Final answer:
The work done on a particle by a force as it moves between two points is calculated using the dot product of force and displacement for a constant force, or through integration for a force that changes with position.
Step-by-step explanation:
The question asks about the work done on a particle by a force as it moves between two points in space. Work is defined as the dot product of the force vector and the displacement vector. In these examples, we are given specific forces acting on particles, as well as their movement paths, and are asked to calculate the resulting work done.
For a constant force F1 = (3 N)i + (4 N)j, the work done as the particle moves from (0 m, 0 m) to (5 m, 6 m) is calculated with W = F1 · d, where d is the displacement vector from the initial to the final point. Similarly, for a force that changes with position F1 = (2y)i + (3x)j, the work done can be calculated using integration along the path of the particle. Additionally, the question involving the particle under the influence of the force F(x) = (3.0/x) N, moving from x = 2.0 m to x = 5.0 m, requires integrating the force over the path to find the total work done.