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| If the same error is repeated every time in a series of observations, the error is said to be constant error. |
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| Constant error is due to faulty calibration of the scale of a measuring instrument. |
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| In order to minimise the constant error, measurements are made with all possible methods. |
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| Relative and percentage errors |
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| Relative error is the ratio of absolute error to true value given by |
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| Absolute errors |
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| It is the magnitude of difference between the true value and the measured value. |
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 Absolute error = True value - measured value |
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| If arithmetic mean is regarded as true value, absolute error in ith measurement is given by |
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| It is expressed in the units of measured value. Final absolute error is taken as the arithmetic mean of the absolute error in various measurements. |
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Considering a single measurement, the
result of measurement will lie between
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| Gross errors |
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| This error occurs due to a person's mistake while performing the experiment. It cannot be corrected. It is of three types:- |
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 Neglect of the source of error: This is a result of negligence of sources of errors. |
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| E.g., plotting field of a magnet, improper setting of magnet along NS line. |
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 Reading the instrument incorrectly: Sometimes 1 ampere might have been divided in 20 or 5 parts instead of 10. The experimenter may read the instrument without paying attention to the value of 1 division. |
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 Improper recording of reading: This mistake is done by the person performing the experiment when he records the reading incorrectly. Eg., He may record 21.3 in place of 23.1. |
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| Random errors |
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| These are irregularly occurring errors, which are at random, in magnitude and direction. They are not due to any definite cause and hence, are called accidental errors. A person may get different readings while experimenting, say, for E.g., while measuring the diameter of a wire with a screw gauge. This happens due to many reasons. Therefore, if the observation is repeated a number of times, the arithmetic mean of all readings is found to be very close to the most accurate reading. Therefore, in an experiment, it is advisable to take readings, a number of times and then take their arithmetic mean. |
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| If a1, a2 ………….. an are n different readings in an experiment, their arithmetic mean is given by |
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| Systematic errors |
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| These errors always have the same sign. These can be eliminated by detecting the source of error and the rule governing this error. They are of 4 types: |
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| Instrumental errors |
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| These are inherent errors of the measuring instruments and apparatus. E.g., zero error of a measuring instrument. All instrumental errors come under this category. Instrumental errors, if any, can be detected by interchanging two similar instruments or by using different methods for measuring the same physical quantity. |
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| Errors due to external causes |
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| These are caused by external conditions (wind, temperature, etc.,). These errors can be prevented by applying suitable corrections. |
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| Errors due to imperfection |
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| Sometimes, even if the error is known, it cannot be corrected due to imperfection in the experimental arrangement. |
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| e.g., Loss of heat due to radiation in calorimetry. |
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| These errors always exist, but observations can be corrected. |
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| Observational errors |
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| This arises from the mode of observation of the person, taking the reading. e.g., Parallax error. These errors can be minimised by obtaining several readings and taking their arithmetic mean. |
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