114 STATICS OF KIGID BODIES 



horizontal table so as to be just in contact along an edge. A smooth 

 sphere of weight w and radius r is supported between them, being in con- 

 tact with a face of each. Prove that for equilibrium it is necessary that 



wcota I W\ 



> r < 2 a sin a tan a [ H -- , 

 \ wl 



where /* denotes the coefficient of friction and 2 a is the length of either 

 base. 



24. A seesaw consists of a plank of weight w laid across a fixed rough 

 log whose shape is that of a horizontal circular cylinder. The inclination 

 to the horizontal at which it balances is increased to a when loads W, W 

 are placed at the lower and higher ends respectively; and the inclination 

 is reduced to /9 when these loads are interchanged. Show that the inclina- 

 tion of the plane when unloaded is 



w) (W'a - 



w' being the weight which, placed at the higher end, would balance the 

 plank horizontally. 



25. A chain is formed of 2 n exactly similar links, the contacts between 

 consecutive links being perfectly smooth. The two end links can slide on 

 a horizontal wire, the contact here being rough, coefficient of friction fj.. 

 Show that in the limiting position of equilibrium, the inclination of either 

 of the upper links to the vertical is 



tan- 1 ^- 



, 2n -1 



26. Two equal circular disks of radius r, with smooth edges, are placed 

 on their flat sides in the corner between two smooth vertical planes inclined 

 at an angle 2 a, and touch each other in the line bisecting the angle. 

 Prove that the smallest disk which can be pressed between them without 

 causing them to separate is one of radius r(sec a 1). 



27. How is the result of the last question modified if all the contacts 

 are rough, the angle of friction at each being e? 



28. Two uniform ladders are jointed at one end and stand with their 

 other ends on a rough horizontal plane. A man whose weight is equal to 

 that of one of the ladders ascends one of them. Prove that the other will 

 slip first. 



If it begins to slip when he has ascended a distance x, prove that the 



coefficient of friction is tan a. a being the length of each ladder 

 2a + x 



and a the angle each makes with the vertical. 



29. A weightless ladder rests against a smooth cube of weight W t 

 standing on smooth ground, with the foot of the ladder tied to the middle 





