Question

For a conservative force, the work done in taking a mass from one position to other does not depend on path taken. Can I prove this statement with the help of Variation? By substituting δW=0, I wanted to know the condition for it.

Here is how you proceed,
δW12=δ∫²₁ F. ds.

Answer 1

The work done by a conservative force only depends on the initial and final positions of the object, not on the path taken. This can be mathematically expressed as the integral of the force dotted with the infinitesimal displacement equals the difference in potential energy between the initial and final positions.

Step-by-step explanation:

A conservative force is one for which the work done is independent of path. Equivalently, a force is conservative if the work done over any closed path is zero. For a conservative force, the infinitesimal work is an exact differential. This implies conditions on the derivatives of the force's components. The component of a conservative force, in a particular direction, equals the negative of the derivative of the potential energy for that force, with respect to a displacement in that direction.

To prove that the work done in taking a mass from one position to another does not depend on the path taken, we can consider the potential energy function associated with the conservative force. If the potential energy function is well-defined and continuous, then the work done by the conservative force only depends on the initial and final positions of the object, not on the path taken.

Mathematically, this can be expressed as:

∫²₁ F ⋅ ds = U(P₂) - U(P₁)

where ∫²₁ F ⋅ ds represents the work done by the conservative force along a path from point P₁ to point P₂, and U(P) represents the potential energy at point P.