A cutting tool under microprocessor control has several forces acting on it. One

Question

A cutting tool under microprocessor control has several forces acting on it. One force is F? =??xy2j^, a force in the negative y-direction whose magnitude depends on the position of the tool. The constant is ?. Consider the displacement of the tool from the origin to the point x= a, y= a.

1. Calculate the work done on the tool by F

if this displacement is along the straight line y=x that connects these two points.

2. Calculate the work done on the tool by

F if the tool is first moved out along the x-axis to the point x= a, y=0 and then moved parallel to the y-axis to x= a, y= a.

3.Compare the work done by F

along these two paths. Is F conservative or nonconservative?

Explanation / Answer

F =-A x* y^2 j,

dW = F. dr.

dr = dx i + dy j

dW = (-A x* y^2 j,).

for x direcion , W = 0 (as force and displacemnt will be perpendicular,

W = integration ( -Axy^2 .dy ) from y = 0 to a,

also x= y ,

W = -A*a^4/4

2.for the first travel W = 0 (as force and displacemnt will be perpendicular,)

for second ,

W = integration ( -Aay^2 .dy ) from y = 0 to a,

W = -A a^4/3

3.

W is not the same for both case , hence force is non conervative

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