Work and Potential Energy — Question 7

Question 7

Testing a proposed potential assignment

A report assigns $U_P=5$ J, $U_Q=14$ J and $U_R=8$ J to one conservative interaction. It reports work $W_{PQ}=-9$ J, $W_{QR}=+6$ J and $W_{PR}=+3$ J.

1. Check the first two work entries.

2. Determine the correct P-to-R work and identify the inconsistent entry.

3. Could adding one constant to every reported potential value fix the inconsistency?

Question 7 - Solution

Testing a proposed potential assignment

(a) Compare each work with the negative change.

The first two entries satisfy $$W_{PQ}=U_P-U_Q=5-14=-9\ \mathrm J,\qquad W_{QR}=14-8=+6\ \mathrm J.$$

(b) Check both direct and composed work.

Both endpoint subtraction and path composition give $$\boxed{W_{PR}=5-8=-3\ \mathrm J=(-9)+6}.$$ The reported $+3$ J has the wrong sign.

(c) Test a reference shift.

A common reference shift cancels: $$(U_P+C)-(U_R+C)=U_P-U_R=-3\ \mathrm J.$$ No constant can repair the discrepancy. The exact work datum or the conservative-interaction model must change.