102 



KINETICS OF A PARTICLE. 



[189. 



189. That particular case of the problem of the conical pendulum in 

 which the particle moves in a horizontal circle can be treated directly 

 in an elementary manner. It finds its application in 

 the theory of the governor of a steam engine. 



Let O (Fig. 27) be the point of suspension, OP 

 the length of the pendulum rod, 2f. QOP= 6 the con- 

 stant angle of inclination of the rod to the vertical OQ 

 The only forces acting on the particle are its weight mg 

 and the tension of the rod. As both these forces li< 

 in the normal plane of the path, the tangential accel- 

 eration is zero, and the particle moves uniformly in the 

 circle. 



The radius of curvature of the path is the radius 

 m fl f QP /sin 9 of the horizontal circle. The resultant R 

 of mg and N must act along the radius ; its magnitude 

 is seen from the figure to befi = mgtanO. Hence th^ equation (2) 

 of Art. 169 gives v 



Q 



R\ 



Fig. 27. 



7' 2 



/sin0 



= mgtan. 0, 



or, 



The figure also shows that the tension of the rod is 



(16) 



COS0 



190. Exercises. 



(1) Show that the time of revolution T of the conical pendulum 

 (Art. 1 86) is the same as the time of one complete oscillation of a 

 simple pendulum of length /cos 0. 



(2) Show that the angular velocity with which the vertical plane 

 of the rod turns about the vertical axis OQ (Fig. 27) is inversely! 

 proportional to the cosine of the angle 0. 



(3) A conical pendulum makes n = 60 revolutions per minute : 

 (a) What is the height of the cone ? () If the mass of the bob bej 

 m\ oz., and the length of the rod / = i ft., what is the tension 

 of the rod? (^=32-2.) 



