56 FOECES ACTING ON A SINGLE PARTICLE 



Clearly the weight is equal to the resultant of the tensions TI, T 2 , T n . 



Let us take any three rectangular axes in space. Let the direction cosines 

 of the first string be /i, mi, n\ ; let those of the second string be J 2 , ^2, n 2 ; and so 

 on. Then, resolved along the axes, the components of the first tension will be 



The components of the other tensions are similar expressions, so that if 

 X, y, Z denote the three components of the resultant, we have 



X=liTi + 1 2 T 2 + + l n T n , 



Y = miTi + m 2 T 2 + + m n T n , 

 Z = niTi + n 2 T 2 + . . . + n n T n . 



Since the magnitude of the resultant is equal to W, we have 



W 2 - X* + F 2 + Z 2 



1 2 T 2 + . + Z n r n ) 2 + (miTi + m 2 T 2 + 



mifOl'+ ml + *fi + + 



= Tf + T| + + 2 TiT 2 cose 12 + , 

 which gives the result required. 



GENERAL EXAMPLES 



1. ABC is a triangle, with a right angle at ^4; AD is the perpendicular 



on BC. Prove that the resultant of forces, acting along AB and 



1 AB AC 



acting along AC, is acting along AD. 

 AD 



2. At a point O there acts a force P, whose line of action is in the plane 

 determined by two lines OA, OB, meeting at 0. The resolved part of P in 

 the direction OA is represented in magnitude and direction by OX, that 

 in the direction OB by OF. Show that the force P is represented in mag- 

 nitude by the diameter of the circle OXY, and find its direction. 



3. Forces P 1? P 2 , , P M acting in one plane at a point are in equi- 

 librium. Any transversal cuts their lines of action in points L l , L 2 , , L n ; 

 and a length OL { is considered positive when the direction from to L t is 

 the same as to P t . Prove that ^PJOL, = 0. 



4. A body is sustained on a smooth inclined plane by two forces, each 

 equal to half the weight, the one acting horizontally, and the other along 

 the plane. Find the inclination of the plane. 



5. The angle of a smooth inclined plane is 30, and a force P acting 

 horizontally sustains a body. In what other direction can P act and sup- 

 port the body ? Compare the pressure upon the plane in the tw r o cases. 



