ON THE MOTIONS OF CONNECTED BODIES. 65 



This may be illustrated by two balls fixed at the end of a rod, with a centre 

 of suspension moveable to any part of the rod, for as the centre approaches 

 the middle of the rod, the vibrations are rendered extremely slow. (Plate V. 

 Fig. 75.) 



The rotatory motion of bodies not fixed on an axis might be considered 

 in this place, but the subject involves in its whole extent some intricacy of 

 calculation, and, except in astronomy, the investigation is scarcely ap- 

 plicable to any problems which occur in practice. We may, however, 

 examine a few of the simplest cases. If two bodies be supposed to be con- 

 nected by an inflexible line, and to be moving with equal velocities in 

 parallel directions ; if an immoveable obstacle be applied, so as to form a 

 fulcrum, at the common centre of gravity, they will, as we have already 

 seen, be wholly stopped ; but if the fulcrum be applied to any other part of 

 the line, one of the bodies will move forwards, and the other backwards, with 

 a velocity which may easily be determined by calculating their rotatory 

 power with respect to the fulcrum. If the fulcrum be applied at a point 

 of the line continued beyond the bodies, the one will lose and the other 

 gain velocity ; since the quantity of rotatory power will always remain 

 unaltered : that point only which is denominated the centre of oscillation 

 retaining its original velocity. Now the same inequality in the motion of 

 the bodies, and consequently the same angular velocity of rotation will be 

 produced, if the connected bodies be initially at rest, and the fulcrum be 

 applied to them with the same relative velocity. For example, if a straight 

 rod or wire receive an impulse at one end in a transverse direction, the 

 centre of oscillation, which is at the distance of two thirds of the length 

 from the end struck, will at the first instant remain at rest, consequently 

 the centre will move with one fourth of the velocity of the impulse, and 

 this must be the velocity of the progressive motion of the rod, since the 

 centre of gravity of any body which is at liberty moves always with an 

 equable velocity in a right line, while the whole rod will also revolve 

 equably round its centre, except such retardations as may arise from 

 foreign causes. In a similar manner the computation may be extended to 

 bodies of a more complicated form. Thus it has been calculated at what 

 point of each planet an impulse must have operated, in order to communi- 

 cate to it at one blow its rotation and its progressive motion in its orbit.* 



Those who have asserted that the motion of the centre of gravity of a 

 body can only be produced by an impulse which is either wholly or partly 

 directed towards it, have obviously been mistaken. The centre of oscilla- 

 tion is the only point which remains at rest with regard to the first effect 

 of the stroke, and the centre of gravity, which never coincides with the 

 centre of oscillation, moves in the direction of the impulse, while the parts 

 beyond the centre of oscillation begin to move in a contrary direction. 

 Hence it is that a thin stick may be broken by a blow on the middle, with- 

 out injuring the glasses on which it is supported : fo'r the ends of the stick, 

 instead of being depressed by the stroke, would rise with half the velocity 

 of the body which strikes them, if the two portions were separated without 



* John Bernoulli, Op. vol. 4, 284. Consult Whewell, Dynamics, 1823, c. 8. 



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