Vibration of the Gravest Mode oj a Thin Rod. 261 



Now to apply Lord Rayleiglfs method*, we require to 

 assume some form for the deflexion curve of the axis of the 

 rod. We shall make two different assumptions and compare 

 the results obtained. 



Let y be the displacement of the axis at distance z from o. 



First assume 



where 3r) is the displacement of the free end and for small 

 vibrations in the plane yoz is a function of t. 



This is the simplest algebraic curve that satisfies the end 

 conditions for a uniform rod, encastred at one end and free 

 at the other — viz., 



dy_ ( ;>fors = 0, 

 dz~ V J 



The last two equations become indeterminate for the rod 

 tapering to a knife-edge, as I is itself zero for z = l, and 

 hence we should not expect this assumption for the deflexion 

 curve to give a very accurate value for this case. This will 

 in fact emerge in the course of the work. 



We require to calculate the kinetic and potential energies 

 of the rod. For the small vibrations in the plane yoz, we 

 have kinetic energy due to y and potential energy due both 

 to the bending and to the tension. 



[-52 Jo 292-] 



* Luc. cit. 



w lie re Lv 



he**. 



