Physics Times – July 2019

Similarly we get

xz  ii

F F

x z

 

 

 

&  iii

Fy Fx

x y

 

 

 

For a given force vector F F i F j F k  x y z  

the force is said to be conservative if it satisfies

all the three equations. For a two dimensional

force F F i F j x y  the force is said to be

conservative if it satisfies the 3rd equation

Fy Fx

x y

 



 ^ Fz^0 .

(ii) Non-conservative force:
The work done by a non conservative force
depends on the path followed by a body.
e.g: Friction, air resistance etc..

1. Verify whether the given force is conservative
   or not
   F xy i yx j ^2 ˆ ˆ^2

1.Sol: Method-I
If the work done by a force is independent of
the path followed by the body then it is a
conservative force.
Consider any closed path. For simplicity
consider a triangle OAB as shown in the figure.
If the work done along O A B  and
O B are equal then the given force is
conservative.

Here the coordinates of A and B can be taken
anywhere.
Path (OA)
Along OA x - changes from 0 to 2
 dx 0
y - remains constant

y dy   0 0

W Fdx F dyOA  x y

2 0

2 2

0 0

x y

x y

xy dx yx dy

 

 

  

 x dx x(0)^2  0( )0^2

(^)   0 0 0
Path (AB)
Along AB x = constant = 2  dx 0
y changes from 0 2
dy 0

W Fdx F dyAB  x y

2 2

2 2

2 0

x y

x y

xy dx yx dy

 

 

  

2

2 2

0

 2( ) (0) (2 )y y dy

2 2 2

0 0

4

4 4 4 8

2 2

y

ydy

 

     

 



WO A B   8.

Path (OB)

OB is a straight line

x changes from 0 2 (^) dx 0
y changes from 0 2 dy 0
The relation between x and y is
y mx  y x
(ii) Non-conservative force:
e.g:

1. 1.Sol: Method-I
   Path (OA)
   Path (AB)
   Path (OB)
   2 0
   1
   2 0
   m
     
      
     

W Fdx F dyOB  x y

2 2

2 2

0 0

x y

x y

xy dx yx dy

 

 

  

2 2

3 3

0 0

  x dx y dy

4 2 4 2

(^4040)
   x y
    
   