A force is represented by a vector: from F= ˆi (3xy) - ˆj (2x^2 *y) . Calculate

Question

A force is represented by a vector: from F= ˆi (3xy) - ˆj (2x^2 *y) .
Calculate the work done by this force in a displacement from the origin (0,0) to (the point (5, 10) along the following
paths:
A2
(i) First a displacement along the x-axis from (0,0) to (5, 0), .and then a displacement parallel to the y-axis from (5,0) to (5, 10).
(ii) First a displacement from (0,0) to (0,10) along the y-axis, then a displacement parallel to the x-axis from (0,10) to (5,10).

A2.
(i) A single displacement from (0,0) to (5,10) along the line y = 2x
(ii) Is this force conservative? Explain your answer (clearly).

Explanation / Answer

Given force, F = ˆi (3xy) - ˆj (2x^2 *y) .
A2. (i) The first displacement between the points (0,0) and (5,0) is       x1 = (5 - 0)^2 = 25i m The second displacement between the points (0,5) and (5, 10) is       x2 = (5-0)^2i + (10 - 5 )^2j = 25i + 25j So the net displacement, s = x1 + x2 = (50i + 25j) m Now the work done is, W = F.s = [ˆi (3xy) - ˆj (2x^2 *y)] . (50i + 25j)    So, W = [ (150xy) i - (50x^2y)j ] J (ii) The first displacement between the points (0,0) and (0,10) is       x1 = (10 - 0)^2 = 100j m The second displacement between the points (0,10) and (5, 10) is       x2 = (5-0)^2i + (10 - 10 )^2j = 25i   So the net displacement, s = x1 + x2 = (25i + 100j) m Now the work done is, W = F.s = [ˆi (3xy) - ˆj (2x^2 *y)] . (25i + 100j)    So, W = [(75xy) i - (200x^2y)j ] J The first displacement between the points (0,0) and (0,10) is       x1 = (10 - 0)^2 = 100j m The second displacement between the points (0,10) and (5, 10) is       x2 = (5-0)^2i + (10 - 10 )^2j = 25i   So the net displacement, s = x1 + x2 = (25i + 100j) m Now the work done is, W = F.s = [ˆi (3xy) - ˆj (2x^2 *y)] . (25i + 100j)    So, W = [(75xy) i - (200x^2y)j ] J A2. (i) A single displacement from (0,0) to (5,10) is given by            s = (5 - 0)^2 + (10 - 0)^2 = ( 25 i + 100 j ) m Now the workdone, W = F.s = [ˆi (3xy) - ˆj (2x^2 *y)] . (25i + 100j) So, W = [(75xy) i - (200x^2y)j ] J (ii) The force is not conservative from the above cases.

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