112 STATICS OF EIGID BODIES 



4. Two heavy uniform rods have their ends connected by two light 

 strings, and the whole system is suspended by the middle point of one rod. 

 Prove that in equilibrium either the rods or the strings are parallel. 



5. Two rods AB, CD lying on a smooth table are connected by stretched 

 strings AC, BD. If the system is kept in equilibrium, by forces acting at 

 the middle points of the rods, prove that if the strings are not parallel 



(a) the rods must be parallel ; 



(&) the tensions must be proportional to the strings. 



6. ABCD is a parallelogram and E is the intersection of the diagonals 

 AC, BD. Show that parallel forces 7, 5, 16, 4 at A, B, C, D respectively 

 are equivalent to other parallel forces, 8 at the middle point of CD, 10 at 

 the middle point of BC, and 14 at E. 



7. A solid cube is placed on a rough inclined plane of angle a with two 

 edges of its base along lines of greatest slope. The angle of friction is e. 

 Prove that if a > 45 it will at once topple over, while if e < a < 45 it will 

 slide down the plane. If a is less than either e or 45, find the friction 

 brought into action. 



8. A uniform rod of length 2/ and weight W rests over a smooth peg 

 at distance h (< Z) from a smooth vertical wall at an angle 6 with the hori- 

 zontal, its lower end pressing against the wall, and its upper being held by a 

 vertical string. Find the tension of the string and show that it vanishes if 



9. Two equal uniform spheres, each of weight W and radius a, rest in 

 a smooth hemispherical bowl of radius b. Find the pressure between the 

 two spheres and also the pressure of each on the bowl. 



10. A uniform rod rests with its two ends on smooth inclined planes, 

 inclined to the horizontal at angles a and p. Find the inclination of the 

 rod to the horizontal. 



11. In the last question a weight equal to that of the rod is clamped to it. 

 At what point must it be clamped in order that the rod may rest horizontally? 



12. A uniform circular ring of weight W has a bead of weight w fixed 

 on it and hangs on a rough peg. Show that if sin e > , then the ring 



can rest without slipping, whatever point of it rests on the peg, e being 

 the angle of friction. 



13. A pentagon ABCDE, formed of equal uniform heavy rods connected 

 by smooth joints at their ends, is supported symmetrically in a vertical 

 plane with A uppermost, and AB, AE in contact with two smooth pegs 

 in the same horizontal line. Prove that if the pentagon is regular, the 

 pegs must divide AB and AE each in the ratio 



1 + sin j 1 ^ TT : 3 sin T ^ TT. 



