88 STATICS OF SYSTEMS OF PAKTICLES 



12. A weightless rod, to which are fixed two rough beads of masses m 

 and m', lies on an inclined plane and is free to turn about an axis through 

 the rod perpendicular to the plane. If it be in a horizontal position, show 

 that it will not begin to slip round unless 



ma ~ m'l) , 

 /A < - tan a, 

 ma + m'b 



where a is the angle of the plane, and a and b the distances of m and mf 

 respectively from the axis. 



13. A bead of weight W, run on a smooth weightless string, rests on an 

 inclined plane of angle a, the coefficient of friction between the bead and 

 plane being /A. The ends of the string are tied to two points A , B in the 

 plane at the same height. Show how to find the positions of limiting 

 equilibrium for the bead, and show that in such a position P, the tension 



of the string is 



\W sec \ APR - (tan 2 a - M 2 )*. 



14. A uniform string is placed on a rough sphere so as to lie on a hori- 

 zontal small circle in altitude a. Prove that, if the string be on the point 

 of slipping along the meridians, the tension is constant and equal to W cot 

 (a + e), where W is the weight of a length of the string equal to the radius 

 of the circle, and e is the angle of friction. 



15. A weightless string is suspended from two fixed points and at given 

 points on the string equal weights are attached. Prove that the tangents 

 of the inclinations to the horizon of different portions of the string form 

 an arithmetical progression. 



16. A smooth semicircular tube is just filled with 2n equal smooth 

 beads, each of weight W, that just fit the tube, and stands in a vertical 

 plane with the two ends at equal height. If R m is the pressure between 

 the rath and (m + l)th beads from the top, show that 



17. In the last question, let the beads be indefinitely diminished in 

 size. Prove that the pressure between any two beads is proportional to 

 the depth below the top of the tube. 



18. A heavy string hangs over two smooth pegs, at the same level and 

 distance a apart, the two ends of the strings hanging freely and the central 

 part hanging in a catenary. Show that for equilibrium to be possible, the 

 total length of the string must not be less than ae. 



