Path Independence of Conservative Forces — Question 5

Question 5

A loop test with finite measurement bounds

Three signed works measured around A to B to C to A are $[4.8,5.2]$ J, $[-2.1,-1.9]$ J and $[-3.2,-2.8]$ J. Treat their hard bounds as independent.

1. Find the interval for total loop work.

2. Is a conservative model ruled out?

3. If the third interval becomes $[-2.2,-1.8]$ J instead, what conclusion changes?

Question 5 - Solution

A loop test with finite measurement bounds

(a) Add allowed interval endpoints.

Independent hard bounds add endpoint by endpoint: $$W_{\min}=4.8-2.1-3.2=-0.5,\qquad W_{\max}=5.2-1.9-2.8=0.5\ \mathrm J.$$ $$\boxed{W_{\mathrm{loop}}\in[-0.5,0.5]\ \mathrm J}.$$

(b) Test inclusion of zero.

The conservative requirement is compatible with the measurements: $$0\in[-0.5,0.5].$$ It is not proved: compatibility on one loop does not establish zero work on every loop.

(c) Repeat with the new independent bounds.

With the revised third interval, $$\boxed{W_{\mathrm{loop}}\in[4.8-2.1-2.2,\ 5.2-1.9-1.8]=[0.5,1.5]\ \mathrm J}.$$ Zero is excluded. Under the stated bounds and assumptions, this loop rules out the conservative model.