from the same height ; you will find that the two marbles reach the ground together. Suppose for simplicity that the height at the moment when the marbles are released is 4 feet, then the time taken by one of the marbles in falling is half a second. But as both marbles reach the ground together, the experiment has really proved to us that a body at the height of 4 feet from the ground will if pro jected horizontally reach the ground in half a second. This is equally true whatever be the magnitude of the velocity, i.e., whether it be 5, 10, or any other number of feet per second. We have now studied the effect of gravity upon a body which has been projected either in a vertical line or in a horizontal line. We have found that in each case the effect of gravity is to bring the body ^gt 1 feet nearer the surface of the earth in t seconds than it would have been had gravity not acted. We are therefore tempted to inquire whether the same statement would not be true for a body projected in any direction. In every way in which this suggestion can be tested it has been found to be verified, and there cannot there fore be the slightest doubt that it is true. To illustrate this principle we may apply it to the case of a body projected in any direction, and deduce the form of the path in which the body moves. Let (fig. 2) be the point from which the body is projected, and let P 5 OP be the direction in which the body would move after projection if it were not for gravi tation. In consequence of the first law of mo tion, we should find that if it had not been for the action of gravity the ball would reach P : in one second, P 2 in two seconds, &c. , where m> _ P P P p 1 M r 2 1 2 1 3 Gravitation will, how ever, make the body swerve from the direc tion OP^, &c., so that at the end of one
second the body is really found at the point Aj, at the end of two seconds at A 2 , at the end of three seconds at A 3 , &c. The curve drawn through the points A 1( A 2 , A 3 , &c. , which is actually de scribed by the body, can be readily constructed. Take, for example, t 3. If gravity had not been acting, the body would in three seconds have reached the point P 3 . We can find where the body actually is by taking a point A 3 , which is vertically beneath P 3 at the distance 16 1 x 3" feet. Similarly we can find where the body is after any other specified number of seconds, and thus we obtain the points Aj, A. 2 , &c. The equation of the curve is thus found. Take the line OP as the axis of x, and let x denote the number of seconds during which the motion has lasted ; then, if y denote the vertical distance through which the body has been deflected by gravity, we must have This curve, being of the second degree, represents a conic section ; and as the highest terms form a perfect square, the conic section must be a parabola.
II. The Centre of Gravity.
§ 6. Centre of Gravity of a Plate.—In studying the effect of gravity upon a body which is at rest, it will be convenient to commence with a simple illustrative experiment which can be easily tried. Out of a piece of cardboard or tin plate a figure of any shape, ABODE (fig. 3), is to be cut. A few holes, A, B, 0, D, E, are to be punched quite at random in this plate. In the wall is a nail, and the plate can be suspended by passing the nail through any of the holes A, B, and C. From the nail is suspended, in front of the plate, a cord AH, which is kept in the vertical direction by the plummet attached to it at H. As the plate is not supported in any other way, it hangs quite freely from the nail ; and if it be displaced and then released, it will, after a few oscillations, settle down again in the position which it occupied at the first. In order to mark this position, it is desirable to draw a line on the plate in the direction AP, indicated by the plummet line which is hanging in front. If the plate be blackened, this can be neatly done by chalking the plummet line and then giving it a flip against the plate.
When the line has been drawn the plate maybe removed from the nail, and again suspended by one of the other holes in its margin, for example B. The plate again assumes a definite position, and again the line and plummet is to be hung on, and a second line drawn as be fore. The two lines drawn on the plate intersect at a point P. When the plate is hung from a third hole, C, and a third plummet line is drawn, a very remarkable result is perceived. It is found that the line drawn on the third occasion passes through the intersection of the two former lines, that is to say, all the three lines pass through the point P. Repeating the operation, with other holes, D, E, &c. , it is found that all the lines drawn in the way we have described pos sess the remarkable property of passing through one definite point of the plate. It is therefore manifest that the point P possesses a very special property, for it is always situated vertically beneath the point of suspension when the plate is hanging at rest. If a hole be actually punched at the position of the point P, and if the plate be suspended by passing this hole over the nail, we then find that the plate will remain at rest in any position whatever. This peculiarity of the point P will be more readily perceived if we make a hole in the plate at a point Q near to P. When the plate is sus pended from Qit will only be at rest in one position, i.e., when P is vertically beneath Q. It must surely be regarded as a matter worthy of careful notice that any plate of any figure, regular or irregular, should contain one specific point which enjoys the unique properties which the experi ments show to be possessed by the point P. This point has received a name ; it is called the centre of gravity.
§ 7. Centre of Gravity of a Rigid Body.—In the illustration we have just given, we have spoken merely of a thin plate, because the experiments were more easily conducted in a body of this nature than in one of entirely irregular form. It must not, however, be supposed that a thin plate of uniform thickness is the only kind of body which possesses a point having the properties we have de scribed. No matter what be the shape or materials of which a rigid body is composed, it possesses a centre of gravity. Let ADBC (fig. 4) be a body of any kind, and let it be suspended by a cord from a point A. Then when the body is at rest it assumes a certain position. We may suppose that a verti cal hole A B is drilled through the body in the direction of the cord by which the body is suspended. If we now suspend the body by another point on its surface, C, the body will come to rest in the position which is represented in fig. 5. It will be found that these two straight holes intersect in the interior of the body at G. In fact, if we thrust a knitting needle through one of the holes, and then attempt to thrust a second knitting needle through the other hole, we shall find that the way is stopped in the interior of the body by the first knitting needle.
If the body be now suspended from any other point on its surface, and if a similar hole be made through the point and in the direction of the string by which the body is suspended, it will be found that this hole also passes through the intersection of the two former holes. From each and every point of suspension the same result is obtained, and thus we are led to the conclusion that in a rigid body of any shape or materials whatever there is one point which possesses the remarkable property thus stated: When a body suspended by a cord from a fixed point is at rest, there is one special point which is ahoays vertically beneath the point of sus- 2>ension, whatever may be the point of the body to which the cord is attached. This point is called the centre of gravity, In the case of a homogeneous body of regular shape, the centre of gravity is determined from the most simple considerations of symmetry. In the case of a sphere it is obvious that the centre of gravity must lie at the centre, for there is no other point symmetri cally related to the figure. In the case of a parallelepiped the centre of gravity is also situated at the centre of volume. This is found by joining the opposite corners of the figure, and thus making a diagonal ; joining another pair of opposite corners we have a second diagonal ; and the intersection of these two lines gives the centre of gravity of the mass.
§ 8. Gravitation of a Rigid Body reduced to One Force.—A body of any description may be considered to be composed of an innumerable multitude of small particles of matter. Each of these particles is acted upon by the attraction of the earth. Each particle is there fore urged towards the earth by a certain force which tends towards the earth's centre. The centre of the earth being nearly 4000 miles distant, the directions of these forces may, for all practical purposes, be regarded as parallel. Even if two particles were a mile distant, the inclination of the directions of the two forces is under a minute. We may therefore treat the forces as parallel without making any appreciable error. Two parallel forces may be compounded into a single force which is parallel to the two components. The forces acting on the two particles of the body may therefore be replaced by a single force. This force may be similarly compounded with the force acting on a third particle of the body, this resultant with the force on a fourth particle, and so on until all the particles
