310 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII 



*274. Examples. 



1. In the initial motion of a chain under gravity prove that the tension 

 satisfies the equation 



1 ("1 ^ _ =0 

 ds \m ds / mp 2 



2. A uniform chain hangs under gravity with its ends attached to two 

 rings which are free to slide on a smooth horizontal bar. Prove that, if the 

 rings are initially held so that the tangents to the chain just below them 

 make equal angles y with the horizontal, and are let go, the tension at the 

 lowest point is changed in the ratio 2J/' : 2Jf' -f M cot 2 y, where M is the 

 mass of the chain, and M' that of either ring. [Of. Example 5, p. 281.] 



3. If the ends of the chain of Example 2 are held fixed, and the chain 

 is severed at its vertex, prove that the tension at a point where the tangent 

 makes an angle < with the horizontal immediately becomes 



^ Mg sec cos y/(cos y + y sin -y). 



4. Impulsive tensions T a , Tp are applied at the ends of a piece of chain 

 of mass M hanging in the form of a common catenary with terminal tangents 

 inclined to the horizontal at angles a and /3. Prove that the kinetic energy 

 generated is 



1 tan a -tan /3 ((T a cos a- Tp cos /3) 2 . . , _/| 

 -^ a _ p L +(7V J smacosa-:7y j sm/3cos/3)j. 



EXAMPLES. 



1. A ball is projected vertically with velocity v l from a point in a rigid 

 horizontal plane, and when its velocity is v z a second ball is projected from 

 the same point with velocity v l ; assuming the restitution in each impact to 

 be perfect, prove (i) that the time between successive impacts of the two 

 balls is VI/<JF, (ii) that the heights at which they take place are alternately 

 (3v 1 --y 2 )( v i+' y 2)/% an d (3tfi+*)(0i fj)/^) (iii) that the velocities of the 

 balls at the impacts are equal and opposite and alternately ^(v 1 v 2 ) and 



2. Two equal balls of radius a are in contact and are struck simultaneously 

 by a ball of radius c moving in the direction of their common tangent ; prove 

 that, if all the balls are of the same material, the impinging ball will be 

 reduced to rest if the coefficient of restitution is 



3. Two equal balls lie in contact on a table. A third equal ball impinges 

 on them, its centre moving along a line nearly coinciding with a horizontal 

 common tangent. Assuming that the periods of the impacts do not overlap, 

 prove that the ratio of the velocities which either ball will receive according 

 as it is struck first or second is 4 : 3 - e, where e is the coefficient of resti- 

 tution. 



