Final answer:
A conservative force is one which ensures that work done depends only on initial and final positions, with the relation between components satisfying (dFx/dy) = (dFy/dx). The magnitude of such a force at a specific point is calculated by integrating the force's components. Work done by a constant force can be determined using the dot product of force and displacement.
Step-by-step explanation:
A conservative force is a force where the work done does not depend on the path taken but only on the initial and final positions. When dealing with such forces, if the conditions (dFx/dy) = (dFy/dx) are met for the force components, the force is conservative. In a two-dimensional scenario, if a force satisfies the condition (dFx/dy) = (dFy/dx) = (4 N/m³) xy, to find the magnitude of this force at a point where x = y = 1 m, one can integrate the force components from point zero to the given point.
For example, if we consider a particle affected by a constant force F₁ = (3 N)Î + (4 N)Ĵç, the work done by this force when the particle moves from the origin to the point (5 m, 6 m) can be calculated using the formula Work = F • d, where F is the force vector and d is the displacement vector. In this case, the work done is the dot product of the force vector and the displacement vector. It's important to note that conservative forces are related to the potential energy via the negative gradient.