Units, Errors and Graphs

1.1. Units. Every physical experiment requires the measurement of one or more quantities. To express a quantity completely we must give the unit in which it is measured and the number of times this unit is contained in it. There are three fundamental units, namely, those of length, mass and time. The systems of fundamental units in common use are:

1. The M.K.S. system. The units of length, mass and time on this system are metre, kilogram and second respectively.

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The metre is the distance between the centres of two transverse lines engraved upon the polished surface of a platinum-iridium bar at the temperature of melting ice, kept at the International Bureau of Weights and Measures at Sevres near Paris.

The General Conference on Weights and Measures (1960) defines the metre as the length equal to 1,650,673.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels p10 and 'd5 of krypton -86. The wavelength of radiation being 6057.802 × 10-10 metre.

The international kilogram is the mass of a cylinder of platinum-iridium kept at the International Bureau of Weights and Measures at Sevres near Paris.

The mean solar second is 1 / (24 x 60 x 60) = 1 / 86400 th part of the mean solar day.

2. The C.G.S. system. The units of length, mass and time on this system are centimetre, gram and second respectively.

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A centimetre is one hundredth (1 / 100) part of a standard metre.

A gram is one thousandth (1 / 1000) part of the international kilogram.

For all practical purposes one gram is the mass of 1 cc of pure water ar 4°C.

3. S.I. units. The International system of units briefly writter as S.I. las six fundamental units

The metre, kilogram and second have already been defined.

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The ampere is the strength of that constant current which flowing through each of the two parallel and straight conductors of infinite length and negligible cross-section placed exactly one metre apart in vacuum exert a mutual force of 2 x 10-7 newton per metre of their length.

The kelvin is 1 / 273.16 of thermodynamic temperature of the triple point of water.

The candela is the luminous intensity in a perpendicular direction of a surface 1/600,000 square metre of a black body at a temperature of freezing platinum under a pressure of 101.325 newton per sq. metre.

Most of the quantities are measured in units, the magnitude of which depends upon those of fundamental units of length, mass and time. These units are called derived units. As an example, the S.I. unit of area is a square metre which is the area of a square of side one metre in length. Some of the derived units are given special names. For example, the S.I. unit of force is called a newton and the S.I. unit of work a Joule. When no special name has been given to a unit it is expressed in terms of the fundamental units, e.g., acceleration is stated as 'metre per second per second'.

1.2. Errors. There is some error or the other in every measurement we make. The errors are of two kinds:

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1. Errors due to known causes (systematic errors).
2. Errors due to unknown causes (random errors).

1. Errors due to known causes. Some important causes of such errors are given below :-
1. Error due to the temperature of the measuring scale being different from that at which it was graduated.
2. Error due to buoyancy of air which arises when a large body is weighed.

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3. Error in measuring time with a stop watch, the watch running either too slow or too fast.
4. Error due to radiation loss or gain in calorimetric experiments.
5. Zero error in various measuring instruments.

All such errors can be eliminated by suitable methods since we know the cause e.g., the error due to buoyancy can be calculated from the density of the weights, density of the substance and the density of air; the error in a stop watch can be checked by comparison with a standard chronometer; the radiation error can be minimised by suitable means or calculated by the preliminary experiment and the zero error can also be measured.

2. Errors due to unknown causes. If an observation is repeated a number of times by the same person under similar conditions, it is found that every time a different reading is obtained, even though the instrument used is very sensitive and accurate and the observer is an experienced one. These errors are not due to any definite cause. We, therefore, cannot depend upon a single observation.

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When a large number of observations are taken it is likely that some of them may have a value slightly greater than the correct value and an equal number may have a value slightly less than the correct value. This is why it is recommended that each observation must be repeated at least three times.

The effect of random error may, therefore be minimised by taking a number of measurements of the quantity to be determined and using the arithmetic mean of the measured values as the best estimate of the true value of the quantity.

Thus if n measurements of the quantity are made, all equally reliable and the measured values are X1, X2, ...... Xn, then

Arithmetic mean X = (1 / n) (X1 + X2 + ...... + Xn) = (1 / n) SXi (from i=1 to n)

The true value is defined as the mean value of an infinite set of measurements made under constant conditions and is denoted by µ. In practice, it is not possible to make an infinite number of observations. Hence when the number of observations is sufficiently large,

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True value µ = Mean value X

The precision with which a physical quantity is measured depends inversely upon the deviation or dispersion of the set of measured values Xi about their mean value X. If the values, are widely dispersed or the observed values have a large deviation from the true value, the precision is said to be low.

The deviation di = Xi - X

Average deviation. The average value of the deviation of all the individual measurements from the arithmetic mean is known as average deviation and is denoted by a

... Average deviation a = (|X1 - X| + |X2 - X| + ..... + |Xn - X|) / n = (S|di|) / n (from i=1 to n)

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Standard deviation. The square root of the mean square deviation for an infinite set of measurements is known as standard deviation (or root mean square deviation) and is denoted by s. When the number of observations n is sufficiently large

s = v((X1 - X)² + (X2 - X)² + ...... + (Xn - X)²) / (n - 1) = v(Sdi²) / (n - 1) (from i=1 to n)

For a fairly large sample of measurements which have a reasonably normal distribution about the mean value X

Average deviation / Standard deviation = a / s ˜ 0.80

Standard error. The quantity s / vn is known as standard error and is denoted by sm

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... Standard error sm = v(S(Xi - X)²) / (n(n - 1)) = v(Sdi²) / (n(n - 1)) (from i=1 to n)

The normal law of errors predicts that the probability that the mean value obtained from a finite number of observations n may be in the interval µ ± sm is 0.68, that it may lie between µ ± 2sm is 0.95 and for it to lie between µ ± 3sm is 0.99.

Probable error. The probable error is a quantity e such that it is an even chance whether true value of the quantity measured differs from the mean value by an amount greater or less than e. For example, if 5.45 is the mean of all the determinations of the density of the earth and 0.25 is the probable error, then it means that there is an even chance that the density of earth lies between 5.20 (5.45 - 0.25) and 5.70 (5.45 + 0.25). It also implies that the chance that the value of the mean density of the earth may differ from the mean value by more than 4 or 5 times the probable error is very remote.

On the basis of the theory of probability it can be shown that

Probable error = ± 0.6745 v(Sd²) / (n(n - 1)) = ± 0.6745 standard error

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where 0.6745 is a natural constant.

Let us consider the following values for the radius of curvature of the surface of a convex lens with a spherometer.

Mean radius of curvature = 15.32

Sd² = d1² + d2².... + dn² = 0.0370

Standard deviation s = v(Sd²) / (n - 1) = v(0.0370 / 5) = 0.0860

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Standard error sm = v(Sd²) / (n(n - 1)) = v(0.0370 / (6 x 5)) = 0.03512

Hence probable error ± 0.6745 x sm = ± 0.6745 v(0.0370 / (6 x 5)) = ± 0.0237

Radius of curvature = 15.32 ± 0.0237

1.3. Degree of accuracy. In an experiment all observations must be taken to the same "degree of accuracy". It is no use taking some observations to a much higher degree of accuracy than the rest of the observations because it will not make the result more accurate. The accuracy of the result is the same as that of the 'least accurate observation'. It does not mean that the observations should not be taken accurately, but a careful sense of proportion must be used. For example, in the determination of specific heat by the method of mixtures it is useless to find the weight of the liquid in the calorimeter correct up to a fraction of a milligram. It is because apart from the errors due to the defects in the balance and the weights, it is not possible to measure the temperature to the desired accuracy. Great care, therefore, should be taken in reading the thermometer. To increase the accuracy practice should be made in estimating the temperature to at least 1/10th of a degree.

Percentage error. The students should have an idea of percentage error. For example, in the above experiment if the weight of the calorimeter is 50 gm an error of 5 mgm in the weight will mean only an error of .01%. The rise in the temperature is, however, only about 10°C and even if measured accurately up to one-tenth of a degree will cause an error of 1% which is 100 times as large as the error in the measurement of the weight.

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1.4. Effect of combining errors. All observed values are subjected to mathematical treatment to find the results. It is not possible to produce greater accuracy by mathematical manipulation. The following are some of the important points:

1. Addition and subtraction. Suppose the quantities having their true values a and b have measured values a ± da and b ± db respectively where da and db are their absolute errors. To find the error dQ in the sum Q = a + b, we have

Q ± dQ = (a ± da) + (b ± db) = (a + b) + (± da ± db)

dQ = da + db

To find the error in the difference Q = a - b, we have

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Q ± dQ = (a ± da) - (b ± db) = (a - b) + (da ± db)

The maximum error in Q is again given by

dQ = da + db

Thus we find that when two quantities are added or subtracted the absolute error in the final result is the sum of the absolute errors in the quantities.

Example 1. Resistance R1 = 100 ± 2 ohm

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Resistance R2 = 200 ± 3 ohm

Equivalent resistance when connected in series

R = (100 ± 2) + (200 ± 3) = 300 ± 5 ohm

Example 2. Mass of a bulb with air = 66.928 ±.001 gm

Mass of empty bulb = 66.682 ±.001 gm

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Mass of air = 0.246 ±.002 gm

Though the mass of air has been found by subtraction the errors are added.

2. Multiplication and division. Suppose the quantity Q = ab and measured values of a and b are a ± da and b ± db respectively, then

Q ± dQ = (a ± da) (b ± db) = ab + b da ± a db ± da db.

Dividing L.H.S. by Q and the R.H.S. by ab, as Q = ab, we have

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1 ± dQ / Q = 1 ± da / a ± db / b neglecting ± (da db) / (ab) which is the product of two very small quantities. Hence the maximum error in Q is given by

dQ / Q = da / a + db / b and expressed as percentage error

(dQ / Q) x 100 = (da / a) x 100 + (db / b) x 100.

If the quantity Q = a / b, then also

Q ± dQ = (a ± da) / (b ± db) = (a / b) (1 + (da / a)) / (1 + (db / b)) ˜ (a / b) (1 + (da / a) - (db / b))

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Dividing L.H.S. by Q and R.H.S by a / b as Q = a / b we have 1 + (dQ / Q) = 1 + (da / a) - (db / b)

Hence the maximum error in Q

dQ / Q = da / a + db / b and expressed as percentage

(dQ / Q) x 100 = (da / a) x 100 + (db / b) x 100.

Thus if the result Q is the product or quotient of two measured quantities a and b the fractional or percentage error in Q is the sum of the fractional or percentage errors in a and b.

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Example 1. Capacity of a capacitor C = 2 ± 0.1 Farad

Applied voltage V = 25 ± 0.5 Volt

Charge on the capacitor Q = CV = 2 x 25 = 50 coulomb

Percentage error in C = (0.1 / 2) x 100 = 5%

Percentage error in V = (0.5 / 25) x 100 = 2%

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Percentage error in Q = 5 + 2 = 7%

Error in Q = 50 x (7 / 100) = 3.5 coulomb

Hence charge on the capacitor Q = 50 ± 3.5 coulomb.

Example 2. Mass of an object M = 345.1 ± 0.1 gm

Volume of the object V = 41.55 ± 0.05 cm³

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Density of the object D = M / V = 345.1 / 41.55 = 8.31 gm/cm³

Percentage error in M = (0.1 / 345.1) x 100 = 0.03%

Percentage error in V = (0.05 / 41.55) x 100 = 0.12%

Percentage error in D = (0.03 + 0.12) = 0.15%

Error in density D = 8.31 x (0.15 / 100) = 0.012 gm/cm³

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Hence density of the object = 8.31 ±.012 gm/cm³

As a corollary, it follows that if the result Q is some power n of a measured quantity a, then the fractional or percentage error in Q is n times the fractional or percentage error in a.

3. General case. The final result of an experiment is generally calculated from a set of observations taken with a number of measuring instruments and connected by means of a formula, the process requiring the use of multiplication and division. It can be shown that all the quantities do not affect the result equally. Some affect more than others do. For example, consider a quantity Q which depends upon other quantities a, b and c connected by the relation

Q = k a^x b^y c^z where k is a constant and x, y, z are numerical values, then

dQ / Q = x (da / a) + y (db / b) + z (dc / c)

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or Percentage error in Q = x times the % error in a plus y times the % error in b plus z times the % error in c.

Thus the effect on the final result is greater for the factor having higher power than that having smaller power. If in the above case x is the higher power the quantity a must be measured with the greatest possible accuracy to make da as small as possible.

The area of a square is given by

A = l²

dA / A = 2 (dl / l)

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In other words, if the length of the side of a square is measured and found to be 10 cm and this measurement has a possible percentage error of ± 0.5, then the value of the area will have a possible error of 2 x 0.5 = 1%.

To find the value of g by a simple pendulum, we have

g = 4p² (l / T²)

dg / g = (dl / l) - 2 (dT / T)

Hence a slight error in the measurement of the time period will produce double the error in the result. The time period, therefore, should be measured very accurately.

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To find the viscosity of a liquid by Poiseuille's formula, we have

? = (p p r4) / (8 l v)

d? / ? = (dp / p) + 4 (dr / r) + (dl / l) + (dv / v)

In the case of random errors i.e. errors due to unknown causes the sign should be chosen in such a manner as to give maximum error.

d? / ? = (dp / p) + 4 (dr / r) + (dl / l) + (dv / v)

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It is to be noted that an error in the measurement of the radius of the capillary tube is magnified 4 times in the final result.

This is why in experiments where the quantities to be measured are raised to some power, the quantity with the highest power should be measured with greatest possible accuracy.

Significant figures. The number of significant figures to which the final result of an experiment should be stated depends upon the nature of the experiment and the accuracy with which the various measurements have been made. For example, in the measurements of the value of g at a place it is not desirable to write it as 980.432 cm/sec because the length and the time period are not measured with the desired accuracy. Moreover, the various other sources of error, e.g, friction due to air, amplitude not in one plane etc., cannot be removed.

In cases where the numbers dealt with are very large as 1053000000000 it may be written as 10.5 x 10¹¹ if the accuracy is 1 in 100, and as 10.53 x 10¹¹ if the accuracy is 1 in 1000.

Thus the number of significant figures automatically gives the accuracy of the result.

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Result. Students often attach a great importance to the result of an experiment. They are usually alarmed when they find a large error in the result. But it is the percentage error that matters and not the absolute error as is clear from examples given below:

1. In an experiment on the determination of 'g' the acceleration due to gravity, the result is 9.6 ms-2. This result has an error of 9.8-9.6 = 0.2 ms-2.
2. In an experiment on the determination of specific gravity of common salt the result is 2.37 instead of 2.17. This result has an error of 2.37-2.17=0.20.

It is clear that the absolute error in both the experiments is the same, but it does not mean that percentage accuracy is the same.

The percentage accuracy in the first case is (0.2 / 9.8) x 100 ˜ 2% (nearly)

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and in the second case it is (0.20 / 2.17) x 100 ˜ 9% (nearly)

1.5. Graph. Physical laws express relationship between various physical quantities. These relationships can easily be expressed by means of graphs.

A graph is a pictorial representation of experimental data in the form of a curve which makes visible, at a glance, the main features of the relationship between two variables.

One of the two variable quantities is varied at will, generally in convenient equal steps and the corresponding values of the second variable quantity are observed experimentally. The corresponding values of the two quantities obtained are plotted on a squared paper and a graph is obtained. The quantity which is varied at will is called independent variable and the other which varies as a result of variation in the first is called dependent variable. As a general rule independent variable is plotted along the X-axis while the dependent variable is plotted along the Y-axis.

Plotting of a graph. The following rules must be observed to plot a good graph.

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1. Find the independent and the dependent variables. Represent the independent variable along the X-axis and the dependent variable along the Y-axis.

Examples. (i) In an experiment with a simple pendulum the time period is measured for different lengths by changing the length. The length is the independent variable and the time period the dependent variable.

(ii) In the verification of Boyle's law where the volume is measured for various values of pressure, the pressure is the independent variable and volume (or 1/V) the dependent variable.

(iii) In the experiment on the determination of Young's modulus where the extension of the wire is measured for various loads, load is the independent variable and extension the dependent variable.

2. Determine the range of each of the variables and count the number of big squares available for each along the two axes.

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3. Scale. Choose a convenient scale for both the variables. It is not essential to have the same scale for both. The scale should neither be too narrow nor too wide. If the scales are too wide the irregularities due merely to experimental errors will become very much magnified and the shape of the graph will not be proper. If the scales are too narrow the points corresponding to accurate observations cannot be plotted with the same degree of accuracy. For convenience one big square should represent 1, 2, 10 or their multiples by any positive or negative power of 10.

A study of the specimen graph between P and I/V for air in Fig. 0.1 shows that the pressure varies from 64.00 to 88.75 cm. Since pressure is the independent variable it is represented along the X-axis. The range of pressure lies roughly between 60 cm and 90 cm. There are seven big squares and the most convenient scale is to represent one big square by 5 cm thus utilising 6 squares.

Reciprocal of volume 1/V is the dependent variable and is represented along the Y-axis. The value of 1/V varies from .045 to.062. There are 10 big squares and the most convenient scale is to represent one big square by .002 thus utilising 9 squares.

Students sometimes unnecessarily trouble themselves in plotting and also make it less easy to read the values from the curves by selecting an inconvenient scale, e.g., in the above case one big square equal to 4 will utilise more squares but will be an unwise choice.

4. Origin. If the relation between the two variables begins from zero or if it is desired to find the zero position on one of the variables where actual determination is not possible, then zero must be taken at the origin on both the scales.

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In all other cases it is not necessary to take zero at the origin. When it is desired simply to test the proportionality of one quantity to the other the origin on both the axes should represent a quantity a little less than the smallest value of the corresponding variable.

In the specimen graph of Boyle's law the origin should be 60 along the X-axis and .044 along the Y-axis which represent pressure P and the reciprocal of volume I/V respectively.

Of course, it will be more complicated to take the actual least value of the two variables at the origin.

5. The quantities to be plotted along X-axis and Y-axis should be given on the top of the graph in a tabular form.
6. Do not write all the values along the respective axes but only mark the ends of the thicker lines to indicate the values of the variables in round numbers. For example, in the graph Fig. 0.1 round values of pressure 60, 65, 70 etc. and those of 1/V .044, 046, 048 etc., are indicated along the respective axes.

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7. Draw a small circle round each of the plotted points or put a cross mark neatly. Join the plotted points by a regular free hand curve and not by zig-zag lines from one point to the other. Maximum number of points should lie on the curve and the remaining points should be situated symmetrically on either side of it.
8. Take at least six observations (preferably more) whenever a graph is to be plotted. The range of observations should be as wide as possible.

Uses. 1. The graphs are used to extrapolate certain quantities beyond the limits of observations of the experiment, e.g. absolute zero can be determined by plotting a graph between pressure and temperature at constant volume. By extrapolating, the temperature at which pressure becomes zero is found and that gives the value of the absolute zero. In this case, it is evident that origin must be zero on both the axes.

2. Verification of certain laws can best be represented by graphs. There is always a mathematical relationship between the two variables. It may be simple or a complex relation. For the purpose of verification of a law the graphs may be divided into two groups.

1. Straight line graphs. A straight line graph is very easy to interpret. The equation of a straight line is

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y = mx + c

where y is the dependent variable, x the independent variable, m the slope or the gradient and c the intercept on the Y-axis.

If the graph plotted is a straight line, we can evaluate the constants m and c and thus we can find the relation that exists. For example, if we plot a graph between E the potential difference across the ends of a conductor and I the current through it the graph is a straight line passing through the origin. If the slope of the straight line is represented by a constant R (intercept c being zero) the relation can be put in the form

E = R (a constant)

This is the statement of Ohm's law.

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2. Graphs which are not straight lines. If the graph is not a straight line, there still may be a simple equation that gives the relation between the independent and the dependent variables. The relation can be converted into linear one by changing the variable, thus giving a straight line graph.

It should, however, be clearly understood that we do not at all change the quantity which is variable in the experiment. For example, in the experiment on verification of Boyle's law the independent variable is the pressure P and the dependent variable is the volume V. According to Boyle's law

PV = k (constant)

The graph between P and V will not be a straight line. To get a straight line the relation is put in the form

P = k (1 / V)

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which shows that a graph between P and 1/V will be a straight line. It is for this reason that the specimen graph has been plotted between P and 1/V.

Similarly in the verification of Newton's law of cooling a straight line graph is obtained by plotting log of excess of temperature and time and thus the law is verified.

In practice the matters often are not as simple as given above particularly when we do not know the relation that exists and we are required to find it out. If y varies as some power of x, say

y ? x^m

then the best method is to plot log y (dependent variable) against log x (independent variable). Slope of the straight line gives the power of x in the original relation. To understand the theory suppose the relation is

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y ? x^m

or y = k x^m

then log y = m log x + log k

or log y = m log x + c ...(1)

where c is another constant.

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Hence a graph between log y as

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