M = K236y = - PgA^x^6y. 



Substituting for x and A yields: 

 ^ c w ' 



1 12 2 



KH„, = - pgA X = - pgA ^ [x + X, ] = T" pg[x - X, ] . (A-4) 



23 ^ w c * w 2 ■• a b-" 2 "'- a b -■ 



Roll Motion. 



The analysis of roll motion- induced forces and moments is compli- 

 cated by the fact that the body is assumed to rotate about the origin 

 of the coordinate system and not the centroid of the waterplane. 



The problem is illustrated in the figure. Here, line 2, the water- 

 line after rotation through and angle 66 must pass through the inter- 

 section of the y' coordinate axis and the initial waterline. Equations 

 for lines 1 and 2 may then be obtained. 



Line 1: y = c. 



Line 2 : y = mx + b . 



The slope of line 2 is: 



Ay 

 m = -r^ = - tan 66. 

 Ax 



Line 2 must also pass through the point P so that: 



and 



x = + c tan 66 

 P 



y = c, 

 P 



These equations yield the relationship: 



b = c(l + tan^6e) . 



To find the force acting on the body as a result of the rotation, 

 the net lost or gained volume is needed. 



6V = (mx + b - c) dx 



•^x 

 a 



""b 2 



[(- X tan 60 + c(l + tan 6 6) - c] dx 



76 



