Final answer:
To calculate the work done by a force field, integrate the dot product of the force vector and the displacement vector over the particle's path, using the equation W = ∫a^b mathbfF ⋅ dmathbfr. Variable forces can be integrated with respect to the displacements in the Cartesian coordinates, as in dW = Fxdx + Fydy + Fzdz, to obtain the total work.
Step-by-step explanation:
To calculate the work done by a force field on a particle as it travels along a trajectory with a certain parameterization, one needs to integrate the dot product of the force vector and the displacement vector over the path taken by the particle. The integral of the force vector F with respect to displacement vector dr is used to find the total work done, as shown in the equation W = ∫a^b mathbfF ⋅ dmathbfr. When the force components are functions of the position along the path, and the displacements are based on the equations of the path, the work done can also be expressed in terms of the Cartesian coordinates using the equation dW = Fxdx + Fydy + Fzdz.
To apply this concept, consider a variable force acting on a particle from a position-dependent force field, such as F = (2y)i + (3x)j. As an example, the integration of the work would be performed from the origin to a point 5 meters on the x-axis. The total work is found by computing the integral of the product of the force components by the corresponding differential displacements over the interval of the trajectory.