THE FIRST -ORDER FORCES AND EQUATIONS OF MOTION 



From Bernoulli's equation, the linearized pressure on the body is 

 9$ 



P = - P 3^ - Pgz 



94>t d<^A. Sff'r r ^<^A 



- Pg- - P^- p-9^- P^ - PA[^^+ ge^^ sin(KR cos 9 - .t)J 



- pgz + p|r(z)cos9 + pi|iR(z)(z-Z(--)cos9 - pAco^e R cos 9 cos cot 



+ pgAe^^ sin wt - pgAKe^^R cos 9 cos cot + 9(R^ log R) 

 = - pgz + p|R (z)cos9 + pijjR(z)(z - Zq)cos9 + pgAe sin ut 



- Zpcj^Ae^^R cos 9 cos wt + 0(r2 log R) [21 ] 



The force and moment exerted on the body by the fluid are obtained 

 by integrating the pressure over the surface. In the absence of any other 



external forces, the force or moment must equal the respective accelera- 

 tion times the mass or mioment of inertia of the body. Thus, with n the 

 unit normal vector into the body, the equations of motion are 



m^ = // pcos(n,x)dS [22] 



^i'i + g) = // P cos(n,z)dS [23] 



14* = //p[(z-ZQ)cos(n,x)-xcos(n, z)]dS [24] 



where m is the body's mass, I its moment of inertia about the center of 

 gravity, and the surface integrals are over the submerged surface of the 

 body. 



In computing the pressure integrals over the body surface, it is ex- 

 pedient to employ the (x',y',z') system, fixed in the body. The direction 

 cosines are 



