126 PERIOD OF VIBRATION OF STEAM VESSELS. 



ROD IN TENSION. 

 Figs. 4 and 5, Plate 59. 



Let a weightless rod of definite length be suspended from a point C, with lower 

 end at D. When a weight W is attached to the end and gradually lowered, the rod 

 will stretch until the end of the rod reaches some point O (Fig. 4, Plate 59). Call 

 the stretch 5. 



In Fig. 5, Plate 59, lay out DO equal to the amount of the stretch of the rod from 

 condition with no weight attached to condition with weight W attached and at rest. 

 Let OP, DL represent to scale the weight W. Join D and P. Then, if the weight 

 is raised a distance 5"i to point A, the portion of the weight carried by the rod will 

 be represented by AE. If the weight is now set free, it will fall to some point Z be- 

 low 0, and then will rise to A, if friction is neglected, and continue the vibratory 

 motion about O as a mean position. When the weight is at point B, distant 5" above 

 O, coming down, the accelerating force F will be represented by MA'', the difference 

 between the total weight and the tension of the rod. By construction, 



DL h h ^^^ 



The work done by the weight in falling from A at the time of passing O is WSx, 

 represented by the area AOPG. Of this, the area AOPE represents the work done 

 in extending the rod, and the area PGE represents the work done in accelerating the 

 weight. 



The area 



PGB = yiEG X GP= VzW^x S, = ^W^ (26) 



This will be equal to the kinetic energy of the weight when passing O, with 

 velocity Vo, or 



W^=%W^ (27) 



and Fo=5,J| (^«> 



Similarly, the work done in accelerating the weight in falling from A io B will 

 be represented by the area MNGE, and this will equal the kinetic energy of the 

 weight in passing B with velocity V, or 



w^=%w^-y2W^ = y2W ^' . ^ (29) 



2g 



From this we obtain the value of V : — 



V=JiiSipSl_ (30) 



^ 



