Problem 3 :
A particle is acted upon by a two dimensional force (2i+3j). The particle moves along a straight line in xy  plane, defined by the equation, ax + 3y = 5, where "a" is a constant. If work done by the force is zero, then find "a".
Solution : As both force and displacement are nonzero, the work can be zero, only if force and displacement are perpendicular to each other. In order to compare the directions of force and displacement, we write equation of a straight line, which is parallel to line of action of force. From the expression of the force, it is clear that its slope, "
m
1
m
1
", is given by :
m
1
=
y
x
=
3
2
m
1
=
y
x
=
3
2
The equation of a straight line parallel to the line of action of force is :
y
=
m
1
x
+
c
=
3
2
x
+
c
y
=
m
1
x
+
c
=
3
2
x
+
c
Now, rearranging equation of trajectory as given in the question, we have :
y
=

a
3
+
5
3
y
=

a
3
+
5
3
Let "
m
2
m
2
" be the slope of the line representing trajectory. Then,
m
2
=

a
3
m
2
=

a
3
For two lines to be perpendicular, the product of slopes is
m
1
X
m
2
=

1
m
1
X
m
2
=  1
,
(
3
2
)
(

a
3
)
=

1
⇒
a
=
2
(
3
2
)
(

a
3
)
=

1
⇒
a
=
2