Question

A force field (in newtons) is defined by F(x,y)=[y2x​+x,−y3x2+1​]T where coordinates x and y are in meters.

(a) Evaluate F(0,1),F(1,1),F(−1,−1),F(2,−1) and sketch them on the xy-plane.
(b) Show that the force field is conservative.
(c) Evaluate a scalar potential function E(x,y) such that F=∇E.
(d) Hence, compute the work done by F(x,y) from point (0,1) to point (1,1), along the curve y=1+x−x2.

Answer 1

The question requires evaluating a conservative force field at given points, showing the condition of conservativeness, finding a potential scalar function, and then using the potential function to compute the work done by the force along a specified curve.

Step-by-step explanation:

The student's question revolves around the concept of a conservative force field and its implications in physics, specifically related to work done by forces that vary, potential energy, and force fields. A conservative force field is one where the work is independent of the path taken and can often be represented by a scalar potential function.

1. To evaluate F(0,1), F(1,1), F(-1,-1), F(2,-1), we substitute the coordinates into the force vector F(x,y). For example, F(0,1) is calculated by plugging x = 0 and y = 1 into the formula, leading to [y(2x) + x, -y(3x^2 + 1)]^T = [1(2*0) + 0, -1(3*0^2 + 1)]^T = [0, -1]^T, and similarly for the other points.
2. To show that the force field is conservative, we apply the strategy of checking if the partial derivative of Fx with respect to y is equal to the partial derivative of Fy with respect to x. If they are equal, the force is conservative.
3. The scalar potential function E(x, y) can be found by integrating the force components such that F=∇E. This involves finding functions whose derivatives with respect to x and y give Fx and Fy, respectively.
4. The work done by the force when the particle moves along a path is the line integral of the force vector dot product with an infinitesimal displacement vector. For the given curve y = 1 + x - x^2 and along the line y = 1, this integral is calculated over the path.