Path Independence of Conservative Forces — Question 3

Question 3

One matched pair of routes is not a universal proof

A planar position force is $\vec F=(y^2\,\hat i)$ N, with coordinates in meters. Between (0,0) and (2,0), compare paths that rise vertically to y=+1 m or y=$-1$ m, cross horizontally, then return vertically.

1. Find work on each of these two paths.

2. Find work along the straight path y=0.

3. What does this imply about using only the first two measurements to establish a potential?

Question 3 - Solution

One matched pair of routes is not a universal proof

(a) Only horizontal segments contribute.

Only the horizontal legs contribute because $\vec F\cdot d\vec r=y^2dx$: $$\boxed{W_{+}=\int_0^2(+1)^2dx=2\ \mathrm J},\qquad\boxed{W_{-}=\int_0^2(-1)^2dx=2\ \mathrm J}.$$

(b) Test a third route.

On the straight route, $y=0$ throughout: $$\boxed{W_0=\int_0^2 0^2dx=0}.$$

(c) Use the counterexample logically.

One detour followed by a straight return forms a counterexample: $$\oint\vec F\cdot d\vec r=W_+-W_0=2\ \mathrm J\ne0.$$ Agreement between two selected routes does not establish path independence. No single-valued potential reproduces the field throughout this region.