1'VNAMICS OF A POINT. 



tremity being at the highest point of the circle; prove that, in the 

 iiing of the motion, the resultant vertical pressure on the 

 circle : the resultant horizontal pressure :: TT* 4 : -i. 



IV. Motion of uniform strings. 



1 I P. A heavy uniform string PQ of which P is the lower 

 extre in motion on a smooth circular arc in a vertical 



. Wing the centre and OA the horizontal radius; prove 

 that the tension at any point R of the string is 



nr y (sin y . sin a -.) 



IT - j '- COS (y + 0) COS (a + 0H, 



0, 2a, 2y are the angles AOP, POQ, POR respectively, and 

 IT is the weight of the string. 



1111. A portion of a heavy uniform string is placed on the 

 arc of a four-cusped hypocycloid, occupying the space between two 

 :it eu.-ps and runs off the curve at the lower i u ; 

 ut at which is vertical; prove that the velocity which the 

 string will have when the whole of it has just left the curve, will 

 be the velocity due to nine-tenths of the length of the string. 



1 1 niform string is placed on the arc of a smooth 



in a vertical plane, and moves under the action of gr .; 

 prove that 



d's . 



./,); 



/ 1). in;' the length of tho string, the arc dr-cribcd by any point 



' : ""' !>> y t ^ 1C ( ^'i lt ' ' '"^ hi-l.i\\- a fixed 



horizontal straight linr. 



lit. '5. A unifunn heavy iced on tho arc of a 



smoot i, \vhose axis is vertical and ids; deter- 



mine the motion, and prove that, so long as th lly in 



Contact with ; !, the tension at any given p 



string .hroughout the motion, <i maximum at 



the middle point. 



