Serway & Jewett — Energy of a System: Problem 32.

(a) Suppose a constant force acts on an object. The force does not vary with time or with the position or the velocity of the object. Start with the general definition for work done by a force \[W = \int_{i}^{f} \vec{F} \cdot d\vec{r}\] and show that the force is conservative. (b) As a special case, suppose the force \(\vec{F} = (3\hat{i} + 4\hat{j})\text{ N}\) acts on a particle that moves from \(O\) to \(C\) in Figure P7.31. Calculate the work done by \(\vec{F}\) on the particle as it moves along each one of the three paths shown in the figure and show that the work done along the three paths is identical. (c) What If? Is the work done also identical along the three paths for the force \(\vec{F} = (4x\hat{i} + 3y\hat{j})\), where \(F\) is in newtons and \(x\) and \(y\) are in meters, from Problem 19? (d) What If? Suppose the force is given by \(\vec{F} = (y\hat{i} - x\hat{j})\), where \(F\) is in newtons and \(x\) and \(y\) are in meters. Is the work done identical along the three paths for this force?