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The given force is conservative.
(Method-II)
The given force F Fi F j Fk x y zˆ ˆ ˆ is said to
be conservative if it satisfies the following three
equations,
Fx Fy
y x
(1)
Fy FZ
z y
(2)
Fz Fx
x z
(3)
For a two dimensional force
F Fi F j x yˆ ˆ then the force is conservative if
it satisfies eq (1)
Given F xy i yx j ^2 ˆ ˆ^2
2
F xyx
2
F x yy
Fx ( )xy 2
y y
x y( )^2
y
x y xy(2 ) 2
Fy ( )yx 2
x x
y x( )^2
y
y x xy(2 ) 2
The given force is conservative
- Consider the gravitation force between two
masses m and M
2 ˆ 3
F r rGmM GmM
r r
Where r is the radial vector drawn from the
centre of M to the centre of m. Verify whether
it is a conservative or not.
2.Sol : 2 ˆ
F rGmM
r
(Method-II)
2.Sol :
r xi yj ˆ ˆ
r x y ^22
r i jˆ cos sinˆ ˆ
r i jˆ x yˆ ˆ
r r
2 2 2 ˆ ˆ
F C x yi j
x y r r
(where C = GmM)
2 2 3/2ˆ 2 2 3/2ˆ
F Cx i Cy j
x y x y
x 2 2 3/
Cx
F
x y
^
y 2 2 3/
Cy
F
x y
2 2 3/
Fx Cx
y y x y
2 2 3/
Cx x y
y
3 2 2 5/
(2 )
2
Cx x y y
Fx 3 Cxy x y 2 2 5/
y
2 2 3/
Fx Cy
y x x y
2 2 3/
Cy x y
x
