Figure 8-3 shows four situations in which the same box is pulled by an appiled force vec(F) up a frictionless ramp through (and then past ) the same vertical distance In each situation, The force has a magnitude of 10 N. In situations (b) and (d), the force is directed along the plane: in situations (a) and (c). it is directed at an angle phi=37^(@) to the plane. as shown. Rank the situations according to the work done on the box in the vertical distance by the appiled force. Also, discuss whether answer depends on the initial speed of box or the presence of other forces.

Solution :KEY IDEAS
Situations (b) and (d) will involve more work than (a) and (c), respectively, because though displacement in (b) and (a) is same and similarly displacement in (d) and (a) is also same, but in (b) and (d) force is acting ALONG displacement. Thus, scalar product of force and displacement will be larger.
CALCULATIONS: Now, we compare magnitude of displacement in (b) and (d), we can see easily that displacement is larger for (d). Thus, by similar ARGUMENT, we can prove that displacement of (c) is larger than (a).
Hence, `W_(d) gt W_(b)`, and `W_(c) gt W_(a)`
Numerical calculations also reveal that our arguments are correct.We find the relation between`W_(b)` and `W_(c)`.
In order to deriver the reltions for work done, we first need to calculate the displacement, that is length of

Figure 8-3 Boxes being pulled on INCLINES by applying forces in different directions.
the ramp in terms of virtual distance Also, in situations the force (F) needs to be resolved along the direction of displacment to determine the work done.
`W_(d) = Fh cosec 37^(@) = (5Fh)/(3)`
`W_(b) = Fh cosec 53^(@) = (5Fh)/(4)`
`W_(c) = (F cos 37^(@))(h cosec 37^(@))=(4 Fh)/(3)`
`W_(d) = (F cos 37^(@))(h cosec 53^(@))= Fh`
Hence `W_(d) gt W_(c) gt W_(b) gt W_(a)`