Uniform and Varying Sectional Area. 121 



To determine G we have that, since the total force on the 

 bar vanishes, 



(B.^f'ydir-0, (9) 



y-i. Jo 



that is 



«. o 



.-. a 



28 yx 



3r ' 



and hence the equation to be assumed is 



/, 14 x 70 x* _«• 28* 6 \ 



^^""ST + l F" 28 P + IP7 



Equation (2) becomes for this case 



dhj pay y\ C x . x , 



and substituting for m, 



= | I ^-5^-7 7 +-7 F -6 r + -16 F ). 



Integrating twice and evaluating the second constant by- 

 virtue of (9) 



-y=^[ij'i '001,993,14 Z 4 — -00925 Z 3 a?+ 0416 # 4 - "038 j 



+ -0138 y ~ "00925|- 9 + -00185 y-l 

 and XT 22-399 kJJ 



The more correct value is 22*373, so that in this case the 

 first approximation gives a ratio of the two values as near 

 as 0-9988. 



6. Clamped- Free Bar of Varying Breadth. (b = A%.) 



Let b be the breadth of the bar, and d its depth in the 

 plane in which the vibration occurs. 



