Encyclopædia Britannica, Ninth Edition/Gravitation/II. The Centre of Gravity.

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.

Fig. 3.

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.

Fig. 4.Fig. 5.

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 have been included. "We thus come to the important conclusion that the effect of gravity upon a body of any shape is to produce one, force which acts vertically downwards. It remains to be shown that the direction of this force passes through the centre of gravity of the body. Suspend a body of any shape by a cord, then when the body is at rest the centre of gravity must lie vertically beneath the point of suspension. If the direction of that one force which constitutes the effect of gravity does not pass through the centre of gravity, then its line of action cannot coincide with the direction of the cord of sus pension ; but it is impossible that two forces should equilibrate unless their lines of action are coincident, whence we are led to the import ant conclusion that the effect of the attraction of the earth upon a rigid body is to produce a single force which passes through the centre of gravity of the body and acts vertically downwards.