18 KINETICS OF A PARTICLE. [35, 



II. Rectilinear Motion. 



35. The motion of a single particle presents a comparatively 

 simple problem, because the forces, being in this case all applied 

 at one and the same point, have a single resultant which is 

 readily found by geometrically adding the forces (Part II., Art. 

 96). Let this resultant be denoted by F, the mass of the par- 

 ticle by m, and its acceleration by j ; then, according to the 

 definition of force (Part II., Art. 60), we must have 



mjF. 



This equation merely expresses the fact that the force F pro- 

 duces in the mass m an acceleration /, which agrees with F in 

 direction and sense, and is inversely proportional to m. 



36. The forces, whose resultant is F, are usually called the 

 impressed forces. Both F and/ are, in general, variable. If at 

 any time t the particle m were acted upon by a force = mj, 

 in addition to the impressed forces, it would evidently be in 

 equilibrium. The product mj of the mass of the particle into 

 its acceleration at any instant is called the effective force of the 

 particle at this instant. It can, therefore, be said that the im- 

 pressed forces are at any instant in equilibrium with the effective 

 force reversed. 



This obvious proposition forms the fundamental idea of a 

 most important method of treating the dynamical equations of 

 motion, known as d'Alembert's principle, which will be discussed 

 more fully later on (see Arts. 97-103, 383-386). It makes it 

 possible to apply to kinetic problems the statical conditions of 

 equilibrium. Thus, in the case of a single particle, if the re- 

 versed effective force, mj, be combined with the impressed 

 forces, we have a system of forces acting on the particle which, 

 at the instant considered, is in equilibrium, and must satisfy 

 the conditions of equilibrium for concurrent forces (Part II., 

 Arts. 97, 101), viz. mf+F=o; or, resolving/ into its com- 







