CENTRE OF GRAVITY. Owing to gravita tion (q.v.), all bodies on the surface of the earth are beimg acted ou by forces drawing them toward the centre of the earth. These forces are all sensibly parallel, owing to the large size of the earth compared with that of most natural objects. and produce what is ordinarily called 'weight.' The weight of any particle of matter whose mass is of is therefore equal to mg. where g is the acceleration which the body would have toward the earth if allowed to fall freely. The value of g is proved by experiment to be the same for all and quantities of matter, but to vary from point to point on the earth's sur face. Any large body may be considered as made up of parts, all beMg acted upon by paral lel forces, and the resultant of these forces he the weight of the body, and will be a force whose value, and direction are given by the ordinary laws for compounding parallel forces. The point in the body (or in space con nee it'd with the body) through which this re sultant always passes, however the body is turned nr placed, is called its 'centre of gravity.' Thus, if and arc the masses of two small bodies, which may be called 'particles.' kept at a distance h apart. their centre of gravity may be calculated as follows: Through any point O in the vertical plane in eluding the particles, draw 0%0 straight lines, oX and Or, at right angles and parallel re speetively to the vertical representing the weight, of vrt, and in„. Let i, a ml be the perpendicular distances of in, and ni, from Or then, by the law of parallel forces, the resultant of the two forces Irs,y and nt,y is a parallel force 1m, + tn,)y at a /MCI' from Ur, where 211% X (at, + 7/1 Hence In,x, T WI at, Now, if the two bodies be moved in space.
still keeping their distance Is apart, oX and Or moving with them, they can be so placed that OX is now vertical, as in the diagram. ('all the distances of on, and 711, from OX y, and y,.
The resultant now is a force int, ni,„)y parallel to and at a distance g such that 71q91 + " 29Y 2 y= * uo,)fi Hence " 70, -r 70, It is evident from geometry that in both eases the resultant passes through a point on the line joining the two particles whose distance from the one of mass In is + an, This may be shown by choosing 0 to coincide with the particle whose mass is m. In that case = 0, y,== 0. and therefore = 7„ 7/1 115 T Its J.
-19d so. by similar triangles. these condi tions are satisfied by a point C on the line 00' such that in 2 01%= This point is. there fore. the of gravity, being inde pendent of the direction of the line 00'. It is evident. further, from these equations for x and g, that the centre of gravity coincides N‘ith the 'centre of inertia' (q.v.). The centre of gravity of any number of particles mav be fmmnd in a perfectly similar way. For a uniform straight wire or rod the centre of gravity is its middle point for a triangular plane figure it the intersection of the three bisectors of the sides from the opposite vertices: for a homogeneous pyramid it is the point of inter section of the lines drawn from each vertex to the centre of gravity of the opposite face.
If a solid body is suspended by a string fas tenol to it. or if it is balanced on a point, the Iille of action of this upward supporting force most pass through the centre' of gravity, if the body is at rest. 'Fins gives. a din,' loothod of determining the position of the centre of gravity of a solid by experiment : Suspend it by a string or balance it on a point, draw in the body a vertical line passing through the point of support ; suspend the body by fastening the string to a different point. or balance it with another portion of the body resting on the pivot, draw in the body a vertical line through the new point of support: the intersection of these two lines is the centre of gravity.