2 = <; COS e - 5 sin fl = ? - CO [4] 



Because of the motions 5 and the mooring cable will be inclined at a 

 o " 



small angle y from the vertical. Assuming the cable length is c, the 

 cable is straight; and the lower end of the cable is at a fixed point 

 (i.e. the anchor). Then 



CY + £9 5 Cq [5] 



Y = Cq/c - ae/c [6] 



We can now write the equations of motion, for the longitudinal 

 force and pitch moment, as follows: 



- F = M 'i + //p cos(n,x) dS + T (5 /c - £e/c) = [7] 



- M ^= I 6 + //p [<; cos(n,Q - 5 cos(n,c)] dS 



+ T£ [(1 + £/c)e - ?^/c] 



Here dots denote time-derivatives, p is the fluid pressure, cos(n,x) and 

 cos(n,z) are the direction cosines of the (outward) normal to the body 

 surface, and the surface integrals are over that surface, representing 

 the negative of the pressure force and moment exerted by the fluid on the 

 body. The last term in Equation [7], equal to T y, is the horizontal com- 

 ponent of the force exerted on the body by the cable. In Equation [8], I 

 is the body-pitch moment of inertia about the origin; the term involving 

 T is the pitch moment due to the cable tension T and the moment arm l; and 

 the last term is the moment due to the weight of the body acting vertically 

 downward. 



To proceed further we must know the pressure p, and it is here 

 that the slender-body assumption becomes necessary. From the Bernoulli 

 equation 



2 

 P = - P [^ + jl V4.1 + gz] . [9] 



