ILLUSTEATIVE EXAMPLES 201 



ILLUSTRATIVE EXAMPLES 



1. Two particles of masses mi, m 2 are placed on two inclined planes of angles 

 a, /3, placed back to back, and are connected by a string which passes over a smooth 

 pulley at the top of the planes. If the coefficients of friction between the particles 

 and the planes are /*i , /xa , find the resulting motion. 



If motion occurs at all, one particle, say mi, must move down its plane, while 

 the other, mg, will move up. Since the string is inextensible, the acceleration 

 of each will be the same, say / in the direction in which motion is taking place. 



The forces acting on the 

 first particle are 



(a) its weight mig ver- 

 tically down ; 



(6) the tension of the 

 string, say T, up the plane ; 



(c) the reaction with 

 the plane. Let this be re- 

 solved into components E, 



fj.E normal to and up the 



, FIG. llo 



plane. 



Since the particle mi has no acceleration normal to the plane, the component 

 of the resultant force in this direction must be zero. Resolving in this direction 



we obtain 



R mig cos a = 0. 



Resolving down the plane, 



mig sin a /J.R T = mi/, 



and if we eliminate the unknown reaction 22, we obtain 







migr (sin a p. cos a) T = mi/. (a) 



A similar equation can be obtained for the motion of the second particle, 



namely 



m z g (sin p + /j. cosjS) T = m 2 /. (6) 



Solving equations (a) and (6) for/, we obtain 



_ mj(sinor /j, cos a) mgKsinff -t- /* cos/3) 



mi +. m 2 

 giving the acceleration. 



If this value of / comes out negative, we see that the acceleration cannot be 

 in the direction in which motion has been assumed to take place. 



If the system starts from rest, motion in the direction assumed is found to 

 be impossible, and we must proceed to examine whether motion in the opposite 

 direction is possible. If this also is found to be impossible, the system will 

 remain at rest. 





