130. Empirical Data Problems. We shall content ourselves with pointing out the method of solution of some very simple problems in determining the constants in the equation of a harmonic curve. The treatment of empirical data problems in which the points representing a given table of values appear to lie on a harmonic curve furnishes sufficient material for a book involving considerable higher mathematics. EXAMPLE. Determine the constants a and b if the graph of y = a sin x + b sin 2x is to pass through the points (π/6, 1 + √3/2) and (π/4, 1 + √2). Since these points are to lie on the graph their coördinates must satisfy the given equation. Hence 1 + and √3 = 1 If several points are given which appear to lie on or near the graph of an equation of the form given in the example, we may determine a pair of values of a and b from each pair of points. If the values of a agree closely, and also those of b, by using the average value of the a's and that of the b's we obtain a good approximation of the law connecting the coördinates of the points. EXERCISES 1. Show that the equation y = a sin (bx + c) may be put in the form 2. If y = A sin bx +B cos bx, find the values of a and c for which the given equation has the form y = a sin (bx + c). 3. Show that y = a sin (bx + c) may be put in the form y a cos (bx + c'), where c' C- π/2. 4. On the same axes = sketch the graphs of (a) sin x, sin x, 3 sin 1x. = 5. Determine the phase, period, and amplitude of each of the following functions, and sketch the graph. 6. Sketch the following compound harmonic curves by the addition of NOTE. Simple Harmonic Motion. Let a radius of a circle, OP, revolve uniformly at the rate w, in radians per second (see Example 1, Section 128), and let M be the projection of P on a fixed diameter. The motion of M is called simple harmonic motion. The point M moves back and forth along the diameter, making one complete oscillation for every revolution of OP. Let the fixed diameter be chosen for the x-axis, and suppose that P starts on the positive x-axis. Then the abscissa of P, which gives the position of M at any time, is s = a cos wt, where a is the radius of the circle. If P starts at the point for which OP makes an angle of Oo with the x-axis then s = a cos (wt + Oo). If the diameter on which M moves is taken for the y-axis, the distance OM is given by s = a sin (wt + Oo). 8. A point moves according to one of the laws below. Find the time of one oscillation, and when and where the distance, velocity, and acceleration have maximum or minimum values. Describe the motion. 9. If s = a sin (wt + 0。) show that (a) the acceleration is proportioned to the distance, (b) that the acceleration and distance have maximum or minimum values when the body is at rest, (c) that the body is at the center of its motion and has no acceleration when it is moving fastest. 10. The wind has left long swells moving along the surface of a lake when a slight breeze produces a ripple. The equations of cross-sections of the waves before and after the breeze are y 2 sin x/4 and y = 2 sin x/4 + sin 2x. Plot both curves on the same axes. = 11. Determine the constants a and b if the graph of the given function is to pass through the given points. Sketch the graph. (a) a sin x + b sin x/2, (π/2, 3 – √2), (π, − 2). (b) a sin x/2 + b sin 2x, (π/3, 1 – √3/4), (2π/3, √3 + √3/4). (c) a sin x + b sin 2x, (.157, .309), (.332, .632), (.611, 1.04). MISCELLANEOUS EXERCISES 1. A and B are two points on the opposite sides of a hill which are to be connected by a tunnel. Both points are visible from a third point C which is 1356 feet from A and 2582 feet from B. If LACB = 47°.34 and if the angles of elevation of A and B at C are respectively 12°.35 and 9°.82, find the length of the tunnel and the inclination of the tunnel. 2. Construct a table of analogous properties (see Section 56) of the functions e, log x, and sin x. 3. Contrast the formulas for ea+b and tan (a + b) by expressing each entirely in terms of the notation f(x). 4. Find approximate values of the real roots of the following equations (see Exercise 12, page 261). 5. To find the distance between two towers on opposite sides of a river a base line 300 feet long was laid off. The following angles were measured, C and D representing the towers and AB the base line: ZABC= 35°.24, ▲ BAD = 29°.61, ZABD = 108°.47, Z BAC = 97°.59. Compute CD. 6. Find the number of acres in a field in the form of a quadrilateral ABCD if AB = 315 yards, ≤DAC = 123°.6, ZCAB = 54°.68, LABD= 65°.23, and LABC 112°.4. = 7. If a projectile is fired at an angle ✪ with the horizontal with an initial velocity of vo, the equation of its path is Find the range as a function of 20, and the value of 0 which gives the maximum range. 8. A corridor turns at right angles. It is 8 feet wide on one side of the turn and 6 feet wide on the other. How long a beam can be carried horizontally around the corner? 9. The path of a point on the rim of a wheel rolling along a level road is given by the equations t = where a is the radius of the wheel, w is the angular velocity of the wheel about the axle, and t is the time. Plot the equation of the path taking π/6W, π/3w, π/2w, etc. Find the velocity of the point at any time, and its maximum and minimum values. Where is the point when it is moving slowest? fastest? A bit of mud is thrown from the highest point of a wheel on an automobile. Compare the velocity with which the mud leaves the Iwheel with that of the automobile. 10. If a is the angle between two lines whose slopes are m1 and m2, show that CHAPTER IX THEORY OF MEASUREMENT 131. Statistical Methods. The object of an experiment is to replace a complex system in which a number of causes are operating by a simpler system in which causes are controlled and allowed to vary only at the will of the investigator. In physics, for instance, after the isolation of two related variables, an endeavor is made to determine the relation between the variables and to express this relation in the form of a function y = f(x). In some sciences it is difficult or impossible to isolate two related variables from the complex system in which they occur. In such cases the variation of each variable is observed independently of the others. Statistics has to deal with variables affected by a number of causes. Some of the methods of statistics which have been developed for the analysis of such variations are discussed briefly in this chapter, three of the principal ends sought being the determination of (1) An average value which will represent the values of a variable. (2) A measure of the variability of the items with respect to the average value chosen. (3) A measure of the extent of the relationship between two variables which are associated. The first few sections are given to some essential prerequisites. To illustrate the fundamental principle on which the following sections are based, consider the EXAMPLE. There are five pitchers and three catchers on a base ball team. In how many ways can a battery be chosen for a particular game? |