Question

hey guys;-

why gravitational force is called as conservative force ???

breif answer required.....

Answer 1

because it doesn't depend upon the initial and final position of the body.
work done by a conservative force may be 0 also

Answer 2

∮F⋅dr=∫A(∇×F)da

Here, you're asking about the line integral around a closed path. When using this equation, you end up making any arbitrary closed path (in other words, you can choose any path as long as it's closed). To choose a path, imagine an object (like a box, for example), and imagine the box under the influence of the gravitational field. Move the box in space in any direction, along any path you so choose. To make a closed path, move the box back to its starting point at the end of the path.

The nice trick about green theorem, instead of calculating the closed line integral (which can be messy since the maths isn't so nice), you can instead calculate

∫A(∇×F)da

which can sometimes be a lot easier. In the case of gravity, electric force, or a spring force, it turns out that

∇×F=0.

So it follows that

∮F⋅dr=∫A(∇×F)da=∫A0da=0

and

∮F⋅dr=0

This is nice because it works for any closed path.

Basically, the curl of a vector field, (∇×F), tells you if a vector field is conservative or not.

If ∇×F=0, like for gravitational and spring forces, the force F is conservative. If ∇×F≠0, like for magnetic forces, the vector-field is non-conservative. If the field is conservative, then

∮F⋅dr=0.

This is true for any conservative vector field.