APPROXIMATIONS 



11 



The term to be subtracted from the first approximation to give 



tJ / Q v O 



the second approximation is -- x (- - ) = 0-00018 given to five 



'> \l^oy 



places of decimals. 



The second approximation is 3-98107. 



Also 3981 is the result correct to four significant figures. 



8. Applications of Hie approximate use of Binomial Theorem. 



(1) To find the position of the crosshead corresponding to a 

 given angular position of the crank. 



Let OA, Fig. 1, represent the crank, of length r, rotating about 

 the fixed centre O, and let AB represent the connecting rod, of 

 length /. 



In the extreme position C of the crosshead, the crank and 



FIG. i. 



connecting rod are in the same straight line and the distance 

 OC = I + r. 



Let 6 represent the angular position of the crank and x the 

 distance of the crosshead from C. 



Then x + I cos (} + r cos = / + r where (3 is the angle the 

 connecting rod makes with the line of centres. 



Also I sin (3 = r sin 



but 



cos 



= V 1 - sin2 



/"** 



= I sin 2 6 approximately, 



M9 



i 



since ^ sin 2 is small compared with 1. 



Then #=r r cos + Z / cos (3 



r 2 

 = r(l - cos 6) + -j sin 2 



r 2 



- r(l - cos 0) + - t (1 - cos 20) 



