The space around the earth in which its gravitational force acts on a body is called the gravitational field. When an object is moved in the gravitational field, the work is done by the gravitational force. If displacement is in the direction of gravitational force, the work is positive. If the displacement. Is against the gravitational force, the work is negative.

Let us consider an object of mass m being displaced with constant velocity from point A to B along various paths in the presence of a gravitational force (Fig. 4.8). in this case the gravitational force is equal to the weight mg of the object.

work-done-conservative_force_1

Fig. 4.8

The work done by the gravitational force along the path ADB can be split into two parts. The work done along AD is zero, because the weight mg is perpendicular to this path, the work done along DB is (-mgh) because the direction of mg is opposite to that of the displacement i.e. θ = 180⁰. Hence, the work done in displacing a body from A to B through path 1 is

WADB = 0 + (- mgh) = -mgh

If we consider the path ACB, the work done along AC is also (-mgh). Since the work done along CB is zero, therefore,

W ACB = -mgh + 0 = -mgh

Let us now consider path 3, i.e. a curved one. Imagine the curved path, the be broken down into a series of horizontal and vertical steps as shown in Fig. 4.9. there is no work done along the horizontal steps, because mg is perpendicular to the displacement for these steps. Work is done by the force of gravity only along the vertical displacements.

 

smooth path

a smooth path may be replaced by a series of infinitesimal x and y displacements. work is done only during the y displacements.

work-done by gravitational field

The net amount of work done along AB path is still (-mgh).

We conclude from the above discussion that.

Work done in the earth’s gravitational field is independent of the path followed.

Can you prove that the work done along a closed path such as ACBA or ADBA (Fig. 4.8), in a gravitational field is zero?

The field in which the work done be independent of the path followed or work down in a closwd path be zero, is called a conservative filed.

The frictional force is a non-conservative force, because if an object is moved over a rough surface between two points along different paths, the work donw against the frictional force certainly depends on the path followed.