which may ant of forces. but a system of couples may have a resultant couple, be found by the same methods as the resultAction and Reaction.-When a man in a boat pushes with his boat-hook against a quay, the boat is driven away from the quay with a force equal to that with which the man pushes against the quay. Similarly, all bodies acted upon exert a force called reaction, which is equal and opposite to the original force. In other words, action and reaction are equal. It follows, therefore, that as every force is accompanied by an equivalent reaction, every machine exerts a reacting force equal and opposite to the force exerted upon it; the force represented by the power being exactly balanced by the reaction due to the weight, as will be abundantly shown in treating of the mechanical powers. And since friction, the weight of the machine, and other resistances have to be overcome in addition to the weight to be moved, the employment of machines, instead of diminishing, has really the effect of augmenting the force necessary to overcome a given resistance. Why, then, are machines used at all? Because it is more convenient to exert a moderate force over a proportionately longer time, than to exert a greater force over a proportionately shorter time. Thus, it is more convenient to carry forty half-hundred-weights a certain distance, occupying forty minutes of time, than to carry a ton the same distance in a single minute. CHAPTER II. GRAVITY. Definition and Divisions.-Every particle of matter has a tendency to draw to itself every other particle, and this tendency is called the force of gravity.1 1 L., gravis, heavy. D The force of gravity may be conveniently studied under three heads; namely: 1. Universal1 Gravity, which is the force exerted by every body upon every other body; 2. Terrestrial2 Gravity, or the gravity of the mass of the earth, as compared with that of bodies upon it; and 3. Relative, or Specific Gravity, or the force exerted upon different kinds of matter by the earth. As this kind of gravity is intimately connected with the subject of Hydro-mechanics, we shall postpone the consideration of it until we treat of that subject. 1. Universal Gravity.-Universal gravity is that force which causes all particles of matter to attract one another. It therefore includes the other two kinds. Its effects are to be seen everywhere. If we place two light bodies, such as leaves, near each other in a basin of water, this force will manifest itself by the movement of the leaves towards each other. Two ships, becalmed, will gravitate, as it is called, towards each other, although miles apart in the first instance. It is the force of universal gravity which keeps the planets from flying off into space. The force in this case is called centripetal, because it tends to draw the body towards the centre round which it moves; as opposed to centrifugal5 force, which is shown in the tendency of a revolving body to fly from the centre. The equilibrium of these two forces keeps the planets in their orbits, and likewise our sun with other similar suns in their own particular paths round the centre of the universe. As all matter possesses this force of gravity, it follows that the sun gravitates towards the earth, and the earth towards the moon. When the moon in her orbit is 4 1 L., universus, the whole. earth. 2 L., terrestris, belonging to the 3 L., species, likeness; relating to bodies of like nature. 5 L., centrum; and • L., centrum, the centre; and peto, to seek. fugo, to fly. L., orbita, a wheel-track. progressing towards the sun, her motion is accelerated; and when from the sun, her motion is retarded, the gravity of the sun becoming in the one case an accelerating, and in the other a retarding force. The planets exert a gravitating force upon each other, hence their orbits are not true geometric figures. When we see a free body moving round a centre, in a path differing from a circle, we conclude that the irregularity is due to the presence of a gravitating force exercised by another body. The motion of the planet Uranus was observed by Adams and Leverrier to be irregular, and they ascribed it to the gravitating effect of an undiscovered planet. After calculating and announcing the position this planet ought to occupy to produce such an erratic motion, it was at once discovered, and is called Neptune. 2. Particular, or Terrestrial, Gravity.-Particular, or terrestrial, gravity is that section of universal gravity exerted by the mass of the earth; or, in other words, that force which tends to draw all bodies towards the earth's centre. The reason why the earth possesses this property is evident when we remember that all the particles of which the earth is composed possess their share of gravity. The result of this is, that the force exerted by the whole earth is the same as if the gravity of every particle were concentrated at or near the centre of the globe, which is therefore called the centre of gravity of the earth. This point is, in fact, that in which the resultants of the force of all the particles meet. As the intensity of the force of gravity of a body is proportional to its mass, it follows that the effect of the gravity of the earth upon any terrestrial body is vastly in excess of that of any terrestial body upon it. This force of the earth is at once manifest when a body is unsupported, it being at once drawn to the earth, or it falls, as we term it. The intensity of the force with which the earth attracts a body is called the weight of the body. The moon is prevented from flying off into space by the equilibrium existing between her centrifugal force and the force of terrestrial gravity. it As every body possesses a force of gravity of its own, may be asked: Why does not the earth move towards the falling body, as it must necessarily be pulled by it? The answer is, that the earth does move towards the falling body; but that the motion imparted to the earth by the falling body is as much less than the motion imparted to the falling body by the earth as the mass of the falling body is less than the mass of the earth. Now, the mass of the earth is 5852 billions of tons; and as the falling body can be at the most but a few tons, and probably does not exceed a few pounds, its effect upon the earth is practically nothing. Parallelism of Gravity.-As the force of gravity acts in straight lines towards the centre of the earth, these lines are necessarily converging lines, the point of convergence being the centre of gravity of the earth. But as the size of the earth is very great, and as we can only observe the effects of gravity upon bodies through a small space, extending from a limited distance above to a limited distance beneath the surface, and covering but a limited area, the amount of convergence is imperceptible. We may therefore consider the force of gravity as acting upon any particular body in parallel lines, without involving any appreciable error. Centre of Gravity.-It has been shown1 that in every system of parallel forces there is a central point, to which, if the resultant be applied in an opposite direction, equilibrium will be produced. This point, with regard to the gravity of a body, is called the centre of gravity of the body, and will be hereafter represented by the letters C G. Method of finding the centre of gravity.-The CG of any body may be thus found. 1 Page 29. Suspend the body by any point, A (fig. 21); a vertical line from this point will pass through the body in the direction AB, and the centre of gravity of the mass must be somewhere in this line. The Now suspend the body by another point, C (fig. 22). vertical from this point will pass A through the body in the direction Fig. 21. CD; and the point, E, where the Fig. 22. two lines intersect is the centre of gravity of the body. A force acting on this point has the same effect as if applied to all the particles of the body collectively. The centres of gravity of sundry important figures are here given : 1. Of a triangle. That point of intersection of two middle lines, or that point in the line joining the middle of the base with the opposite angle, which is one-third of its length from the base. 2. Of a semicircle. At a distance from the base found by dividing two-thirds of the square of the diameter by the circumference. 3. Of a semi-ellipse. Same as a semicircle of the same height. 4. Of a parabola. Three-fifths of the height. 5. Of a cycloid. Seven-twelfths of the height. 6. Of a sector of a circle. At a distance from the centre found by multiplying two-thirds of the radius by the chord, and dividing by the arc. 7. Of a quadrant. At the same distance from either radius as that of the semicircle is from its base. 8. Of the surface of a hemisphere. At the middle point of the height. 9. Of a prism or cylinder. The middle point of the line joining the centres of gravity of the two ends. 10. Of a pyramid or cone. That point on the line joining the centre of gravity of the base with the apex, which is one-fourth of its length from the base. 11. Of a hemisphere. At three-eighths of the radius. To find the C G of two particles, A and B: Join A B. Bisect the line A B in C, so that the weight of A is to the 1 Gr., hemi, half; and sphaira, a ball. |