344 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



through their centres perpendicular to their bases come into contact. Prove 

 that the distance x between their centres at time t is given by the equations 



205. A uniform chain of length I is held by its upper end so that its 

 lower end is at a height I above a fixed horizontal plane, and is let drop on 

 the plane. Prove that when half the chain is on the plane the pressure on 

 the plane is f of the weight of the chain. 



206. A uniform chain of length I is coiled at the edge of a table; one 

 end is attached to a particle of mass equal to that of the chain, and the other 

 end is put over the edge of the table. Prove that immediately after leaving 

 the table the particle is moving with velocity s\/(f gl). 



207. Two uniform chains, lengths , I', masses ml, m'l', are coiled in 

 loose heaps and lie close together. They are projected with velocities v, v 1 

 in directions containing an angle a and move under no forces. Prove that, 

 if ml 2 > m'l' 2 , the former will be uncoiled before the latter, that the tension so 

 long as neither is completely uncoiled is mm' (v 2 +v' 2 2vv' cosa)/(Jm+*Jm') 2 , 

 and that the whole time which elapses before both are uncoiled is 



| (ml 2 + m'l' 2 + 2m'll')/ml*J(v 2 + v' 2 - 2vv' cos a). 



208. A coil of uniform chain of mass m l per unit of length is placed on a 

 smooth table and one end of it is joined by a thread passing over the edge 

 of the table to one end of another coil of mass m 2 per unit of length, which 

 is held just at the edge. Prove that if the second coil is let go the straight 

 parts of the chains increase with uniform accelerations 



g^m 2 l(^m^ + ^m z } and gJm^KJm^Jm^ 

 so long as neither is completely uncoiled. 



209. A chain of length I slides from rest down a line of greatest slope 

 on a smooth plane of inclination a to the horizontal, the end of the chain 

 hanging initially just over the edge. Prove that the time of leaving the 

 plane is A V /{%(l-sina)}log(cot^a). 



210. A smooth circular cylinder is fixed with its axis horizontal and 

 vertically over the edge of a table, on which a length a of a uniform chain 

 of mass ml and length I is coiled, the chain passing over the cylinder and 

 having its free end on a level with the table. Prove that, if this end is 

 slightly displaced downwards, the amount of energy that will have been 

 dissipated by the time the chain leaves the table is \mgcfi '/I. 



211. A smooth circular cylinder is fixed with its axis horizontal and 

 vertically above the edge of a table on which lies a chain of length I and 

 mass /*, one end of which is attached to a thread passing over the 

 cylinder and supporting a body of mass M. If all the chain is off the 



