r- 



CENTRE OF GRAVITY. 



227 



round the point of suspension on every side to a distance equal to that of the 

 centre of gravity from the point of suspension, when the cord is fully stretched, 

 the centre of gravity will be at liberty to move in every direction within this 

 sphere. 



There are an infinite variety of circumstances under which the motion of a 

 body may be restrained, and in which a most important and useful class of 

 mechanical problems originate. Before we notice others, we shall, however, 

 examine that which has just been described more particularly. 



Fig. 9. 



"'IE 



Let P, fig. 9, be the point of suspension, and C the centre of gravity, and 

 suppose the body to be so placed that C shall be within* the sphere already 

 described. The cord will therefore be slackened, and in this state the body 

 will be free. The centre of gravity will therefore descend in the perpendicu- 

 lar direction until the cord becomes fully extended ; the tension will then pre- 

 vent its further motion in the perpendicular direction. The downward force 

 must now be considered as the diagonal of a parallelogram, and equivalent to 

 two forces C D and C E, in the directions of the sides, as already explained 

 in fig. 1. The force C D will bring the centre of gravity into the direction 

 P F, perpendicularly under the point of suspension. Since the force of grav- 

 ity acts continually on C in its approach to P F, it will move toward that line 

 with accelerated speed, and when it has arrived there, it will have acquired a 

 force to which no obstruction is immediately opposed, and consequently by its 

 inertia it retains this force, and moves beyond P F on the other side. But 

 when the point C gets into the line P F, it is in the lowest possible position ; 

 for it is at the lowest point of the sphere which limits its motion. When it 

 passes to the other side of P F, it must therefore begin to ascend, and the 

 force of gravity, which in the former case accelerated its descent, will now, 

 for the same reason, and with equal energy, oppose its ascent. This will be 

 easily understood. Let C' be any point which it may have attained in ascend- 

 ing : C 7 G 7 , the force of gravity, is now equivalent to C' D 7 and C' E'. The 

 latter, as before, produces tension ; but the former, C 7 D 7 , is in a direction im- 

 mediately opposed to the motion, and therefore retards it. This retardation 

 will continue until all the motion acquired by the body in its descent from the 

 first position has been destroyed, and then it will begin to return to P F, and 

 so it will continue to vibrate from the one side to the other until the friction on 

 the point P, and the resistance of the air, gradually deprive it of its motion, and 

 bring it to a state of rest in the direction P F. , 



