Question

A cutting tool under microprocessor control has several forces acting on it. One force is $\overrightarrow{F}=-\alpha x{y}^2\hat{j}$, a force in the negative y-direction whose magnitude depends on the position of the tool. The constant is $\alpha =2.50N/m^3$. Consider the displacement of the tool from the origin to the point $x=3.00m,y=3.00m$. What can you predict about $\overrightarrow{F}$?

• A. Force is non-conservative
• B. Force is conservative
• C. Force is neither conservative nor non-conservative
• D. Data insufficient to conclude

Answer 1

Answer: (A)

Force is non-conservative

Consider, two paths of motion
Case(i): from $(0,0)$ to $(0,3)$ to $(3,3)$ in straight line path.
Work done in process $1=$${\int }_0^3\left(0\right)dy+{\int }_3^3[-2.5(3^2\left)x\right]dx$
$=-101.25J$.
Case (ii):from$(0,0)$ to $(3,0)$ and to $(3,3)$
Work done in process $2=$${\int }_0^3\left(0\right)dx+{\int }_3^3[-2.5(3\left)y^2\right]dy$
$=-67.5J$
Since both works are not the same even the displacement is the same. So, $F$ is a non-conservative force.