EXERCISES XXIV.-COMPOSITION AND RESOLUTION OF VECTORS (F 1. Find the components R, and R, of the resultant of two forces 12, a' = 30°) and (F" = 20, a'' = 60°). = 2. Find the magnitude R and the direction of the resultant of Ex. 1. 3. Find the resultant (R, 0) of three forces (100, 350°), (150, 490°), (200, 720°), where (F, a) indicates a force of magnitude F and direction a. 4. Find the resultant (R, 0) of three velocities (25, 20°), (10, 210°), (18, 325°), where (v, α) indicates a velocity of magnitude v and direction α. 5. If a force of intensity F makes an angle a with any line in space, show that the component of F along that line is Fcosa. Draw a figure. 6. A force is often indicated by stating its two components, in the order (F, F), in parentheses, separated by a comma. Find the magnitude and the direction of each of the following forces: (b) (→ 42.5, 25.64). (c) (-6, 8). (a) (5, — 10).. (d) (-48.6, -72.9). (e) (50.8, 32.9). (ƒ) (— 42.2, — 54.6). 7. Show that, in general, the magnitude F and the direction a of a force whose components on the axes are F and F, are given by while the quadrant in which F lies is given by the algebraic signs of F÷F and cos α = F÷F. sin α = 8. If a force of intensity 12 makes angles whose cosines are 2/3, 1/3, and 2/3 with three mutually perpendicular lines Ox, Oy, and Oz, respectively, show that the components on these three lines are 8, 4, 8, respectively. 9. If the cosines of the angles which a force F makes with three mutually perpendicular lines are cos a, cos ß, and cos y, respectively, show that the components of F on those lines are F cosa, Fcos ß, Fcosy, respectively. 10. If the components of a force on three mutually perpendicular lines in space are 2, 3, and 6, respectively, show that the intensity of their resultant is represented by the diagonal of the rectangular parallelopiped determined by the components, and compute its value. of Ex. 10 11. Show that the cosine of the angle made by the force with the first of the three mutually perpendicular lines is 2/7. Find the cosines of the angles which the force makes with each of the other two perpendicular lines. 12. If a force has components A, B, and C on three mutually perpendicular lines Ox, Oy, Oz, show that the intensity of the force is R = √ A2 + B+ C2. Show that the cosines of the angles which the force makes with Ox, Oy, and Oz are A/R, B/R, and C/R, respectively. 60. Uniform Circular Motion. The importance of the functions of large angles and of negative angles is well illustrated by the simple problem of uniform rotation. Let M be a point of a rotating body, at a distance a = OM (in feet) from the axis of rotation. We have seen (§ 45) that it is convenient to measure the angle 0, between the initial position OA and the position OM, in radians. Then the arc s which subtends is (1) sa0, (0 in radians, a and s in feet); = and is proportional to the time t (in seconds) after M was at A: (2) kt, whence s akt, (k a constant), = motion, the value of 0 will exceed π as soon as t exceeds π/k; hence large angles occur very naturally in (3) and (4). If the rotation is clockwise, it is counted (§ 37) as negative; and all values of and the number k are negative. Then the angles in (3) and (4) are negative. 61. Period. Amplitude. The total time T for one revolution is called the period of the rotation. Since the angular speed is k radians per second, and since one revolution is 2 π radians, T=2π/k. If t is increased by the amount T, the angle is increased by one revolution or 2 radians. Hence, by (3), x and y are not changed since both the sine and the cosine have the same values for any two congruent angles. The quantity a, the radius of the circular path, is called the amplitude. 62. Vibration. Simple Harmonic Motion. A point on a vibrating stretched cord moves back and forth in a manner similar to the motion of the projection of P, § 60, on the x-axis. It is assumed in Physics that any point of such a steadily vibrating body actually moves precisely as this projection: The motion of the projection of P on the y-axis is precisely similar to the preceding; it is The kind of motion described by (1) or by (2) is called simple harmonic motion (S. H. M.). As in § 61, the quantity a is called the amplitude, and T=2π/k is called the period of the S. H. M. The moving point returns to its original position every 2 /k seconds, as in § 61; i.e. T=2π/k is the time of one complete vibration. EXERCISES XXV.-CIRCULAR MOTION VIBRATION 1. Show that the coördinates (x, y) of the point P of § 60, for a uniform rotation of angular speed 1, are x = a cost, y = a sin t. See Ex. 21, p. 89. 2. Show that the period of the rotation of Ex. 1 is 2 π. 3. Find the values of x and y in Ex. 1 with a = 10, when t = 1, 2, π/2, 3, π, 4, 2π, 7. Plot these pairs of values of x and y, and show that the points lie on the circular path. 4. A recording instrument often used in Physical laboratories consists of a cylinder which is covered with a paper painted with lampblack. A fine needle is attached to a vibrating body, such as a tuning fork, and this needle is allowed to touch the lampblacked surface while the cylinder is slowly rotated. If the apparatus is adjusted so that the needle would trace an element of the cylinder if the cylinder were at rest, show that the curve actually traced on the moving cylinder resembles the curve y = sin t. 5. If the cylinder of Ex. 4 rotates so that a point on its surface travels a distance 2 π/k units per second, show that the curve traced on the blackened paper is precisely the curve y = a sin kt, where k is the period of the tuning fork, if the amplitude of the vibration remains constant. 6. If a tuning fork makes 256 complete vibrations per second, show that its period is T = 1/256. 7. Show that the motion of a point on the tuning fork of Ex. 6 is described by the equation y = a sin kt where k = 2π/T 512 π, i.e. by the equation y = a sin 512 πt. 8. Plot the curve which represents the equation x = a sin 512 πt for the value a = 1/100 (ft.) = 1 large unit on the y-axis. 9. If two vibrations in the same direction occur simultaneously, the displacement of a point in the vibrating body is the sum of the displacements due to the two vibrations taken separately. Show that the equation y = a sin kt + b cos kt represents such a compound vibration. 10. Plot the curve which represents each of the following compound vibrations : (a) y = 2 cos 3 t + 3 sin 3 t ; cos 3 t; (b) y = sin 2 t 11. The pitch of a screw is the distance it moves parallel to its axis when the head makes one complete turn; i.e. the distance between two turns of the screw thread. Find the distance the screw moves when the head turns through an angle of 230°, if the pitch is 1/20 in. [NOTE. Instruments of precision for measuring distances accurately are made on this principle, and are called micrometers.] 12. Find the pitch of a screw that moves through a distance 1/8 in. in turning through an angle of 1200°. 13. A spiral stairway has a railing similar to the thread of a screw. If the pitch is 10 ft., find the angular width of steps that rise 7 in. How large must the inner radius of the base be made to make the net width of the inner tread 5 in. ? 14. If the base of a spiral similar to a screw thread has its center on the origin of a pair of rectangular axes Ox and Oy, and passes through a point of Ox at a distance a from O, and if the vertical distance from this base is denoted by h, show that x = a cos 0, y = a sin 0, h = p0/360, where p is the pitch and is the angle, in degrees, through which the point (x, y) turns. 15. If the point of Ex. 14 moves with uniform speed, so that where k is a constant, show that = kt x = a cos kt, y = a sin kt, h = pkt/360. CHAPTER VI THE ADDITION FORMULAS - Formulas. 63. Reduction of A cos a± B sin a. Such expressions as A cos a B sin a arise in various connections: thus a combination of two vibrations gives this kind of a form. Another very different connection in which such expressions arise is in resolution of forces. If a force of magnitude A makes an angle & with the positive x-axis, while another force B makes an angle of a+90° with the x-axis, the x-component R2 of their resultant R is FIG. 63 -X H A cos a+ B cos (α + 90°) and the y-component R, of Ris = = In both these cases, it is possible, and advantageous, to ex press the combination A cos α ± B sin a as the product of a single number times the cosine of a single angle. In the case of forces this is obvious; for in Fig. 63 we have, by § 48, |