296 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



We wish to discover the condition that motion in a horizontal 

 circle, 6 = a, with angular velocity &> may be possible. We have 



sin 2 0<f> = sin 2 &>, 

 so that the energy equation may be written 



., /A, sin 4 a&> 2 \ . 



* a ( + sin 2 Q } - # cos = const. 



Differentiating with respect to the time we obtain the equa- 

 tion 



sin 3 6 a 



Now the steady motion is possible if o> is so adjusted that 

 = when 6 a. This gives us the condition 



ao> 2 = g sec a. 

 (Cf. Example 1, Article 204.) 



If the particle is projected from a point for which 6 is nearly 

 equal to a, in a nearly horizontal direction, with an angular 

 momentum ma?a) sin 2 a about the vertical diameter, then either it 

 tends to remain always very near the circle = a, or to depart 

 widely from it. Supposing it to remain near the circle, we may 

 put 6 = a. + %, expand the terms of equation (1), and reject powers 

 of % above the first. We thus find 



'r a cos a 



showing that the particle oscillates about the state of steady 

 motion in a period equal to that of a simple pendulum of length 



a cos a/(l -t- 3 cos 2 a). 



*261. Examples. 



1. Prove that the steady motion with angular velocity w of a conical 

 pendulum of length I is stable, and that, if a small disturbance is made, 

 oscillations take place in time 



2. A particle describes a horizontal circle of radius r in a smooth parabo- 

 loid of revolution whose axis is vertical and vertex downwards. Prove that, 

 if it is slightly disturbed, its period of oscillation is 



where 4a is the latus rectum. 



