116 STATICS OF EIGID BODIES 



37. The line of hinges of a door is inclined at an angle a to the vertical. 

 Show that the couple required to keep it in a position inclined at an angle 

 /3 to that of equilibrium is proportional to sin a sin ft". 



38. Show that any system of forces acting on a rigid body can be 

 reduced to two equal forces equally inclined to the central axis. 



39. Prove that the central axis of two forces P and Q intersects the 

 shortest distance c between their lines of action and divides it in the ratio 



Q(Q + Pcostf) :P(P + Qcos0), 

 B being the angle between their directions. Also prove that the moment 



of the principal couple is 



. cPQ sin $ 



VP2 + Q2 + O PQ COS 6 



40. Prove that the axis of the resultant of two given wrenches (72 1} HJ 

 and (^ 2 ' ^2)' ^^ e axes f which are inclined to each other at an angle 0, 

 intersects the shortest distance, 2c, between their axes at a point the dis- 

 tance of which from the middle point is 



(R? - Rf) c + 

 R* + 



