urged forward is represented by the length of Om, which is seen to be but 9.3 pounds instead of 10 pounds. The component which tends to lift the sled is represented by On. To apply the test of experiment to the conclusions of the preceding paragraph, let a wagon be placed upon an inclined plane (Fig. 62) the height of which, bc, is equal to half its length ab. In this case the force acting on the wagon is the weight of the wagon, and its direction is downward. Let this force be represented by the line OR. Then, by the construction of the preceding paragraph, the line Om will represent the value of the force which is pulling the carriage down the plane, and the line On the value of the force which is producing pressure against the plane. Now, since the triangle ROm is similar to the triangle abc (for ≤ mOR = ≤ abc, ▲ RmO = ≤ acb, and ▲ ORm = ≤ bac), we have От bc m R FIG. 62. Component of weight parallel to an inclined plane that is, in this case, since bc is equal to half of ab, Om is half of OR. Therefore the force which is necessary to prevent the wagon from running down the plane should be equal to half its weight. To test this conclusion let the wagon be weighed on the spring balance and then placed on the plane in the manner shown in the figure. The pull indicated by the balance will, indeed, be found to be half the weight of the wagon, if there is little or no friction. The equation Om/OR = bc/ab gives us the following rule for finding the force necessary to prevent a body from moving down an inclined plane: The force which must be applied to a body to hold it in place upon an inclined plane bears the same ratio to the weight of the body as the height of the plane bears to its length. 82. Component of gravity effective in producing the motion of the pendulum. When a pendulum is drawn aside from its position of rest (Fig. 63), the force acting on the bob is its weight, and the direction of this force is vertical. Let the force be represented by the line OR. The component of this force in the direction in which the bob is free to move is English mathematician and physicist ; "prince of philosophers"; professor of mathematics at Cambridge University; formulated the law of gravitation; discovered the binomial theorem; invented the method of the calculus; announced the three laws of motion which have become the basis of the science of mechanics; made important discoveries in light; is the author of the celebrated "Principia" (Principles of Natural Philosophy), published in 1687 THE GREAT CANTILEVER BRIDGE AT QUEBEC The channel span of this bridge measures 1800 feet. It is the longest cantilever bridge in the world, being to hold together the pieces of steel On (or On'), and the component at right angles to this direction is Om (or Om'). The second component simply produces stretch in the string and pressure upon the point of suspension. The first component is alone responsible for the motion of the bob. A consideration of the figure shows that this component becomes larger and larger the greater the displacement of the bob. When the bob is directly beneath the point of support, the component producing motion is zero. Hence a pendulum can be permanently at rest only when its bob is directly beneath the point of suspension.* n R m R FIG. 63. Force acting on displaced pendulum 83. Laws of the pendulum. The laws of the pendulum are obtained from the following experiments: Let two pendulums of exactly the same length (about 1 m.), one having a bob of lead and the other a bob of steel or glass, be pulled back and released at the same instant, and their periods observed. We shall thus obtain the first law of the pendulum; namely, 1. The periods of pendulums of the same length swinging through short arcs are independent of the weight and the material of the bobs. Let the two pendulums be set swinging through arcs of lengths 2 cm. and 5 cm. respectively. We shall thus find the second law of the pendulum; namely, 2. The period of a pendulum swinging through a short arc is independent of the amplitude of the arc. Let pendulums one fourth and one ninth as long as the one used before be swung with it. The long pendulum will be found * It is recommended that the study of the laws of the pendulum be introduced into the laboratory work at about this point (see Experiment 12 of "Exercises in Laboratory Physics," by Millikan, Gale, and Davis). to make only one vibration while the others are making two and three respectively. Therefore the third law of the pendulum is 3. The periods of pendulums are directly proportional to the square roots of their lengths. SUMMARY. A gram of mass is a definite quantity of matter which is the same everywhere; whereas a gram of force, which is the pull of the earth at sea level upon a gram of mass, varies slightly with position on the earth. The resultant of a set of forces is that single force which will produce the same effect upon a body as is produced by the joint action of the set of forces. The rules for finding the resultant have been found (§§ 77–79). The component of a force in a given direction is the effective value of the force in that direction. Rules for finding the component have been found (§ 81). A pendulum swings because of the component of its weight in the direction of its motion. The length law of the pendulum expressed as a formula is 1. A boy can row a boat 5 mi. per hour in still water. In a river which flows 2 mi. per hour, how fast would his boat move downstream? upstream? 2. The wind drives a steamer east with a force which would carry it 12 mi. per hour, and its propeller is driving it south with a force which would carry it 15 mi. per hour. What distance will it actually travel in an hour? Draw a diagram to represent the exact path. 3. Represent graphically a force of 30 lb. acting southeast and a force of 40 lb. acting southwest at the same point. What will be the magnitude of the resultant? What will be its approximate direction? 4. An airplane which flies in still air with a velocity of 120 mi. per hour is flying in a wind whose velocity is 60 mi. per hour |