EXA1IFLHB. 



27. A uniform string acted on by a central : 

 the form determine the law.fthe- 



the centre offeree bein, r on OM eireumfcrence of the circle. 



Result. Tho force varies inversely as the cube of the. 



28. A smooth sph- upon a string without w 

 fastened at its extremities to two fixed points; shew t; 



the arc of contact of the string and sphere be not less than 

 1 1*, the sphere may be divided into two e^ual portions 

 \ertical plane without disturbing the e'jui- 

 libriuin. 



29. Shew that if a chain exactly surround < a m 



tical circle, so as to be in contact at the lowest point without 

 pressing, the whole pressure on the circle is double the 

 weight of the chain, and the tension at the highest point is 

 three times that at the lowest. 



30. Two strings without weight of the same length have 

 each of their ends fixed at each of two points in tin- 

 horizontal plane. A smooth sphere of radius r and \\ 



W is supported upon them at the same iViin -ach of 



the given points. If the plane in which each strin 

 maki :lc a with the horizon, prove that the tension of 



each is - cosec a; a being the distance between the points. 



of 



.". 1 . A uniform heavy chain hangs over two smoot 

 a distance 2a apart in the same horizontal plane. AY hen there 

 is equilibrium, 2* is the length of the chain between tin 

 which hangs in the form of a catenary, c is the length <>i a 

 portion of the chain whose weight is equal to i ion at 



the lowest point, and h the length of the end that 1. 

 down vertically. If &? and c/t he the small increments of 

 i /* corresponding to a small uniform expansion of the. 

 chain, .shew that & : Sh^s.c //.'/ : h.c s.a. 



. A uniform heavy chain is placed on a rouirli inelined 

 ; what length of chain must hang over t t the 



, in order that the chain may be on the point of slipping 

 up the piano ? 



