EXAMPLES. 349 



238. A string is laid on a smooth table in the form of a catenary, and 

 an impulse is communicated to one extremity in the direction of the tangent ; 

 prove (1) that the initial velocity of any point, resolved parallel to the 

 directrix, is proportional to the inverse square of the distance of that point 

 from the directrix, (2) that the velocity of the centre of gravity of any arc, 

 resolved in the same direction, is proportional to the angle between the 

 tangents at the ends of the arc directly and to the length of the arc 

 inversely. 



239. A uniform chain lies in a part of the curve r=a$ Q from 0=0 to 

 & = /3 and receives a tangential impulse T at 0=0, the other end being free. 

 Prove that the impulsive tension at any point is 



240. An endless uniform chain lying in the form of a circle receives a 

 tangential pluck at one point A which gives it an impulsive tension T at 

 that point ; prove that the impulsive tension at any point P is 



sinh(27r-#) 

 sinh27r 



6 being the angle which AP subtends at the centre, and that P starts in a 

 direction making an angle (p with the tangent, where 



241. A thin chain of variable density is placed on a smooth table in 

 the form of the curve in which it would hang under gravity, and two 

 impulsive tensions are applied at its extremities, which are to each other in 

 the ratio of the tensions at the same points in the hanging chain. Prove 

 that the whole will move without change of form parallel to the line which 

 was vertical in the hanging chain. 



242. A uniform flexible inextensible chain of density p rests on a smooth 

 plane ; a part of its length is in contact with a smooth circular disc of radius 

 a which lies on the plane, the length of this part being a (o+/3) ; the remainder 

 is in two straight portions which touch the disc at the ends of the arc of 

 contact; and particles of masses m and m are attached at the ends. Prove 

 that, when the disc is suddenly moved with velocity V in a direction making 

 an angle a with the radius to the point at which the portion carrying m 

 leaves the disc, m begins to move with velocity 



M- 1 V [(m! + pi } (sin a + sin /3) + pa {(a + ) sin a + (cos a - cos /3)}] , 

 where M is the mass of the whole system, and I is the length of the straight 

 portion of chain to which m is attached. 



243. A uniform inextensible chain is stretched nearly straight with two 

 ends at the same level ; suddenly one end is released. Prove that, to a first 

 approximation, half the product of the tensions at the other end before and 

 after release is equal to the square of the weight of the chain. 



