38 FORCES ACTING ON A SINGLE P ARTICLE 



law. Having proved this, it will follow that forces are vectors, 

 and may be resolved and compounded according to the general 

 rules already given. 



28. Parallelogram of forces. THEOREM. If two forces are 

 represented in magnitude and direction l>y the two sides of a par- 

 allelogram, their resultant will be represented ly the diagonal of 

 the parallelogram. 



Let AB, AC represent the two forces, and let Ab, Ac represent 

 the accelerations they would produce if they acted on any particle 



separately. Since, by the second law 

 of motion,- the acceleration is propor- 

 tional to the force, we must have 



Construct the parallelograms Abdc, 



ABDC. On account of the proportion 

 FIG. 15 



just obtained, the two parallelograms 

 will be similar, so that AdD will be a straight line, and we shall have 



d = AB: Ab. 



But Ad, the diagonal of the parallelogram of edges Ab, Ac, repre- 

 sents the resultant acceleration. Since AB represents the force 

 necessary to produce acceleration Ab, it follows from the proportion 

 just obtained that AD will represent the force necessary to produce 

 acceleration Ad. In other words, the acceleration of the particle 

 is the same as if it were acted on by a single force represented by 

 AD. Thus AD represents the resultant of the forces AB, A C. 



It now follows that force is a vector, so that forces can be 

 compounded according to the laws explained in 14-16. 



PARTICLE IN EQUILIBRIUM 



29. In statics we are concerned only with particles, or systems 

 of particles, at rest. The resultant force on each particle must 

 accordingly be nil. It is therefore important to consider cases in 

 which the resultant of a system of forces is nil. 



