EXAMPLES 115 



point of one of the lowest edges of the cube ; a man of weight w ascends 

 the ladder. Prove that, if the ladder projects above the top of the cube, 

 the cube will tilt before he reaches the top of the cube unless 



W>2w cos a (sin a cos a), 

 where a is the inclination of the ladder to the horizontal. 



30. Four equal spheres rest in contact at the bottom of a smooth 

 spherical bowl, their centers being in a horizontal plane. Show that if 

 another equal sphere be placed upon them the lower spheres will separate 

 if the radius of the bowl be greater than (2 Vl3 + 1) times the radius of 

 a sphere. 



31. Three equal spheres rest in contact on a smooth horizontal plane, 

 so that their centers form an equilateral triangle, and are bound together 

 by a fine string passing around them on the level with their centers. If 

 another equal sphere be placed symmetrically on them, show that the 



tension of the string is increased by to, where w is the weight of the 

 upper sphere. " 



32. A right circular cone of vertical angle 2 a rests with its base on a 

 rough horizontal plane. A string is attached to the vertex, and is pulled 

 in a horizontal direction with a gradually increasing force. Find in what 

 way equilibrium will first be broken. 



33. A heavy particle is placed on a rough inclined plane whose inclina- 

 tion is exactly equal to the angle of friction. A thread is attached to the 

 particle and is passed through a hole in the plane which is lower than the 

 particle, but is not in the line of greatest slope through it. Show that if 

 the thread be gradually drawn through the hole, the particle will describe 

 a straight line and a semicircle in succession. 



34. A uniform cubical block of weight W rests with one edge horizontal 

 on a rough inclined plane, and against the block rests a rough sphere of 

 weight W of radius less than an edge of the cube. The inclination of the 

 plane is gradually increased. Examine the different ways in which equi- 

 librium may be broken, and determine which will actually occur in a 

 given case. 



35. A rough uniform rod is placed on a horizontal plane and is acted 

 on, at one of the points of trisection of its length, by a horizontal force in 

 a direction perpendicular to its length. Find about what point the rod 

 will begin to turn. 



36. A heavy bar AB is suspended by two equal strings of length /, 

 which are originally parallel. Find the couple which must be applied to 

 the bar to keep it at rest after it has been twisted through an angle in 

 the horizontal plane. 



