Si.] RECTILINEAR MOTION. 27 



^**y2ji" 



effective force for motion upwards is = mg sin + yang cos 6 ; 

 hence, the work done in moving the mass m from P Q to P l is 



sin 4- prngs cos V = mg 



where !=P Q P is the horizontal distance of the final position P l 

 from the starting point P Q . The total work is, therefore, the 

 sum of the work of overcoming gravity through the vertical 

 distance h and the work of overcoming friction through the 

 horizontal distance /. 



50. Work done on a System of Particles. Let there be given 

 any number of particles of masses m-^ m^, ... m^ at the distances 

 .s lf s z , ... s n above a fixed horizontal plane ; and let these masses 

 be raised vertically against gravity so that their distances from 

 the same plane become s^, s 2 ', ... s n f . The centroid of the 

 masses in their original position has a distance s = 2ms/2m 

 from the fixed plane, while in the final position it has the dis- 

 tance s f =*2ms f fm from the same plane. It has, therefore, been 

 raised through a distance s r s. It follows that the total work 

 done in raising the separate masses, viz. 



is equal to the work that would be done in raising the total mass 

 through the distance s' s traversed by the centroid, i.e. to 



51. The Work of a Variable Force is well illustrated by the 

 expansion of gas or steam in a cylinder with a movable piston 

 (Fig. 9). Let r be the radius of the cylinder, / the pressure (in 

 pounds) at any instant of the gas per square inch of surface ; 

 then the total pressure of the gas on the inside of the piston is 

 P rrrr^p pounds, and if P Q be the pressure on the outside (say 

 the atmospheric pressure), the effective force acting on the 

 piston is FPP^ friction being neglected. 



The force F is variable, since the pressure p varies with the 





