EXAMPLES 113 



14. A uniform beam of length / leans against the horizontal rim of a 

 hemispherical bowl of radius a, with its lower end resting upon the smooth 

 concave surface. Find its inclination to the vertical. 



15. A bowl in the shape of a paraboloid of revolution is placed with its 

 axis vertical. A uniform rod rests on a peg at the focus and has its lower 

 end resting on the inner surface. Both contacts are perfectly smooth. 

 Find the inclination of the rod to the vertical. 



16. A uniform beam of weight W rests against a vertical wall and a 

 horizontal plane with which it makes the angle a. Both contacts are 

 perfectly smooth. The lower end of the beam is attached by a string to 

 the foot of the wall. Find the tension of the string. 



17. One end of a straight uniform heavy rod rests on a rough horizontal 

 plane, the other end being connected with a fixed point by a string. If 

 0, 0, \f/ be the inclinations of the string, the rod, and the total reaction of 

 the horizontal plane respectively to the vertical, show that 



cot 6 2 cot - cot ^ = 0. 



18. Two uniform rods AB, BC of the same material but of different 

 lengths are jointed freely at B and fixed to a vertical wall at A and C. Show 

 that the direction of the reaction at B bisects the angle ABC. 



19. A uniform regular-hexagonal board ABCDEF of given weight W 

 is supported in a horizontal position on three pegs, placed at the corners 

 A , B and the middle point of DE. Find the pressures on the pegs. 



20. Two spheres of radii a, b and weights W, W respectively are 

 suspended freely by strings of lengths /, /' respectively from the same hook 

 in the ceiling. If V > I + 2 a, show that the angle which the first string 

 makes with the vertical is 



. Wa 



21. A uniform rod hangs by two strings of lengths I, I' fastened to its 

 ends and to two hooks in the same horizontal line at the distance a. If 

 the strings cross one another and make the respective angles a, a', with 

 the horizontal, show that when the rod is in equilibrium 



sin (a + a') (I' cos a' I cos a) = a sin (a a'). 



22. A uniform plank of length 2 b rests with one end on a rough hori- 

 zontal plane, touches a smooth fixed cylinder of radius a lying on the plane, 

 and makes an angle 2 a with the plane, the angle of friction being e. 

 Show that equilibrium is possible if 



a sin e > b tan a cos 2 a sin (2 a + e). 



23. Two equal and similar isosceles wedges, each of weight W and 

 vertical angle 2 a, are placed side by side with their bases on a rough 



