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Front Matter Pages Jean-Benoit Morin, Scott R. Brown, Matthew R. Pietro E. Cristian Osgnach, Pietro E. The prototypes of the International standard kilogram supplied by the International Bureau of Weights and Measures BIPM are available in many other laboratories of different countries.

While dealing with atoms and molecules, the kilogram is an inconvenient unit. Mass of commonly available objects can be determined by a common balance like the one used in a grocery shop. Large masses in the universe like planets, stars, etc. These may vary from tiny mass of the order of kg of an electron to the huge mass of about kg of the known universe. This implies that the uncertainty gained over time by such a device is less than 1 part in ; they lose or gain no more than 3 s in one year.

The time interval of events that we come across in the universe vary over a very wide range. You may notice that there is an interesting coincidence between the numbers appearing in Tables 2. Note that the ratio of the longest and shortest lengths of objects in our universe is about Interestingly enough, the ratio of the longest and shortest time intervals associated with the events and objects in our universe is also about This number, comes up again in Table 2.

The ratio of the largest and smallest masses of the objects in our universe is about 2. Is this a curious coincidence between these large numbers purely accidental? The result of every measurement by any measuring instrument contains some uncertainty.

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This uncertainty is called error. Every calculated quantity which is based on measured values, also has an error. We shall distinguish between two terms: accuracy and precision. The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured. The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument.

For example, suppose the true value of a certain length is near 3.

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In one experiment, using a measuring instrument of resolution 0. The first measurement has more accuracy because it is. We now use an atomic standard of time, which is based on the periodic vibrations produced in a cesium atom.

This is the basis of the cesium clock, sometimes called atomic clock, used in the national standards. Such standards are available in many laboratories. In the cesium atomic clock, the second is taken as the time needed for 9,,, vibrations of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium atom. The vibrations of the cesium atom regulate the rate of this cesium atomic clock just as the vibrations of a balance wheel regulate an ordinary wristwatch or the vibrations of a small quartz crystal regulate a quartz wristwatch.

The cesium atomic clocks are very accurate. In principle they provide portable standard. The national standard of time interval second as well as the frequency is maintained through four cesium atomic clocks. In our country, the NPL has the responsibility of maintenance and improvement of physical standards, including that of time, frequency, etc. The efficient cesium atomic clocks are so accurate that they impart the uncertainty in time realisation as. Thus every measurement is approximate due to errors in measurement. In general, the errors in measurement can be broadly classified as a systematic errors and b random errors.

Systematic errors The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are : a Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc.

3.2 Measures of Variation

For example, the temperature graduations of a thermometer may be inadequately calibrated it may read C at the boiling point of water at STP whereas it should read C ; in a vernier callipers the zero mark of vernier scale may not coincide with the zero mark of the main scale, or simply an ordinary metre scale may be worn off at one end.

Other external conditions such as changes in temperature, humidity, wind velocity, etc. For example, if you, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce an error due to parallax.

Systematic errors can be minimised by improving experimental techniques, selecting better instruments and removing personal bias as far as possible. For a given set-up, these errors may be estimated to a certain extent and the necessary corrections may be applied to the readings. Random errors The random errors are those errors, which occur irregularly and hence are random with respect. These can arise due to random and unpredictable fluctuations in experimental conditions e. For example, when the same person repeats the same observation, it is very likely that he may get different readings everytime.

Least count error The smallest value that can be measured by the measuring instrument is called its least count. All the readings or measured values are good only up to this value.

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The least count error is the error associated with the resolution of the instrument. For example, a vernier callipers has the least count as 0. Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors.

If we use a metre scale for measurement of length, it may have graduations at 1 mm division scale spacing or interval. Using instruments of higher precision, improving experimental techniques, etc. Repeating the observations several times and taking the arithmetic mean of all the observations, the mean value would be very close to the true value of the measured quantity.

The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement. This is denoted by a. In absence of any other method of knowing true value, we considered arithmatic mean as the true value.

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But absolute error a will always be positive. It is represented by amean. This is because, as explained earlier, it is reasonable to suppose that individual measurements are as likely to overestimate. The relative error is the ratio of the mean absolute error amean to the mean value amean of the quantity measured.

The errors in the measurements are 2. As the arithmetic mean of all the absolute errors is 0. Hence there is no point in giving the period to a hundredth. We indicate this by saying that the measurement has two significant figures. In this case, the two significant figures are 2, which is reliable and 6, which has an error associated with it. You will learn more about the significant figures in section 2. When the relative error is expressed in per cent, it is called the percentage error a.

(PDF) control system engineering (6th edition) solution | Aqeel Ahmad -

At noon by the standard clock, the readings of the two clocks are : Clock 1 Monday Tuesday Wednesday Thursday Friday Saturday Sunday Clock 2 If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer? Answer The range of variation over the seven days of observations is s for clock 1, and 31 s for clock 2. The average reading of clock 1 is much closer to the standard time than the average reading of clock 2.