PERIOD OF VIBRATION OF STEAM VESSELS. 123 



APPENDIX. 



SIMPLE PENDULUM. 

 Figs. 1 and 2, Plate 59. 



Let a weight JV he suspended from a point C by a weightless rod of length / 

 (Fig. I, Plate 59). 



If the weight be raised to A and released, it will swing about the point C, pass- 

 ing through O, vertically below C, and continuing to a point on the opposite side 

 practically as high as A, when its motion will be reversed, and it will swing back 

 toward A again, etc. 



The work done in swinging from A to O will be IVyh, where yi is the height of 

 A above O. Neglecting friction, this work will be the amount of kinetic energy of 



the weight as it passes O with velocity Vo, or Wy^ = W —^. Also, the work done at 



the time of passing B would be W(yi — y), and the kinetic energy W — , the velocity 



2^ 

 at the point B being V. Equating, we have — 



W{y,-y)=lV~ 



and 



V=-^2g{y^-y') (lo) 



If the chord of the arc OA have length S\, and of arc OB, S, then by construc- 

 tion — 



3., = -^^.andy=_ (II) 



Substituting values of yx and y in equation (lo), we have — 



V^^-~y<Vs^^' (12) 



^0=^,^ (13) 



V=^^ S,'-S' (14) 



To determine the period of swing, lay ofif a half circle with radius = Sx (Fig. 

 2, Plate 59), and consider a point on the circumference to move with uniform velocity 

 Fo. When the point is at D, the component of the velocity parallel to OA, V , will be 



F=Fo^ 



