308 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



that the parts of the chain between them may be considered vertical. Elastic 

 threads of natural lengths I and I' and moduluses X and X' are fastened to 

 points P and P of the chain on opposite sides of B and their other ends are 

 fixed to points and 0' vertically below P and P. The system oscillates so 

 that the threads are always stretched and the points P and P' are never for 

 any finite time at rest. Prove that the time of a complete oscillation is 



7. A fine elliptic tube is constrained to rotate with uniform angular 

 velocity o> about its major axis which is vertical, and contains a uniform 

 chain whose length is equal to a quadrant of the ellipse. Prove that, if 

 0,2 4^ where I is the latus rectum of the ellipse, the chain will be in 

 stable relative equilibrium with one end at the lowest point. 



8. A rough helical tube of pitch a and radius a is placed with its axis 

 vertical, and a uniform chain is placed within it, the coefficient of friction 

 between the tube and the -chain being tan a cos e. Prove that when the 

 chain has fallen a vertical distance ma its velocity is <J(ag secasinh2/i), 

 where /u, is determined by the equation 



cot J tanh p = tanh (/A sin e +^m cos a sin 2). 



*272. Initial Motion. When the chain starts from rest in 

 a position which is not one of equilibrium the initial velocities are 

 zero, and the equations of motion are simplified by the omission of 

 d<f>/dt. At the same time the kinematic conditions are altered in 

 form. Since 



dt \pj dsdt ' 



this quantity is initially zero, and we may therefore differentiate 

 equation (1) of Article 269, and write our result 



d*u _ I dv 



dsdt p dt ' 



Writing equations (3) of the same Article in the form 

 du Q , 1 dT 



5T = O H -r- , 



dt m ds 



dt m p J 



differentiating the first with respect to s, multiplying the second 

 by I/p, and subtracting, we obtain an equation 



d ( 1 dT\ IT __dS N 

 ds \m ds) m p 2 ds p 



