Solution Found!
In Section 4.8, I proved that a force F(r) = f (r)i that
Chapter 4, Problem 4.45(choose chapter or problem)
In Section 4.8, I proved that a force F(r) = f (r)i that is central and conservative is automat-ically spherically symmetric. Here is an alternative proof: Consider the two paths ACB and ADB of Figure 4.29, but with rB = rA + dr where dr is infinitesimal. Write down the work done by F(r) going around both paths, and use the fact that they must be equal to prove that the magnitude function f (r) must be the same at points A and D; that is, f (r) = f (r) and the force is spherically symmetric.
Questions & Answers
QUESTION:
In Section 4.8, I proved that a force F(r) = f (r)i that is central and conservative is automat-ically spherically symmetric. Here is an alternative proof: Consider the two paths ACB and ADB of Figure 4.29, but with rB = rA + dr where dr is infinitesimal. Write down the work done by F(r) going around both paths, and use the fact that they must be equal to prove that the magnitude function f (r) must be the same at points A and D; that is, f (r) = f (r) and the force is spherically symmetric.
ANSWER:Step 1 of 2
That \(\mathbf{F}\) is conservative tells us that the amounts of work done along the two paths \(ACB\) and \(ADB\) are equal, \(W_{ACB} = W_{ADB}\).
That \(\mathbf{F}\) is central implies that no work is done along \(CB\) and \(AD\), that is, \(W_{CB} = W_{AD} = 0\).