Let the force F is making an angle q with the surface on the body is moved.

Resolving F into its perpendicular components F_x and F_y as;

 

$F_{x}=F\cos\theta$

$F_{y}=F\sin\theta$

 In case when force and displacement are not parallel then only the x-component $F_{x}$ parallel to the surface causes the body to move on the surface and not the y - component $F_{y}$

 

Hence  W =  $F_{x}\times S$

$W= \left(F\cos\theta\right)S$

$W = FS \cos\theta$

 

Mini Exercise

A crate is moved by pulling the rope attached to it. It moves 10 m on a straight horizontal road by a force of 100 N. How much work will be done if?

 

1. The rope is parallel to the road

Solution:        Force = 100 N

                        Distance = S = 10 m

                           Work = W = ?

$W=F\times S$

$W = 100 x 10 = 1000 J$

 

2. The rope is making an angle of 30⁰ with the road.

Solution:            Force = 100 N

                           Distance = S = 10 m

$\theta =30\circ$

Work = W = ?

$W=FS cos\theta$
$W = 100\times10 cos(30\circ)$
$W = 1000\times (0.866) = 866 J$