Problem of the Week #127 - September 1st, 2014

[Chris L T521]

Here's this week's problem!

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Problem: Suppose that an object moves along a path (avoiding the origin) from $(1,1,-1)$ to $(8,3,5)$ under the influence of an inverse first-power gravitational attraction law $\mathbf{G}(\mathbf{x}) = -(1/\|\mathbf{x}\|^2)\mathbf{x}$. Find the work done by the gravitational force.

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[Chris L T521]

This week's problem was correctly answered by Euge and Kiwi. You can find Euge's solution below.

[sp]Let $r = ||x||$, $x_1 = (1,1,-1)$ and $x_2 = (8,3,5)$. Then $-\log r$ is a potential for $G$ as

$\displaystyle \nabla(\log r) = \frac{x}{r} \frac{d}{dr}(\log r) = \frac{x}{r^2} = -G(x)$.

Hence, the work done by the force is

$-\log ||x_2|| - (-\log ||x_1||) = -\log 7\sqrt{2} + \log \sqrt{3} = \log\frac{\sqrt{6}}{14}$.[/sp]