Solution Found!
.43 suggests two proofs that a central, spherically
Chapter 4, Problem 4.44(choose chapter or problem)
4.43 suggests two proofs that a central, spherically symmetric force is automatically conservative, but neither proof makes really clear why this is so. Here is a proof that is less complete but more insightful: Consider any two points A and B and two different paths ACB and ADB connecting them as shown in Figure 4.29. Path ACB goes radially out from A until it reaches the radius rB of B, and then around a sphere (center 0) to B. Path ADB goes around a sphere of radius rA until it reaches the line OB, and then radially out to B. Explain clearly why the work done by a central, spherically symmetric force F is the same along both paths. (This doesn't prove that the work is the same along any two paths from A to B. If you want you can complete the proof by showing that any path can be approximated by a series of paths moving radially in or out and paths of constant r.)
Questions & Answers
QUESTION:
4.43 suggests two proofs that a central, spherically symmetric force is automatically conservative, but neither proof makes really clear why this is so. Here is a proof that is less complete but more insightful: Consider any two points A and B and two different paths ACB and ADB connecting them as shown in Figure 4.29. Path ACB goes radially out from A until it reaches the radius rB of B, and then around a sphere (center 0) to B. Path ADB goes around a sphere of radius rA until it reaches the line OB, and then radially out to B. Explain clearly why the work done by a central, spherically symmetric force F is the same along both paths. (This doesn't prove that the work is the same along any two paths from A to B. If you want you can complete the proof by showing that any path can be approximated by a series of paths moving radially in or out and paths of constant r.)
ANSWER:Step 1 of 3
The expression for central force is given by,
The expression for work done along the path is given by,
The following figure shows the path ACB and ADB and represents vectors along paths.
