23] 



CONICAL PENDULUM. 



55 



The condition for equal roots is that (&(#) and <!>'(#) have a common 

 root. Now 



c 2 , 



If then (!>' (%) = 0, we have 



together with 

 from which 



We accordingly have for the value of 



dcp _ c 



~dt i*- s * ~ 



We thus obtain for the time of revolution 



*(*.) = 2 fo* + A) (J-V) -- 

 c 2 = 2 to, + A) (Z - V) = g { 



The time of revolution of a conical pendulum 

 is the same as that of a complete oscillation of 

 a plane pendulum of length # performing 

 small vibrations. 



As & approaches a right angle, # and 

 therefore T approaches zero, that is the velocity 

 increases without limit. We have in this case 



Now the centripetal acceleration in the circular 

 motion is ( 10), 



^="9 



An acceleration g directed downward together rig. 17. 



with the reaction R directed toward the center 



of the sphere will compound into an acceleration g tan & in a 

 horizontal direction (Fig. 17). Accordingly if the particle is projected 

 horizontally with the velocity v, it will describe a circle. 



