Let an irregular sheet of zinc be thus balanced on the point of a pencil or the head of a pin. Let a small hole be punched through the zinc at the point of balance, and let a needle be thrust through this hole. When the needle is held hor izontally, the zinc will be found to remain at rest, no matter in what position it is turned. . a n G G n m To illustrate another method of finding the center of gravity of the zinc, let it be supported from a pin stuck through a hole near its edge, that is, b (Fig. 70). Let a plumb line be hung from the pin, and let a line bn be drawn through b on the surface of the zinc parallel to and directly behind the plumb line. Let the zinc be hung from another point a, and let another line am be drawn in a similar way. FIG. 70. Locating center of gravity Since the attraction of the earth for a body may be considered as a single force applied at the center of gravity, a suspended body (for example, the sheet of zinc) can remain at rest only when the center of gravity is directly beneath the point of support (see § 85). It must therefore lie somewhere on the line am. For the same reason it must lie on the line bn. But the only point which lies on both of these i lines is their point of inter section G. The point of inter FIG. 71. Illustration of varying degrees of stability section, then, of any two vertical lines dropped through two different points of suspension locates the center of gravity of a body. 87. Stable equilibrium. A body is said to be in stable equilibrium if it tends to return to its original position when very slightly tipped, or rotated, out of that position. A pendulum, a chair, a cube resting on its side, a cone resting on its base, a boat floating quietly in still water, are all illustrations. In general, a body is in stable equilibrium whenever tipping it slightly tends to raise its center of gravity. Thus, in Fig. 71 all of the bodies A, B, C, D, are in stable equilibrium, for in order to overturn any one of them its center of gravity FIG. 72. Quebec bridge G must be raised through the height ai. If the weights are all alike, that one will be most stable for which ai is greatest. In building cantilever bridges such as the large one over the St. Lawrence River at Quebec (Fig. 72) the engineers build out the cantilever arms equally in opposite directions, so as to keep their centers of gravity constantly over the piers until the parts either meet at the center or are close enough to receive the central span, which is hoisted to place. A B Ꮐ C The condition of stable equilibrium for bodies which rest upon a horizontal plane is that a vertical line through the center of gravity shall fall within the base, the base being defined as the polygon formed by connecting the points at which the body touches the plane, as ABC (Fig. 73); for it is clear that in such a case a slight displacement must raise the center of gravity along the arc of which OG is the radius. If the vertical line drawn through the center of gravity fall outside the base, as in Fig. 74, the body must always fall. FIG. 73. Body in stable equilibrium The condition of stable equilibrium for bodies supported from a single point, as in the case of a pendulum, is that the point of support be above the center of gravity. For example, the beam of a balance cannot be in stable equilibrium, so that it will return to the horizontal position when slightly displaced, unless its center of gravity g (Fig. 3, p. 7) is below the knife-edge C. (The pans are not to be considered, since they are not rigidly connected to the beam.) FIG. 74. Body not in equilibrium 88. Neutral and unstable equilibrium. A body is said to be in neutral equilibrium when, after a slight displacement, it tends neither to return to its original position nor to move farther from it. Examples of neutral equilibrium are a spherical ball lying on a smooth plane, a cone lying on its side, a wheel free to rotate about a fixed axis through its center, or any body supported at its center of gravity. In general, a body is in neutral equilibrium when a slight displacement neither raises nor lowers its center of gravity. A body is in unstable equilibrium when, after a slight tipping, it tends to move farther from its original position. A cone balanced on its point or an egg on its end are examples. In all such cases a slight tipping lowers the center of gravity, and the motion then continues until the center of gravity is as low as circumstances will permit. The condition for unstable equilibrium in the case of a body supported by a point is that the center of gravity shall be above the point of support. QUESTIONS AND PROBLEMS 1. Explain why the toy shown in Fig. 75 will not lie upon its side, but rises to the vertical position. Does the center of gravity rise? 2. Where is the center of gravity of a hoop? of a cubical box? Is the latter more stable when empty or when full? Why? 3. Where must the center of gravity of the with reference to the supporting knife-edge C? Could you make a weighing if C and g coincided? Why? beam of a balance be (Fig. 3, p. 7.) Why? 4. What is the object of ballast in a ship? 5. What is the most stable position of a brick? the least stable? Why? 6. In what state of equilibrium is a pendulum at rest? Why? 7. What purpose is served by the tail of a kite? 8. Do you get more sugar to the pound in Calcutta than in Aberdeen when using a beam balance? when using a spring balance? Explain. 9. What change would there be in your weight if your mass were to become four times as great and that of the earth three times, the radius of the earth remaining the same? FIG. 75 10. The pull of the earth on a body at its surface is 100 kg. Find the pull on the same body 4000 mi. above the surface; 1000 mi. above the surface; 3 mi. above the surface. (Take the earth's radius as 4000 mi.) FALLING BODIES 89. Galileo's early experiments. Many of the familiar and important experiences of our lives have to do with falling bodies. Yet when we ask ourselves the simplest question which involves quantitative knowledge about gravity, such as, for example, Would a stone and a piece of lead dropped from the same point reach the ground at the same time or at different times? most of us are uncertain as to the answer. In fact, it was the asking and the answering of this very question by Galileo, about 1590, which may be considered as the starting point of modern science. Ordinary observation teaches that light bodies like feathers fall slowly and heavy bodies like stones fall rapidly, and up to Galileo's time it was taught in the schools that bodies fall with "velocities proportional to their weights." Not content with book knowledge, however, Great Italian physicist, astronomer, and mathematician; founder of experimental science"; was son of an impoverished nobleman of Pisa; studied medicine in early youth, but forsook it for mathematics and science; was professor of mathematics at Pisa and at Padua; discovered the laws of falling bodies and the laws of the pendulum; was the creator of the science of dynamics; constructed the first thermometer; first used the telescope for astronomical observations; discovered Jupiter's satellites and the spots on the sun. Modern physics begins with Galileo |