1620] 



COUPLES, 



21 



Now moment of R about = R . ON 



= R.OAsm0 

 = R . sin 6 . OA 



= moment about of resolved part 

 at right angles to OA. 



19. Theorem of Moments. The sum (with proper signs) 

 of the moments about a point of two vectors localised at a point 

 A is equal to the moment of their resultant about 0. 



Let Pj and P 2 be the magnitudes of the vectors, l and 2 the 

 angles the lines represent- 

 ing them drawn from A 

 make with AO, R the mag- 

 nitude of the resultant, </> 

 the angle the line repre- 

 senting it makes with AO. 

 Then the magnitudes of the 

 resolved parts at right an- 

 gles to AO are Pj sin lt 

 P 2 sin0 2 , and R sin </>, and 

 we know (Article 10) that 

 R sin </> =P 1 sm0 1 + P 2 sin 2 . 



Now sum of moments of P a and P 2 about 



= OA (P! sin 1 + P 2 sin 2 ) 



= 0-4 . R sin $ 



= moment of R about 0. 



This result can be immediately extended to any number of vectors 

 localised at a point. 



20. Couples. A pair of vectors of equal magnitudes and 

 opposite senses localised in parallel lines is called a couple. 



The plane of the two parallel lines is the plane of the couple. 



The moment of one of the vectors about any point on the line 

 of the other is the moment of the couple. The magnitude of the 

 moment is, of course, a number, which is the product of the 

 numbers expressing the magnitude of the vector and the distance 

 between the lines. 



A line drawn from any point, in a certain sense, at right angles 



