In pointing out the part played by mathematics in the illustration of the scientific method in Example 1, page 79 (see bottom page 80), it was stated that a more satisfactory verification of the law obtained would be indicated later, by deducing from the law some fact that may be verified by an experiment of a different nature. This deduction will be made in Example 2 below. Friction between an object and a plane acts parallel to the plane, and in the direction opposite to that in which the object moves or tends to move. When motion is about to take place, the force of friction is equal to the coefficient of friction for the surfaces in contact (see page 81) multiplied by the pressure of the object on the plane in the direction perpendicular to the plane. It is this law that we wish to verify. EXAMPLE 2. A block of wood weighing 20 grams rests on a horizontal board. If the coefficient of friction is 0.29, the result obtained in Example 1, page 79, at what angle may the FIG. 109. since the pressure on the plane is numerically equal to the resistance of the plane. Resolving W into components parallel and perpendicular to the plane, as in Example 1 above, we get where is the angle at which the block is on the point of sliding. Substituting in (1) the values of F and R given by (2), we get, 20 sin 0 = 0.29 × 20 cos 0. Dividing both sides by 20 cos 0, we obtain sin 0.29. cos (2) This result may be readily tested by experiment. If it is found that when one end of the board is gradually raised the block begins to slide when the inclination is a little over 16°, then we have a verification of the correctness of the deduction by which ℗ was found, and also a verification of the law (1) on which the deduction was based. That is, we have a verification of the law obtained in Example 1, page 79, which was re-stated on page 81. EXERCISES t 1. A boy kicks a football so that it would roll across a street at the rate of 15 feet per second, and simultaneously a second boy kicks it so that it would roll along the street with a velocity of 12 feet per second. Find the actual velocity of the ball (magnitude and direction). 2. A man walks across a canal boat. If his velocity with reference to the earth is 7 feet per second in a direction inclined at 30° to the bank of the canal, find the velocity of the boat and that at which he walks. 3. A ball is thrown into the air at an angle of 40° with a velocity of 30 feet per second. Find the horizontal and vertical components of the velocity. 4. If a ball is placed on a plane inclined at 20°, find the acceleration with which it rolls down the plane, and how fast it will be moving at the end of 3 seconds. 5. A ball is rolled up a plane inclined at 10° with an initial velocity of 20 feet per second. How long will it roll up the plane? 6. What is the inclination of a sidewalk if a ball rolls down it with an acceleration of 8 feet per second per second? 7. A cake of ice weighing 200 pounds is held on an ice slide at an ice house by a rope parallel to the slide. If the inclination of the slide is 45°, find the pull on the rope and the pressure on the slide. 8. An automobile weighing 2500 pounds stands on a pavement inclined at 12°. What force do the brakes exert? 9. Will a box slide down a board inclined at 15° if the friction is 0.3 of the pressure of the box on the plane? 10. A bar of iron weighing 5 pounds rests on a rough board. As one end of the board is gradually raised, it is found that the bar is just ready to slide when the inclination is 20°. Find the force of friction and the coefficient of friction (Definition, page 81). 11. A boy and his sled weigh 50 pounds. What force is necessary to hold them on an icy sidewalk whose inclination is 10° by means of a rope inclined at 30° to the walk? 12. (a) A rifle is fired at an angle of 20° to the horizon. If the muzzle velocity of the ball is 2000 feet per second, find the horizontal and vertical components of the velocity at the muzzle. (b) How long will a ball rise if it is thrown vertically upward with a velocity of 32 feet per second? of 80 feet per second? How long will the rifle ball in (a) rise? How far will it move horizontally in this time? 13. A rifle with muzzle velocity of 1600 feet per second is fired at an inclination of 30°. Find the horizontal and vertical components of the velocity at the muzzle. How long will it rise? When will it hit the ground (see Exercise 13, page 104)? How far from the point where it is fired will it hit? 14. A rope is tied to a heavy weight lying on the ground. A boy pulls on the rope with a force of 50 pounds in such a way that the rope is inclined at 25° to the vertical. Find the horizontal and vertical compo nents of his pull. What is the force with which he tends to lift the weight? To drag it? If the weight weighs 100 pounds, what is the pressure of the weight on the ground when the boy pulls? 15. A body weighing 25 pounds rests on a rough plane inclined at 5o. Find the components of the weight parallel and perpendicular to the plane. What is the pressure of the body on the plane? What force is tending to make the body move down the plane? Why does it not move? weight of a body is the force with which the earth attracts it.) (The 16. Three bales of cotton, a total weight of 1400 pounds, are raised from the hold of a ship by a derrick until they are just above the deck, when they are pulled one side by a horizontal rope. Find the pull of the rope and of the cable on the derrick when the cable is inclined at 12° to the vertical. 17. A stone weighing 300 pounds is raised to the top of a building by means of a derrick on top of the building, while a man on the ground keeps it away from the side of the building by means of a rope tied to the stone. Find the pull on the cable of the derrick and on the rope if the cable is inclined to the vertical at 5° and the rope at 60°. 18. Two cords are tied to a weight of 10 pounds. The weight is held by two boys who pull on the cords. If one cord is inclined at 20° to the vertical and the other at 14°, find the force with which each boy pulls. 19. Two narrow boards are 2 feet long. A boy places their upper ends together and their lower ends on the ground 1 foot apart, and balances a weight of 5 pounds on top of them. Find the pressure along each board. 20. A body weighing 500 pounds is suspended from the center of a horizontal beam. The beam rests on two V-shaped supports, inverted, which are made of pieces of "2 x 4" each 4 feet long. Find the thrust or pressure along each 2 × 4, assuming that the effect is the same as if half of the weight were placed directly over each support. 21. Just above the door on the second story of a barn, a beam projects horizontally for 3 feet. Objects are raised to the second story by a rope which passes over a pulley at the end of the beam and enters the barn over a pulley in the wall at a point 4 feet directly above the beam. Find the thrust along the beam when an object weighing 50 pounds is suspended by the rope. 68. Functions of n90° ± 0. In Section 62 we saw how to express the functions of the complement of 0, 90° - 0, in terms of functions of 0. Let us now consider the functions of 180° — 0, the supplement of 0, employing the same method. Construct the angles 0 and 180° - 0 with their initial lines coinciding with the x-axis, and on their terminal lines take P(x, y) and P(x', y') so that r = r'. Then P and P' are symmetrical with respect to the y-axis (why?), and hence x' = x, Dividing (1) by (2), and applying (2), page 167, we obtain tan (180° - 0) tan 0. (3) Similar formulas may be obtained for the other functions from these formulas by means of the reciprocal relations. was taken acute. But the proof applies without change to the second figure, in which is in the third quadrant. It would also apply to properly constructed figures in which lies in the second or fourth quadrants, so that these formulas hold for all positive values of less than 360°. The proof may be extended to include all values of by using the periodicity of the functions. Any function of one of the angles – 0, 90° ± 0, 180° ± 0, 270° ± 0, 360° – 0 may be expressed in terms of a function of 0. Any one of these forty-eight formulas may be written down by the following rule: If any one of the trigonometric functions be denoted by f(0), and its co-function (page 177) by co-f(0), then angles 0, 180° - 0, 180° + 0, and 360° 0, OX being the initial line of each. On their terminal lines take, respectively, P(x, y), P1(x1,y1), P2(x2, y2), and P3(xs, y3), so that 3 Then P1, P2, and Ps are symmetrical to P with respect to the y-axis, the origin, and the x-axis respectively (why?). Hence the numerical values of x, x1, x2, and x3 are equal, and so also are those of y, Y1, Y2, and yз. Hence the numerical value of any function of one of the angles 180° – 0, 180° + 0, or 360° - 0 |