Resistance of Fluids. 399 



506. It is proposed to determine the angle of inclination at 

 which a current of air acting upon the sails of a windmill will pro- 

 duce the greatest effect. 



Let CD represent the velocity and impulsive force of the F 'g- 24 2- 

 wind against a plane whose section is AB. By reducing this 

 force into the direction CE, we obtain the efficient force with 

 which the fluid strikes against the plane; and by reducing this 

 again into the direction CH, we obtain that force which imparts a 

 rotatory motion to the sail. 



If CD be called a, and the angle CDE, a?, then CE = a sin a?, 

 and CH a sin a; cos x. But as the number of particles which 

 strike against the plane is also diminished on account of its incli- 

 nation in the ratio of rad. to sin a?, it follows that the action of the 

 fluid against the plane is expressed by a sin x 2 cos x. 



To find when this is a maximum, we have, by taking the dif- 

 ferential, 



2 a sin a? cos x 3 d x a sin a? 3 d x = 5 

 or 2 cos x 2 sin x 2 = ; 



hence 

 sin # = V2 cos a?, and - or tang a?, radius being 1 , = y2= 1 ,4 1 4. 



Now 1,414 is the tangent of 54 44' ; therefore 

 x = 54 44'. 



It is obvious that this determination is obtained upon the sup- 

 position that the plane is at rest. We shall now proceed to in- 

 quire at what angle the impulsive force produces the greatest 

 effect, when the plane has acquired a determinate velocity. 



Let CD, as before, represent the direction and impulsive force Fig, 243. 

 of the wind, DG the direction and velocity of the sail's motion. 

 By reducing the velocity of the wind and sail into the direction 

 AB perpendicular to the plane, the relative velocity of the wind, 

 or that with which it strikes the plane, is ED DL. Hence the 

 effect of the wind in producing a rotatory motion will be express- 

 ed by EH DK. 



