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- In the previous sections, we have studied various types of errors, their origins and the ways to minimize them.
- In the previous sections, we have studied various types of errors, their origins and the ways to minimize them.
- Our accuracy is limited to the least count of the instrument used during the measurement.
- Our accuracy is limited to the least count of the instrument used during the measurement.
- Least count is the smallest measurement that can be made using the given instrument.
- Least count is the smallest measurement that can be made using the given instrument.
- For example with the usual metre scale, one can measure 0.1 cm as the least value.
- For example with the usual metre scale, one can measure 0.1 cm as the least value.
- Hence its least count is 0.1cm.
- Hence its least count is 0.1cm.
- Suppose we measure the length of a metal rod using a metre scale of least count 0.1cm.
- Suppose we measure the length of a metal rod using a metre scale of least count 0.1cm.
- The measurement is done three times and the readings are 15.4, 15.4, and 15.5 cm.
- The measurement is done three times and the readings are 15.4, 15.4, and 15.5 cm.
- The most probable length which is the arithmetic mean as per our earlier discussion is 15.43.
- The most probable length which is the arithmetic mean as per our earlier discussion is 15.43.
- Out of this we are certain about the digits 1 and 5 but are not certain about the last 2 digits because of the least count limitation.
- Out of this we are certain about the digits 1 and 5 but are not certain about the last 2 digits because of the least count limitation.
- The number of digits in a measurement about which we are certain, plus one additional digit, the first one about which we are not certain is known as significant figures or significant digits.
- The number of digits in a measurement about which we are certain, plus one additional digit, the first one about which we are not certain is known as significant figures or significant digits.
- Thus in above example, we have 3 significant digits 1, 5 and 4.
- Thus in above example, we have 3 significant digits 1, 5 and 4.
- The larger the number of significant figures obtained in a measurement, the greater is the accuracy of the measurement.
- The larger the number of significant figures obtained in a measurement, the greater is the accuracy of the measurement.
- If one uses the instrument of smaller least count, the number of significant digits increases.
- If one uses the instrument of smaller least count, the number of significant digits increases.
- Rules for determining significant figures
- Rules for determining significant figures
1) All the nonzero digits are significant, for example if the volume of an object is 178.43 cm^3, there are five significant digits which are 1,7,8,4 and 3.
1) All the nonzero digits are significant, for example if the volume of an object is 178.43 cm^3, there are five significant digits which are 1,7,8,4 and 3.
2) All the zeros between two nonzero digits are significant,
2) All the zeros between two nonzero digits are significant,
eg., m = 165.02 g has 5 significant digits.
eg., m = 165.02 g has 5 significant digits.
3) If the number is less than 1, the zero/zeroes on the right of the decimal point and to the left of the first nonzero digit are not significant e.g. in 0.001405, the underlined zeros are not significant.
3) If the number is less than 1, the zero/zeroes on the right of the decimal point and to the left of the first nonzero digit are not significant e.g. in 0.001405, the underlined zeros are not significant.
Thus the above number has four significant digits.
Thus the above number has four significant digits.
4) The zeros on the right hand side of the last nonzero number are significant (but for this, the number must be written with a decimal point),
4) The zeros on the right hand side of the last nonzero number are significant (but for this, the number must be written with a decimal point),
e.g. 1.500 or 0.01500 have both 4 significant figures each.
e.g. 1.500 or 0.01500 have both 4 significant figures each.
- On the contrary, if a measurement yields length L given as L = 125 m = 12500 cm = 125000 mm, it has only three significant digits.
- On the contrary, if a measurement yields length L given as L = 125 m = 12500 cm = 125000 mm, it has only three significant digits.
- To avoid the ambiguities in determining the number of significant figures, it is necessary to report every measurement in scientific notation (i.e., in powers of 10) i.e., by using the concept of order of magnitude.
- To avoid the ambiguities in determining the number of significant figures, it is necessary to report every measurement in scientific notation (i.e., in powers of 10) i.e., by using the concept of order of magnitude.
- The magnitude of any physical quantity can be expressed as A×10n where ‘A’ is a number such that 0.5 A<5 and ‘n’ is an integer called the order of magnitude.
- The magnitude of any physical quantity can be expressed as A×10n where ‘A’ is a number such that 0.5 A<5 and ‘n’ is an integer called the order of magnitude.
(i) radius of Earth = 6400 km = 0.64×10^7 m The order of magnitude is 7 and the number of significant figures are 2.
(i) radius of Earth = 6400 km = 0.64×10^7 m The order of magnitude is 7 and the number of significant figures are 2.
(ii) Magnitude of the charge on electron e = 1.6×10^-19 C.
(ii) Magnitude of the charge on electron e = 1.6×10^-19 C.
Here the order of magnitude is -19 & the number of significant digits are 2.
Here the order of magnitude is -19 & the number of significant digits are 2.
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