50 



FORCES ACTING ON A SINGLE PAETICLE 



The three forces TFsina, Zi, and T 2 must be in equilibrium; hence, by 

 Lami's theorem, each must be proportional to the sine of the angle between 

 the other two. 



In fig. 23, CD, CA, CB are the lines of action of these three forces. The 

 line CZ>, being the line of greatest slope through O, is at right angles to the 

 line AB, which we are told is horizontal. Thus if DC is pr6duced to meet AB 



in P, the angle APC is a right angle. Hence 



sin ACD = sin AC P = cos CAP, 

 and similarly 



sin BCD = sin BCP = cos GBP. 

 By Lami's theorem we have 



Wsina _ TI T 2 



sin AC B ~ sin BCD ~ sinACD 



From the relations just obtained we can 

 obtain these ratios in terms of the angles of 

 the triangle ABC. We have 



Wsina = T T 2 



FIG. 23 sin C cos B cos A 



We can now, if required, express cos J., cos B, and sin C in terms of the sides 

 Zi, ^2, and h of the triangle, by means of the ordinaiy formulae of trigonometry. 



The student is advised to examine for himself the form assumed by the result 

 in the two special cases 



(a) A = B 0, in which ACB are in a straight line ; 



(b) C = 0, A = B = , in which the strings are parallel. 



2. Find the direction and magnitude of the smallest force which will start a 

 body resting on an inclined plane into motion down the plane. 



Let a be the angle of the plane, and fj, the coefficient of friction between it 

 and the body to be moved. Let W be the weight of the body, and let a force F 

 be applied in a direction making 

 an angle with the line of greatest 

 slope down the plane, this force 

 being supposed to be just sufficient 

 to move the body. 



The forces acting on the body 

 consist of 



(a) its weight W ; 



(b) the applied force F ; 



(c) the reaction with the plane. 



FIG. 24 



Let the last force be resolved into two components along and perpendicular 

 to the plane. Taking the latter to be E, the former will be /xR acting up the 

 plane, for, by hypothesis, the body is on the point of motion down the plane. 



