152 WORK 



WORK DONE AGAINST FORCE OBLIQUE TO DIRECTION OF MOTION 



117. So far we have only considered cases in which the force 

 acts in a direction exactly opposite to that in which the particle 

 moves. We may, however, have to calculate the work when the 

 motion makes any angle with the direction of the force. 



When a body is moved at right angles to the force acting on it, 

 the work done will clearly be -nil ; e.g. in moving a weight about 

 on a horizontal surface no work is done against gravity. 



We can now find the amount of work done when a body is moved 

 in a direction making any angle with the force acting on it. Let a 

 body be moved from P to Q, a small distance ds of its path, while 

 acted on by a force R, of which the line of 

 action makes an angle < with QP. Eesolve 

 , R into two components, R cos </> along QP 



and R sin < perpendicular to QP. The work 

 done against the force R is the same as the 



work which would be done if these two forces R cos <, R sin <f> were 

 acting on the body simultaneously. The work done against th( 

 former force would be R cos <f> ds ; that against the latter would 

 be nil. Thus the whole amount of work done is R ds cos c/>. 



118. Let R have components X, Y, Z, and let the element oi 

 path PQ have direction cosines I, m, n. The direction cosines of the 

 line of action of R are 



X Y Z 



> > 

 R R R 



and since this makes an angle TT < with PQ, we must have 



~X~ V 7 



COS(TT - <) = I + m- + n-> 



Hence R ds cos </> = ds (IX + mY + nZ) 



= - (Xdx + Ydy + Zdz), 



where dx, dy, dz are the projections of ds on the axes. This gives 

 an analytical expression for the work done in a small displace- 

 ment. By integration, we can find the work done in any motion. 



