266-268] 



CHAIN MOVING OVER CURVE. 



303 



the velocity, <f> the angle between the vertical and the normal at 

 any point, R the pressure on the curve per unit length, pR the 



Fig. 82. 



friction also per unit length, and suppose that the sense of v is 

 that in which s increases. The equations of motion of an element 

 are found by resolving along the tangent and normal, in the forms 



mdsv = mgds sin < + (T + dT) cos (d<j>) -T- 



v* 

 m ds = mgds cos </> + (T+ dT) sin d<f> Rds, 



P 

 and these are 







mv = mg sin 



+ -^ -- 



.(1), 



(2), 



m = mg cos H R 



P P 



where p is the radius of curvature of the curve of the chain at the 

 point distant s from the chosen particle. 



In the case of a chain with free ends, and on a smooth curve, 

 the first equation (with fjuR omitted) can be integrated, and the 

 integral is the equation of energy. When v is found from this 

 equation we can find the tension at any point by substituting in 

 equation (1), and then equation (2) determines the pressure. 



268. Examples. 



1. A uniform chain of length a is laid out straight on a smooth table at 

 right angles to the edge, and one end is put just over the edge. Prove that, 

 if the edge of the table is rounded off so that the part of the chain which 



