Physics Times 07.2019

(Kiana) #1
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
    
   

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