5, 10, 15, 20, necting s and v for the corresponding values given in the table. In how many feet should the car stop when moving 30 miles an hour? 8. The magnitude of the pressure on a surface perpendicular to the direction of the wind is proportional to the area of the surface and to the square of the velocity of the wind. If the pressure on the side of a building 20 feet wide and 60 feet high is 648 pounds when the velocity of the wind is 15 feet per second, what is the pressure on a building 30 feet by 80 feet when the velocity is 45 feet per second? Draw a graph to show how the pressure on a particular building changes as the velocity of the wind changes. Draw a graph to show how the pressure on various buildings differs for the same wind. 9. The area of the safety valve for a boiler may be determined by allowing 1 square inch of valve to every 2 square feet of grate surface. Find the diameter of the valve as a function of the area of the grate, and plot the graph. What should the diameter be for a grate 10 feet by 6 feet? NOTE. When any thin plane surface is moved through the air so that the direction of the motion makes a small angle with the lower side, the resultant pressure of the air is very nearly normal to the plane surface. The lifting efficiency of the plane is increased by curving it longitudinally, the concave surface being placed so as to meet the air. There is a suction on the upper surface of good wings which amounts to more than twothirds of the total lift. If a plane surface is presented edgewise to the direction of motion, there is no upward pressure, but if a curved wing is presented edgewise there is upward pressure. If the forward edge of the wing is depressed, the upward pressure decreases. When it is depressed to the point where the upward and downward pressures balance, a horizontal chord of the wing may be drawn through the rear edge. In any other position of the wing, the angle which this chord makes with the horizontal is called the angle of incidence, 0. The equation connecting the normal pressure P, in pounds, the area S of the wing in square feet, the velocity v, in feet per second, and the angle of incidence 0, in degrees, is P = KSv20, where K is a constant of proportionality which remains constant only for small values of and variations in 0. If W represents the weight of the aeroplane in pounds, the necessary condition for flight is that the vertical component of the pressure on the plane is equal to W. For small values of 0, the vertical pressure is very nearly equal to the normal pressure, so that approximately, for horizontal flight W = KSv20. (1) = 10. (a) If W = 2500 pounds, S 400 square feet, and K = 0.000087, plot the graph of v as a function of 0. In practice @ varies from about 3°, the lowest safe value, to 12°. For what value of 0 will the aeroplane have the greatest horizontal velocity? If additional power is given by the motor the aeroplane will not increase its horizontal velocity but will rise, and if the motor is shut off, the aeroplane begins to fall, but the horizontal velocity remains the same. Explain. (b) The term loading is applied to the quantity W/S. If 0 = 5° and K = 0.000087, plot the graph of v as a function of the loading. How will v change if S is constant and W is quadrupled? How will is constant and S is reduced by ? change if W (c) If W, S and have the values used above, how will vary as K, the efficiency coefficient, changes? As the efficiency decreases, will the power required increase or decrease? Why not make aeroplanes with a lifting coefficient that is inefficient and fly very fast? 11. The total resistance to the motion of the aeroplane is made up of two parts, the horizontal component of the pressure on the planes, given by W0, and the resistance of the struts, body, etc., given by csv2, where c is a constant and s is a theoretical surface which would have the same resistance as the various elements of the aeroplane except the wings. If the value of v from (1) is substituted, the total thrust, t, is For a given aeroplane W, c, s, K, S are constant. The single constant f, called the fineness, is defined by the equation 1/f2 = cs/KS. Making this substitution, we have If ƒ = 5, and W = 2500, plot t as a function of 0. Determine approximately the most efficient value of 0. How does an increase in S affect the ratio S/s, hence the fineness, and hence the most efficient angle? How does the thrust vary as the fineness increases? How does the thrust vary with respect to s? with respect to the plane area, S? the lifting efficiency? 12. The power required for horizontal flight at velocity v is Substitute the value of v obtained from (1) in (3) and plot P as a function of 0. As decreases and approaches the critical angle 3o, which increases most rapidly, P, t, or v? What can be said of the power required for high speeds at small angles? How is P affected by an increase in W? a decrease in S? an increase in K? an increase in ƒ? CHAPTER IV TRIGONOMETRIC FUNCTIONS AND THE SOLUTION OF TRIANGLES (ANGLE MEASUREMENT) 57. Introduction. One of the most useful applications of mathematics has been the systematic location and relocation of points and lines on the earth's surface, and the measurement of portions of the surface the art of surveying. In surveying an extensive region, as a state, two points several miles apart, where the intervening surface is level, are selected for the extremities of a so-called base line. The length of this base line is measured with great accuracy. A number of stations are selected to serve as the vertices of a network of triangles stretching across the country, and the angles subtended at each station by every pair of visible stations are carefully measured. From these measurements the sides of all the triangles can be calculated. A C B C FIG. 75. The principle underlying the computations of the surveyor is that the homologous sides of similar triangles are proportional. Its use is illustrated in EXAMPLE 1. To find the width of a river. Take a base line AC a short distance from one bank, and measure its length. Note the point B directly opposite C', so that ZACB is a right angle. From a point B' on AB draw B'C' perpendicular to AC, and measure AC' and B'C'. Since the right triangles ACB and AC'B' are similar, B'C' B'C' = AC. AC The width would then be obtained by subtracting from BC the distance CD from C to the bank of the river. The solution of the example depends upon finding the ratio of two sides of a right triangle which has the same acute angle BAC as the given triangle. It would be tedious, and sometimes difficult or impossible, if every time we wished to determine an inaccessible side of a triangle we had to construct a similar right triangle whose sides could be measured and their ratio determined. To avoid this difficulty, tables have been constructed which give the ratios of the sides of a right triangle with a given acute angle. Let us consider the construction of such a table. Suppose that one acute angle of a right triangle is 30°, and that the triangle be placed in the position OMP, with the vertex of this angle at the origin of a system of coördinates, the adjacent leg lying along the positive part of the x-axis, and the hypotenuse falling in the first quadrant. The triangle is placed in this position so that the procedure will be in accord with fundamental definitions to be given in Section 59. Let the coördinates of P be x = OM and y = MP, and let OP = r. We seek the values of the rations y/r, x/r, and y/x. Since the angle at O is 30°, that 130° y х M FIG. 76. at P is 60°. Hence AOMP is half an equilateral triangle OPQ, and therefore r = 2y and x2 + y2 = r2. (1) The ratios required may be determined by solving these equations for any two of the sides in terms of the third. We already have r in terms of y. Substituting in the second equation, x2 + y2 = 4y2, whence x = √3 y. We may now find the values of the ratios: As all right triangles whose acute angles are 30° and 60° are similar, these results are true for all such triangles. Problems in geometry which can be solved by means of equations (1) may be solved more expeditiously by means of the ratios (2), as in EXAMPLE 2. Find the height of a flag pole MP if the pole subtends an angle of 30° at a point 0 100 feet from the foot M. By the method of plane geometry we obtain the equations r = 2y, 1002 + y2 = r2. The solution is completed by solving these equations for y. Using the third of the ratios (2), which contains the known and required sides, we have The values of the ratios y/r, x/r, and y/x for two other angles may be readily obtained by the methods of plane geometry. If ▲ MOP = 45°, we have y = x and x2 + y2 = r2. (3) These equations may be solved for yo and r in terms of x, and the values of 60 x M P FIG. 79. the ratios determined. If ▲ MOP = 60°, we have r = 2x and y2+ x2 =r2, (4) which may be solved for y and r in terms of x, and the ratios determined. The details of the work are left as exercises. |