OP FORCES IX OXE PUUCE. 



80. Two equal heavy beams of sufficient length. ad 

 connected by a hinge, are supported by two smooth pecs in 

 the same horisonl a sphere U placed between them, 



determine the position of equilibrium. 



tig the sides ItC, CA, AD of 



a triangl ^ultant jinsses through the centres of the 



inscribed and circumscribed circles ; shew that 



cos-cosO : cos C- cos 4 : cos 4- cos & 



the position of equilibrium of a uniform beam 

 resting in a vertical plan* .c end pressing agai 



1 supported by the convex are 



of a vertical parabola whose vertex is at the loot of the wall 

 and axU horizontal 



83. A uniform beam P(j of given weight and length rests 



in contact with a fixed vertical circle whose vertical diameter 



such a manner that strings AP, BQ attached to the 



rod and circle are tangents to the circle at the points A and & 



tensions of the strings, and shew that the conditions 



uire that the inclination of the beam to the 



vertical must be less than sin" 1 ~ . 



Shew that no uniform rod can rest partly within and 

 parti .* a fixed smooth hemispherical bowl at an incli- 



nation to : /.on greater than sin" 1 -^. 



85. The sides of a rigid plane polygon are acted on by 



forces at right angles to the siaea ana proportional to them in 



magnitude, all the forces acting in the plane of the polygon, 



.vards; also the sides tal o same order 



are severally divided by the points of application in the con* 



lo of p to q\ shew that the system of forces is eqii- 



to a couple whose moment is 



-f) v 



8 fia represents the force applied to any side a of the 

 polygon. 



