--> // Finding magnitude of F
--> //When F is applied at point B,refer fig 2.18(a)and(b)
--> //From triangle AOC
--> OC=300-150
 OC  = 

   150.
--> AO=300
 AO  = 

   300.
--> alpha=acosd(OC/AO)  //degree
 alpha  = 

   60.
--> //from triangle AOB using geometry we get angle OBA=30 degree
--> //Resolving the forces we get
--> R=2000/cosd(30)  //N
 R  = 

   2309.4011
--> F=R*sind(30)  //N
 F  = 

   1154.7005
--> printf("Least force through point B is F=%.1f N",F)
Least force through point B is F=1154.7 N--> //Least force required through the centre of roller
--> //Assume that F makes an angle theta with the horizontal
--> //Refer fig. 2.19 (a) and (b)
--> //Resolving the forces we get
--> //F*cosd(theta)=R*sind(60)...(1)
--> //F*sind(theta)+R*cosd(60)=W...(2)
--> //Solving (1) and (2) we get
--> //sind(theta)+cotd(60)*cosd(theta)=W/F
--> //For obtaining maximum value of W/F we differentiate W/F w.r.t. theta and we get
--> theta=acotd(cotd(60))  //degree
 theta  = 

   60.
--> //Least value of F is observed when it is at right angle to reaction R
--> Fmin=2000*sind(60)  //N
 Fmin  = 

   1732.0508
--> printf("\nLeast force through the centre of roller is Fmin=%.0f N",Fmin)

Least force through the centre of roller is Fmin=1732 N