If the pairs of values be plotted, it may happen that the points so obtained lie very nearly on a straight line. If such is the case, it is reasonable to assume that the graph of the relation is a straight line, and hence that the relation connecting the two variables is a linear equation.

When graphical methods are being used, all that is needed is the graph of the relation. Any straight line which passes through or near to each of the points will serve approximately as the graph. A method frequently used by engineers to get the line giving the best approximation is to stretch a rubber band over two pins stuck in the drawing board, and move the pins about until the stretched band appears to make the average distance from the band to each of the points as small as possible. When algebraic methods are employed, a method of obtaining the equation of a line which answers well as the graph is illustrated in the following examples.

W

FIG. 44.

EXAMPLE 1. In an experiment dealing with friction of wood on wood, a block of wood with various weights on it was placed on a horizontal board. A string fastened to the block ran over a pulley at the end of the board, and a pan was tied to the hanging end. Weights were placed in the pan until the block was just on the point of moving. Using W to represent the combined weight of the block and the weights on it, and W' for the

weight of the pan and of the 40,11.2) weights in it, corresponding values of W and W' were found as indicated in the table,

(30,8.1)

W

30

40 W

FIG. 45.

W'

10, 20, 30, 40

3.1, 5.8, 8.1, 11.2'

the unit of weight being the gram. Determine approximately the relation between W and W'.

Plot the points whose coördinates are the pairs of values of The four points obtained lie

W and W', using values of W as abscissas.

very nearly on a straight line through the origin, and hence we assume that the graph of the relation is a straight line through the origin.

(That the graph ought to pass through the origin is clear, for if W = 0, so also must W' = 0.) The required relation must therefore be of the form W' = mW, where m is the slope of the line.

The slopes of the lines joining the origin to each of the four points are respectively

=

From these values it appears that m may have any value between 0.27 and 0.31, and the line W' mW would be a fair approximation to the graph required. We shall choose as a good value of m the average of the four values of m, namely,

In dividing 1.15 by 4, the second decimal figure is 8, but the quotient is nearer to 0.29 than to 0.28; hence the former value is chosen. Hence the relation desired is represented approximately by the equation

W' = 0.29W.

% of

error

(1)

The accuracy with which this equation represents the given data may be determined by constructing the table adjoined. The first two columns give the observed values of W and W', the third, the values of W' computed by means of equation (1) from the observed values of W; the fourth, the error in the com

- 6.4

W W' 0.29W Error

10

3.1

2.9

- 0.2

20

5.8

5.8

0.0

0

30

puted values of W', which are obtained by subtracting the second column

The percentage of

from the third; and the fifth, the percentage of error. error is found by dividing the error by the observed value of W'. Thus 0.2/3.1 = 0.064, or 6.4 %.

Note the part played by mathematics in this illustration of the scientific method. The given data are obtained by observation. The principles of graphic representation enable us to put the data in a form which makes reasonable the hypothesis that the graph of the law under investigation is a straight line, and that the law is represented by a linear equation. By deductive processes we determine the numerical values of the coefficients of the equation, and the accuracy of the representation of the given data by the equation found. The verification

would consist, in part, of repeating the experiment a number of times, varying the values of W, and seeing if the law remained approximately the same. But a more satisfactory verification would be to deduce from this law some other law which could be verified by an experiment of a different sort. This will be done in a later section.

The law obtained in Example 1 can be stated in a more convenient form. The force tending to make the block slide is equal to W', because the tension of the string is the same at all points. This force acts in the direction of the surfaces in contact, and when motion is about to begin, it is numerically equal to the force of friction which is preventing the block from moving. W is the pressure on the board acting perpendicularly to the surfaces in contact. Hence the result obtained may be stated as follows, using the language of Section 21:

The friction of wood on wood varies as the pressure perpendicular to the surfaces in contact.

If different kinds of wood, or other substances, be used in the experiment, the value of m obtained would not be the same. But extensive experiments have shown that we are reasonably justified in stating the law:

When motion is about to take place, the friction between two surfaces varies as the pressure perpendicular to the surfaces. Hence if F denotes the force of friction and P the perpendicular pressure

F mP.

The constant m is called the coefficient of friction, and it is equal to the ratio of the friction to the pressure perpendicular to the surfaces in contact. The coefficient of friction for the

block and board in Example 1 is 0.29.

EXAMPLE 2. In an experiment with a block and tackle, the pull P necessary to raise a weight W, both measured in pounds, was found for

W | 100,

P

37,

200, 300, 400,
59, 87, 111,

each of the values of W in the table.

Find approximately the equation expressing P as a function of W.

Plotting the points representing the pairs of numbers in the table, using values of W as abscissas, it is seen that they lie very nearly on a

straight line. This line ought not to pass through the origin, because if W = 0, a certain force P is necessary to raise the lower block. Hence P is approximately a linear function of W of the form

P = mW+b.

A good value of m is the average of the slopes of the lines determined by each pair of points.

Denoting the points by A, B, C, D respectively, the slope of the line AB is

In like manner we find the value of m for each of the lines determined by two of the points A, B, C, D. These values are given in the table

In like manner, we find the value of b on the assumption that each of the points B, C, D, lies on the graph of (2). table.

The results are given in the

The accuracy with which this equation represents the observed data is shown by the table below. The first two columns give the observed values of W and P. The third column gives the values of P computed

from the observed values of W by means of (3). The fourth gives the error in the computed value of P as compared with the observed value, while the last column gives the percentage of error, which is obtained from the second and fourth columns.

A property of a linear function is illustrated in the table, which gives

5

Ay

2

Ax

3

1

1

1

3

9

1

2

4

11

2

values of x, y, Ax, and Ay for the function y = 2x +3. The values of x being such that the successive values of Ax are equal, it appears that the successive values of Ay are also equal. That this is always the case follows from the fact that the value of Ay/Ax is always the same for points on a straight line.

Now suppose, as in the examples above, the points representing a given table of values appear to lie on a straight line. To test the accuracy of this assumption, find the successive values of Ax and Ay. If the values of Ax are equal, and if those of Ay are nearly equal, we are justified in believing that y is indeed a linear function of x.

The labor involved in computing an average may be lessened by means of the following rule: Assume a number x which appears to be a reasonable "guess" for the average desired. Subtract x from each of the given numbers, find the average of the differences, and add it to x, paying due regard to signs throughout.

85

0

The process is illustrated for the average of the - 15 numbers in the first column, the value of x being 5 chosen as 85. The differences obtained by subtracting 85 from each of the numbers are given in 10 the second and third columns. Their average is obtained by adding them and dividing by 6, the - 30 number of numbers to be averaged, and is found to be - 2.5. This must be added to 85, which gives 82.5 as the average.

The advantage of the method lies in the fact that 85 + (2.5) = 82.5 it involves only relatively small numbers, which decreases the probability of error, and also enables

one to handle simple cases mentally.