Path Independence of Conservative Forces — Question 4

Question 4

Completing a conservative work network

For a conservative interaction, $W_{AB}=4$ J, $W_{BC}=-10$ J and $W_{CD}=3$ J. Points A, B, C and D are distinct configurations.

1. Find $W_{AC}$ and $W_{AD}$.

2. Find $W_{DA}$ and $W_{DB}$.

3. An extra sensor reports $W_{DB}=8$ J exactly. Can all the data describe the stated conservative interaction?

Question 4 - Solution

Completing a conservative work network

(a) Compose paths with common orientation.

Conservative endpoint work composes along the chain: $$\boxed{W_{AC}=W_{AB}+W_{BC}=4-10=-6\ \mathrm J},$$ $$\boxed{W_{AD}=W_{AC}+W_{CD}=-6+3=-3\ \mathrm J}.$$

(b) Reverse and reconnect.

Reverse the full path and reconnect to B: $$\boxed{W_{DA}=-W_{AD}=+3\ \mathrm J},\qquad\boxed{W_{DB}=W_{DA}+W_{AB}=3+4=7\ \mathrm J}.$$

(c) Close a consistency loop.

The additional exact reading produces $$W_{BC}+W_{CD}+W_{DB,\mathrm{reported}}=-10+3+8=1\ \mathrm J\ne0.$$ The data cannot all describe the stipulated conservative interaction.