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XXI. On the Period of Vibration of the Gravest Mode of a 

 Thin Rod, in the form of a Truncated Wedge, ivhen in 

 liotation about its Base*. By Lorna M. Swain, Lecturer 

 in Mathematics, Newnham College, Cambridge^ . 



THIS problem was suggested to the author, in connexion 

 with work undertaken at the Royal Aircraft Establish- 

 ment, under Mr. R. V. Southwell, to estimate the effect of 

 centrifugal force on the periods of vibration of airscrew 

 blades. The effect was worked out rigorously for the case 

 of a uniform rod by Mr. Arthur Berry J and for the case of 

 a rod tapering to a knife-edge by Mr. H. A. Webb and the 

 author§. As explained in the latter report, the mathematical 

 work for a rod tapering to an arbitrary depth would appear to 

 be somewhat involved, and it seemed worth while to consider 

 if the problem could not be tackled by some other method, 

 likely to give results sufficiently accurate to be of use. 



This paper contains an attempt to apply Lord Rayleigh's 

 method for the calculation of the period of vibration of the 

 gravest mode|| to this case and also gives some estimation of 

 the accuracy likely to be attained. 



R*. 1. 



Consider a thin rod in the form of a truncated wedge, 

 symmetrical about an axis [oz), encastred at the base, 

 which contains the origin o, and rotating with uniform 

 angular velocity co about a perpendicular axis (oy). The 

 breadth, measured perpendicular to the plane yoz, is uniform, 



* For the case of no rotation, see also a paper bv J. Morrow in Phil. 

 Mag-, vol. x. p. 124 (1905). 



t Communicated by the Author. 



% Advisory Committee for Aeronautics. Reports and Memoranda, 

 No. 488. 



§ Advisory Committee for Aeronautics. Reports and Memoranda. 

 No. 626. 



|| Rayleigh, 'Theory of Sound,' vol. i., §§ 88, 89. 



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