14. Given the equation y2: = 2x 1, take the following values of y, namely, 0, ± 0·5, 10, 15, 20, 25, and 30. Calculate the corresponding values of x, plot the points, and draw a fair curve through them. Name the curve. Unit = 1 inch.

15. The equation to an ellipse is 0·49x2 + y2 = 1.96. Take the following values of x, namely, 0, ± 0·4, ± 0·8, ±1·2, 16, 18, and ± 20. Calculate the corresponding values of y plot the points and draw a fair curve through them. Unit 1 inch.

=

16. Draw the rectangular hyperbola xy = 200 between the limits x = x 40. Unit = 0.1 inch.

5, and

17. A cubic foot of gas at a pressure of 100 pounds per square inch expands isothermally until its volume is 5 cubic feet. Draw the expansion curve which shows the relation between the pressure and volume of the gas. Scales,Pressure, 1 inch to 20 pounds per square inch. Volume, 1 inch to 1 cubic foot. 18. Plot the curves, y = x, and y = x3, between the limits = 2. Scale for x, 1 inch to 1 unit. Scale for y, 1 inch to 10 units.

19. Plot the curves, y

=

x2, y

=

x23, and y = x3, between the limits, x = 0 and 3. Scale for x, 1 inch to 1 unit. Scale for y, 1 inch to 5 units. 20. Plot the points whose co-ordinates are given in the following table, and find the equation to the fair curve joining them.

Take the scale for x as 1 inch to 1 unit and the scale for y as 1 inch to 5 units. 21. Plot the points whose co-ordinates are given in the following table, and find the equation to the fair curve drawn through them.

=

22. Plot the curve y(x +0.75)1.32 6, between the limits x = 0·75 and x = x and y being in inches.

4,

23. The volumes (v cubic feet) and corresponding pressures (p pounds per square inch) of a pound weight of steam as it expands are given in the following table. Plot the expansion curve and test whether it approximately follows the law pvn = c, and if it does, find the values of n and c.

24. Draw the first and second convolutions of an Archimedean spiral, having given that a certain radius vector cuts the first convolution at 0.7 inch from the pole and the second at 2 inches from the pole. Draw the tangent to the curve at the point which is 1.5 inches from the pole.

25. The equation to an Archimedean spiral is, r = 0 inches. Draw two convolutions of the spiral.

26. Draw a triangle POQ. OP = 1.1 inches, OQ = 1.4 inches, and the angle POQ = 45°. P and Q are points on the first convolution of an Archimedean spiral of which O is the pole. Find the initial line and draw the first convolution of the curve.

27. The equation to a logarithmic spiral is, r = 1.3 inches. Draw one convolution of the spiral and draw a tangent to the curve inclined at 60° to the initial line.

28. In a logarithmic spiral the common ratio of two radii containing an angle of 30° is 8:9. The longest radius vector in the first convolution is 3 inches. Draw one convolution of the spiral.

You may obtain points on the curve by calculation, using mathematical tables; or by projection from the logarithmic spiral, r = ae-0·1750,

0.175 X 2

1.10

€2.7181 = 1:3 very nearly.

Observe that in this spiral the ratio of the lengths of two radii which differ in direction by 360° = e [B.E.] 30. The curve (Fig. 267) representing the curtail of a stone step is constructed of circular arcs, each of 90°. Set out the figure to the given dimensions, which are in centimetres. Scale . [B.E.) 31. The given volute (Fig. 268) of one convolution is constructed of circular arcs each of 90°. Set out the curve, when the dimensions are as figured, in centimetres.

[B.E.]

32. Fig. 268 shows a construction for describing a volute of one convolution. Set out a similar curve of one and a half convolutions, when the dimensions, in centimetres, are as in Fig. 269.

34. Solve, graphically, the simultaneous equations:

[B.E.]

CHAPTER X

PERIODIC MOTION

126. Periodic Motion.-If any position of a moving body be selected and it is found that after a certain interval of time the body is again moving through that position in the same direction and with the same velocity, the body is said to have periodic motion. The interval of time between two successive appearances of the body in the same position moving in the same direction with the same velocity is called the periodic time or period of the motion. Periodic time is generally measured in seconds and will be denoted by T.

The simplest case of periodic motion is that of a body revolving about a fixed axis with uniform velocity. In this case the periodic time is the time taken to make one complete revolution. A swinging pendulum has periodic motion, but it has different velocities in different positions. In general all the parts of a machine have periodic motion although some parts may have very variable velocities and very complicated motions during a period.

The number of periods in the unit of time is called the frequency and is the reciprocal of the periodic time, or frequency = N = 1/T.

127. Simple Harmonic Motion.-P (Fig. 270) represents a point which is moving with a uniform velocity V along the circum

ference of the circle XYX,Y, whose centre is C. M is another point which is moving backward and forward along the diameter YY, of the same circle in such a manner that PM is always at right angles to YY1, in other words, M is the projection of P on YY,. Under these circumstances the point M has simple harmonic motion. If XX, is a

diameter at right angles to YY, and if PN is perpendicular to XX, then the point N also has simple harmonic motion.

CP is the representative crank of the harmonic motion of the point M or the point N, and the circle XYX,Y, is the auxiliary circle or circle of reference.

The amplitude of a harmonic motion of a point is the greatest displacement of the point from its mean position and is equal to the radius of the representative crank.

Denote the angle PCX by 0 and let CP =r, CN = x, and CM=y. Then, x = r cos 0, and y = r sin 0.

If the angular velocity of CP be w = V/r, and if P moves from X to P in the time t, then = wt. Hence, ar cos wt, and

y = r sin wt. If the time t be plotted along the straight line OL and the corresponding displacement y be plotted at right angles to OL as shown in Fig. 270, the harmonic curve OEFGL is obtained. The length OL represents the periodic time and this has been divided into twelve equal parts. The circumference of the circle has been divided into the same number of equal parts to obtain the corresponding values of 0.

1

The distance y is to be taken as positive when measured above XX, and negative when measured below. Also, the distance x is to be taken as positive when measured to the right of YY, and negative when measured to the left.

Since is proportional to t, lengths along OL will also represent values of 0 and the curve OEFGL is then called a sine curve. If is in degrees OL represents 360°, but if is in radians OL represents 2π.

The tangent to the harmonic or sine curve at any point Q is determined as follows. Draw Qq parallel to OL to meet the circle at Draw qn perpendicular to XX1. Draw fS Make Qf FO. at right angles to OL. Make ƒS to any convenient scale and make ƒH = 1 on the same scale. Make fu Cn. Draw uK parallel to HS to meet ƒS at K. KQ is the tangent required.

The mean height

=

2r

of the curve OEF is and the mean depth

2r

π

of the curve FGL is also or 0.637r. In other words, the mean ordi

π

nate for half a period is 0.637r. If is the mean of the squares of the ordinates for half a period, then s2 = r2 and therefore 8 = 0·707r. These results are of importance in connection with alternating electric currents.

Consider the equation y = 2 sin 1.2t. simple harmonic curve for which r =

per second. The periodic time is T =

curve may be drawn as shown in

This is the equation to a 2 feet, and w = 1.2 radians

2π π

=

1.2 0.6
Fig. 270.

= 5 236 seconds. The

OL will represent

5.236 seconds and will be measured on a time scale.

If OL and the

circle be divided into twelve equal parts the successive values of t will be

π

2π

Зп

etc. This harmonic curve may also be 12 × 0.6' 12 × 0·6' 12 × 0.6' constructed by calculating values of y from the equation y = 2 sin 1-2t. In this case values of t increasing by, say, 0.5 second may be taken. The values of t would be multiplied by 1.2 to obtain the values of which would be in radians. Taking the values of the sines of these angles from a table of sines and multiplying them by 2 would give the corresponding values of y. t would be plotted along OL on a time scale while y would be plotted at right angles to OL on a distance scale.

Consider the equation y = 2·5 sin 0. This is the equation to a sine curve which may be constructed as in Fig. 270. The radius of the circle is 2.5 on any convenient scale and y will be measured on the same scale. If the circle be divided into twelve equal parts the successive values of 0 will be 30°, 60°, 90°, 120°, etc. and these will be plotted along OL to any convenient scale. The corresponding values of y are projected from the circle as shown, This sine curve may also be constructed by calculating values of y from the equation y = 2.5 sin 0, values of sin being taken from a table of sines.

The exact position at any instant of the point M (Fig. 270) which has simple harmonic motion is defined by the angle which is called the phase angle. The term phase is generally used to designate the position of the point M at any instant by stating the fraction of its period of motion which it has performed reckoned from its middle position when moving in the positive direction. The first or zero phase is when M is passing through C towards Y. For this position = 0. For a complete period = 2π or 0

360° and for any position the phase is the

2 period phase. For example, when is equal to the phase is the half period phase.

A well known mechanism which gives simple harmonic motion is shown in Fig. 271. The crank pin of the crank works in the slot on the end of the piece which is made to reciprocate. The reciprocating piece has simple harmonic motion when the crank rotates with uniform velocity.

FIG. 271.

128. Lead and Lag.-Instead of using the phase angle to fix the position of a point having harmonic motion it is frequently more convenient to give the angular position of the representative crank with reference to a fixed radius of the auxiliary circle which is inclined at an angle a to the zero position of the representative crank. In Fig. 272 CX is the zero position of the representative crank and CA is a fixed radius inclined at an angle a to CX. CP is any position of the crank and is the angle which CP makes with CA. The angle a is positive when measured in the anti-clockwise direction and negative when measured in the clockwise direction.