92 KINETICS OF A PARTICLE. [174. 



174. Exercises. 



(1) Show that when the given forces are zero and there is no friction, 

 the particle moves uniformly on the curve, and the pressure on the curve 

 is proportional to the curvature of the path. 



(2) A particle of mass m moves down a straight line inclined to the 

 horizon at an angle 0, under the action of gravity alone. 



(a) If there- be no friction, we have by Art. 169, since p=oo (see 

 Fig. 24). 



dv . n 



m = mg sin 0, 

 dt 



o = mg cos Q N. 



The first of these equations is the 

 same as that derived in kinematics 

 for motion down an inclined plane 

 (see Part L, Arts. 164-166). The 

 second equation gives the normal 

 reaction of the line JV= mg cos 6 ; 

 hence, the pressure on the line, 

 N, is constant. 



(b) If the line be rough, the second equation remains the same, while 

 the first must be replaced by the following, 



m = mg sin 9 pN= mg(sm /A cos 6). 

 dt 



As the acceleration is constant whether there be friction or not, the 



motion is uniformly accelerated, unless sin /A cos 9 = o, i.e. /A = tan 6. 



Find v and s ; show that, in the exceptional case /x = tan 0, the motion 



is uniform unless the initial velocity be zero ; show that, for motion up 



the plane, the first equation becomes dv/dt=. "(sin + //.cos0), the 



motion being uniformly retarded until /= v /g(sm -\- /A cos 0) when the 



, particle either begins to move down the line or remains at rest. 



(3) A string of length / (ft.) carries at one end a mass of m Ibs. 

 while the other end is fixed at a point O on a smooth horizontal table. 

 The mass m is made to describe a circle of radius / about O on the 

 table, with constant velocity = v ft. per second. Show that the tension 

 of the string is = mi? /I poundals. 



