346 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



219. A chain of length a is coiled up on a ledge at the top of a rough 

 plane of inclination a to the horizontal, and one end is allowed to slide down. 

 Prove that, if the inclination of the plane is double the angle of friction (X), 

 the chain will be moving freely at the end of a time *J{6a/(g tan X)}. 



220. One end of a uniform chain of length I and mass m is fixed to a 

 horizontal platform of mass (2& l)m, the chain passes over a smooth fixed 

 pulley, and is coiled on the platform. As the platform descends vertically, 

 the chain uncoils, rises vertically and passes over the pulley. Prove that at 

 any time t before the chain is completely uncoiled the depth x of the platform 

 satisfies an equation of the form x 2 = a+ftx + ye~ kx ''\ where a, /3, and y are 

 constants. 



221. A uniform chain is placed on the arc of a smooth cycloid whose 

 axis is vertical and vertex upwards. Show that so long as the chain is 

 wholly in contact with the cycloid the tension at any point of the chain 

 is constant throughout the motion, and is a maximum at the middle point. 



222. A string without weight is coiled round a rough horizontal uniform 

 solid cylinder of mass M and radius a which is free to turn about its axis. 

 To the free extremity of the string is attached a uniform chain of mass m 

 and length I ; if the chain is gathered close up and then let go, prove that 

 the angle, 6, through which the cylinder has turned after a time t, before the 

 chain is fully stretched, satisfies the equation Mla6=m (^fft 2 - a6) 2 . 



223. A piece of uniform chain is placed on a four-cusped hypocycloid 

 occupying the arc between two cusps the tangents at which are horizontal 

 and vertical, and the chain runs off the curve at the lower cusp. Prove that 

 the velocity of the chain when it has just all run off is that due to falling 

 through T % of its length. 



224. An elastic circular ring of which the radius when unstrained is a 

 rests on a smooth surface of revolution whose axis is vertical in the form of 

 a circle of radius r. Prove that the period of the small oscillations in which 

 each element moves in a vertical plane is the same as for a simple pendulum 

 of length Z, where 1/1 = sinacosa/(r - a) sec a/p, p being the radius of 

 curvature of the meridian curve at a point on the ring, and a the incli- 

 nation of the normal to the vertical. 



225. An endless flexible and inextensible chain, of which the mass per 

 unit length is p through one continuous half and // through the other, is 

 stretched over two equal rough discs each of mass M and radius a, which 

 can turn freely about their centres at a distance b apart in a vertical line. 

 Prove that the time of a small oscillation of the chain under gravity is 



the discs being rough enough to prevent slipping. 



