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Question

A central force is one of the form $\mathbf{F}=f(\|\mathbf{r}\|) \mathbf{r}$, where f has a continuous derivative (except possibly at $\|\mathbf{r}\|=0$ ). Show that the work done by such a force in moving an object around a closed path that misses the origin is 0.

Solution

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Answered 2 years ago

Step 1

1 of 2The work done by the force ${\bf F}$ in moving an object around a closed path $C$ is given by the line integral:

$W=\oint_C{\bf F}\cdot{\bf T}ds$

Applying the Stoke's Theorem we have that:

$W=\oint_C{\bf F}\cdot{\bf T}ds=\iint_S(\nabla\times{\bf F})\cdot{\bf n}dS$

where $C$ is the curve defined by the open surface $S$.

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