prove that central force is conservative?
A central force is one in which the direction of the force (attractive or repulsive) is directed towards or away from a fixed point in space, which may be called the centre of the force. In addition, the magnitude of a central force depends only on the distance between the point where the force is being measured and the centre.
Without loss of generality, the origin can be taken as the centre of the central force. Then, the direction of the force at any point would be along the position vector of that point and its magnitude would be a function of only the magnitude of the position vector of that point.
⇒ The central force can be written as F = F ( r ) r ^ .
For an attractive force such as the force of gravity, F ( r ) is negative and for a repulsive force such as that between two like charges, F ( r ) is positive.
A conservative force is one in which the work done in moving a particle from one point to another depends only on the positions of the two particles and is independent of the path taken while moving the particle.
Force is the negative gradient of the potential energy.
⇒
F
(
r
)
=
−
d
U
d
r
U
=
−
∫
F
(
r
)
d
r
.
⇒
Work done,
W
=
∫
r
1
r
2
F
⋅
d
r
→
=
∫
r
1
r
2
F
(
r
)
=
∫
r
1
r
2
F
(
r
)
d
r
=
−
∫
r
2
r
1
F
(
r
)
d
r
=
U
(
r
1
⇒ The work done depends only on the difference between the potential energy at the final position and the initial position.
⇒ The work done is independent of the path.
⇒ The central force is a conservative force.
