Find the work done by the force bold upper f equals xy bold i plus left parenthesis y minus x right parenthesis bold jf=xyi+(y−x)j over the straight line from left parenthesis 0 comma 0 right parenthesis(0,0) to left parenthesis 1 comma negative 3 right parenthesis(1,−3).

Workdone = 5 units

Step-by-step explanation:

Given:-

- The force is defined in cartesian coordinates:

                         F = xy i + (y-x) j

- Acts for the line between two points:

                        ( 0 , 0 )   &   ( 1 , -3 )

Find:-

Find the work done by the force over the straight line.

Solution:-

- The work-done by an force is defined as a dot product of Force (F) and the path at which the force acts (ds) or a path integral of force.

- The path (ds) is defined as straight line between two points. We will determine the equation of line:

                           y = mx + c

 Where,              m = ( -3 - 0 ) / ( 1 - 0 ) = -3

                           c = 0

                           y = -3x

- We will substitute the value above into Force F:

                           F = x(-3x) i + (-3x-x) j

                           F = -3x^2 i - 4x j

- Now evaluate the integral for Work done:

the formula of a slope:

we have the points (-2, 7) and (2, 3). substitute: