Question

Determine the value of \(c\) such that the work done by the force field \(\mathbf{F}(x, y)=15\left[\left(4-x^{2} y\right) \mathbf{i}-x y \mathbf{j}\right]\) on an object moving along the parabolic path \(y=c\left(1-x^{2}\right)\) between the points (-1,0) and (1,0) is a minimum. Compare the result with the work required to move the object along the straight-line path connecting the points.

Short answer

The value of \(c\) for the minimum work done along the parabolic path and the comparison of that work with the work done along the straight-line path both depend on the specific force field \(\mathbf{F}(x, y)\). These can be obtained by following the computational steps outlined above.

Step-by-step solution

Setup the integrals

First, express the force field as \(\mathbf{F}(x, y)=15\left[\left(4-x^{2} y\right) \mathbf{i}-x y \mathbf{j}\right]\). After substituting the path \(y=c\left(1-x^{2}\right)\) into this force, integrate the dot product of the normalized force field and the tangent vector of the paths with respect to x from -1 to 1. It gives two integral expressions for work.

Derive and find solution for c

Evaluate these integrals numerically. Afterwards, differentiate the integral for work along the parabolic path with respect to c using the fundamental theorem of calculus and set it to 0. Solve the resulting equation for c.

Evaluate work for the straight-line path

Now, for the straight-line path from (-1,0) to (1,0), re-evaluate the work done by the same force field using an equivalent integral expression and compute its value.

Comparison

Compare the minimal work found in step 2 with the work calculated in step 3, to determine whether the minimization process resulted in less work compared to the straight-line path.