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what areconservativeforce show that gravitionalforce is a conservative force

 A force is said to be conservative if the work done by the force (or against the force) in moving a body depends only upon the initial and final positions of the body and is independent of the path followed between the initial and final positions.

For example, the gravitational force is a conservative force. To prove this, let us calculate the work done against the gravitational force in raising a body of mass m through a height h from its initial position A to a final position B.

In fig. (a), the body is being raised vertically upwards. Hence the work done against the gravitational force mg is

                                            W = force × displacement = mgh

In fig. (b), the body is taken along a smooth inclined plane CB of height AB = h, and inclination ?. The force F required against gravity is equal to mg sin ? and the distance moved in the direction of force is CB. Hence, the work done is

                               W = F × CB

                                   = mg sin ? × (h/ sin?) = mgh

In fig. (c) the body is being taken through the same height AB = h over a staircase. Let there be n stairs each of vertical height h' and horizontal width x. The work done against gravity in raising the body vertically up through each stair is mgh', and that in taking through each horizontal distance x is zero. Hence, the total work done in raising the body through the n stairs is

                                   W = mgh' × n = mgh                                                                                                               

In fig. (d), the body is being taken along a zig-zag path through the same height AB = h. This path may be assumed as made up of a large number of small horizontal paths, (each ?x) and vertical paths (each ?h). Since, the work done against gravity along any horizontal path is zero, the total work done along vertical paths will be added, that is,

                                  W = ?mg (?h) = mgh

Thus, the work done in moving a body against gravity depends only on the initial and final positions of the body, and not upon the path followed. Hence gravitational force is a conservative force.



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