Work and Potential Energy — Question 1

Question 1

Reconstructing an interaction-energy ledger

A conservative interaction does +18 J of work from A to B and $-7$ J from B to C. Choose $U_A=12$ J.

1. Find $U_B$ and $U_C$.

2. Find work from C directly to A.

3. Would choosing $U_A=100$ J change any of these works?

Question 1 - Solution

Reconstructing an interaction-energy ledger

(a) Use the defining sign convention.

For a conservative interaction, $\Delta U=-W$: $$\boxed{U_B=U_A-W_{AB}=12-18=-6\ \mathrm J},$$ $$\boxed{U_C=U_B-W_{BC}=-6-(-7)=1\ \mathrm J}.$$

(b) Close the configuration cycle.

The return work depends only on the endpoints: $$\boxed{W_{CA}=U_C-U_A=1-12=-11\ \mathrm J}.$$ The closed-cycle check is $18-7-11=0$.

(c) Shift all values together.

Changing the reference adds one constant: $$C=100-12=88\ \mathrm J,\qquad(\widetilde U_A,\widetilde U_B,\widetilde U_C)=(100,82,89)\ \mathrm J.$$ $$\widetilde W_{if}=(U_i+C)-(U_f+C)=U_i-U_f=W_{if}.$$ Every work remains unchanged.