Question

6. Imagine the r-axis is an infinitely long heavy wire of uniform density. The gravitational force it exerts on a unit mass placed at a point (w, z) (0,0) in the plane is given by where c is a positive constant. Taking c = 1, find by direct calculation of the line integral (that is, by using a parameterisation of the path C) the work done by the gravitational field in moving a unit mass along each of the following paths C in the yz-plane: (a) the half-line z = 1, y 0 starting from the point (0,1); (b) the circle of radius a with centre at origin, traced counterclockwise; (c) the line from (0,1) to (1,0).

Answer 1

a> We know that work done by a constant force F when it acts on an object in order to displace the object by a displacement 'd' is given by :

W = Fd

Now in our case the constant force F moves the particle around a unit curcle in the xy-plane.

We know that a unit circle has a radius of 1 unit.

If the particle starts form the point say (1,0) then this point is on the x-axis now when the force F is applied to move the particle abound the circle once that is 1 complete revolution then the particle would come back at the start point that is (1,0)

Hence the displacement of the particle is = 0 .

Please note displacement is = The shortest distance between two points

Hence the Work Done = Fd = F*0 = 0

Hence proved.