233 



CENTRE OF GRAVITY. 



Fig. 34. 



Although the centre of gravity takes its name from the familiar properties 

 which it has in reference to detached bodies of inconsiderable magnitude, 

 placed on or near the surface of the earth, yet it possesses properties of a 

 much more general and not less important nature. One of the most remarka- 

 ble of these is, that the centre of gravity of any number of separate bodies is 

 never affected by the mutual attraction, impact, or other influence which the 

 bodies may transmit from one to another. This is a necessary consequence 

 of the equality of action and reaction ; for if A and B, fig. 33, attract each 

 other, and change their places to A' B 7 , the space a a' will have to b b' the 

 same proportion as B has to A, and therefore, by what has just been proved in 

 fig. 33, the same proportion as a C has to b C. It follows that the remainders 

 «' C and b' C will be in the proportion of B to A, and that C will continue to 

 be the centre of gravity of the bodies after they have approached by their 

 mutual attraction. 



Suppose, for example, that A and B were twelve pounds and eight pounds 

 respectively, and that a b were forty feet. The point C must divide a b into 

 two parts, in the proportion of eight to twelve, or of two to three. Hence it 

 is obvious that a C will be sixteen feet, and b C twenty-four feet. Now, sup- 

 pose that A and B attract each other, and that A approaches B through two 

 feet. Then B must approach A through three feet. Their distances from C 

 will now be fourteen feet and twenty-one feet, which, being in the proportion 

 of B to A, the point C will still be their centre of gravity. 



Hence it follows, that if a system of bodies, placed at rest, be permitted to 

 obey their mutual attractions, although the bodies will thereby be severally 

 moved, yet their common centre of gravity must remain quiescent. 



When one of two bodies is moving in a straight line, the other being at rest, 

 their common centre of gravity must move in a parallel straight line. Let A 

 and B, fig. 35, be the centres of gravity of the bodies, and let A move from A 

 to a, B remaining at rest. Draw the lines A B and a B. In every position 

 which the body B assumes during its motion, the centre of gravity C divides 

 the line joining them into parts A C, B C, which are in the proportion of the 

 mass B to the mass A. Now, suppose any number of lines drawn from B to 

 the line A a ; a parallel C c to A a through C divides all these lines in the 

 same proportion ; and therefore, while the body A moves from A to a, the com- 

 mon centre of gravity moves from C to c. 



If both the bodies A and B moved uniformly in straight lines, the centre of 

 gravity would have a motion compounded of the two motions with which it 

 would be affected, if each moved while the other remained at rest. In the 

 same manner, if there were three bodies, each moving uniformly in a straight 

 line, their common centre of gravity would have a motion compounded of that 

 motion which it would have if one remained at rest while the other two moved, 



