304 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



has run off at any time is vertical, the velocity of the chain as the last 

 element leaves the table is *J(ag}. 



2. A uniform chain of length I and weight W is suspended by one end 

 and the other end is at a height h above a smooth table. Prove that, if the 

 upper end is let go, the pressure on the table as the coil is formed increases 

 from 2A W/l to (2A +31) W/L 



3. A uniform chain AB is held with its lower end fixed at B and its 

 upper end A at a vertical distance above B equal to the length of the chain. 

 The end A is released, and at the instant when it passes B the end B is also 

 released. Prove that the chain becomes straight after an interval equal to 

 three-quarters of that in which A fell to B. 



4. Two uniform chains whose masses per unit of length are m 1 and m 2 

 are joined by a thread passing over a fixed smooth pulley. Initially the 

 chains are held up in coils and they are released simultaneously without 

 causing any finite impulse in the thread. Prove that, until one of the chains 

 has become entirely uncoiled, the thread slips over the pulley with uniform 

 acceleration 



and that the portions of the chains which have become straight increase 

 during the interval with uniform accelerations 



5. A uniform chain of length I and weight W is placed on a line of 

 greatest slope of a smooth plane of inclination a to the horizontal so that 

 it just reaches to the bottom of the plane where there is a small smooth 

 pulley over which it can run off. Prove that when a length x has run 

 off the tension at the bottom of the plane is 



6. A uniform chain is held with its highest point on the highest generator 

 of a smooth horizontal circular cylinder, and lies on the cylinder in a vertical 

 plane, subtending an angle /3 at the centre of the circular section on which it 

 lies. Prove that, when the chain is let go, the lower end is the first part of 

 it to leave the cylinder, and that this happens when the radius drawn through 

 the upper end makes with the vertical an angle $ given by the equation 

 J|8 cos (< + j3) =sin + sin < - sin 



*269. Chain moving freely in one plane. Let s be the 



length of the chain measured from a chosen particle to any 

 particle P, s + ds the length up to a neighbouring particle P', u 

 the component velocity of the particle P along the tangent to the 

 chain in the sense in which s increases, v the component velocity of 

 the same particle along the normal inwards, <f> the angle the tangent 

 at P makes with a line fixed in the plane, p the radius of curva- 

 ture of the curve of the chain at P. Then there are two conditions 



