EOOK OP MATHEMATICAL PROBLEMS, 

 length fastened to a point in the wall ; prove that the least angfo 



/0\ 



which the string can make with the wall is tan~ J ( -) . 



1260. A uniform rod of weight IF rests with one end against 

 a rough vertical plane and with the other end attached to a string, 

 which passes over a smooth pulley vertically above the former end 

 and supports a weight P. Find the limiting positions of equili- 

 brium, and prove that equilibrium will be impossible unless P be 

 greater than TFcos c, tan e being the coefficient of friction. 



1261. A heavy uniform rod of weight W rests inclined at an 

 angle to the vertical in contact with a rough cylinder of revo- 

 lution, whose axis is horizontal and whose diameter is equal in 

 length to the rod. The rod is maintained in its position by a fine 

 string in a state of tension, which passes from one end of the rod 

 to the other round the cylinder; prove that the tension of the 

 string must be not less than 



1262. Two weights support each other on a rough double 

 inclined plane, by means of a fine string passing over the vertex, 

 and both weights are On the point of motion; prove that if the 

 plane be tilted till both weights are again on the point of motion, 

 the angle through which the planes must be turned is 



2 tan" 1 p. 



1263. A square lamina has a string of length equal to that 

 of a side attached at one of the angular points ; the string is also 

 attached to a point in a rough vertical wall, and the lamina rests 

 with its plane vertical and perpendicular to the wall; prove that, 

 if the coefficient of friction be 1, the angle which the string makes 



with the wall lies between - and ^ tan' 



42 i 



1264. Two weights P t Q of similar material resting on a 

 rough double inclined plane are connected by a fine string passing 



