Path Independence of Conservative Forces — Question 2

Question 2

Reversing a prescribed path in a stationary field

A time-independent position force does 9 J along a prescribed path A to B. Now traverse precisely the same path backward. Separately, a constant 3 N sliding resistance acts along a path of length 2 m.

1. Find the position-force work on the reversed path.

2. Find resistance work on an outward-and-return trip.

3. Does a zero work sum on one retraced position-force path prove that force conservative?

Question 2 - Solution

Reversing a prescribed path in a stationary field

(a) Reverse every displacement element.

For a stationary position field, path reversal changes only the directed displacement: $$W_{BA}=\int_{B}^{A}\vec F(\vec r)\cdot d\vec r=-\int_A^B\vec F(\vec r)\cdot d\vec r=\boxed{-9\ \mathrm J}.$$

(b) The resistance direction also changes.

Sliding resistance reverses with the motion and does negative work on both legs: $$\boxed{W_{r,\mathrm{cycle}}=-fs-fs=-3(2)-3(2)=-12\ \mathrm J}.$$

(c) Distinguish retracing from arbitrary loops.

Exact retracing gives $$W(C)+W(-C)=0$$ for any single-valued stationary position field. Conservativeness instead requires $\oint_C\vec F\cdot d\vec r=0$ for every allowed closed route, including routes that do not retrace.