Alien artificial gravity

Classical Mechanics Level pending

In your interstellar travels, you have discovered an alien artifact that produces a very strange local gravitational field. You cautiously enter a circular orbit around the device.

As your ship maneuvers into different circular orbits near this device, you notice that the time it takes for you to complete one orbit is entirely independent of your orbital radius from the device!

Intrigued, you want to figure out how the gravitational acceleration caused by this device depends on distance from the device.

What value of $n$ correctly gives the dependence of the artificial gravitational field strength $\vec{g}$ on distance from the device $r$ in the following expression?

$\vec{g} \propto r^n$

The answer is 1.

1 solution

Brian Kardon
Feb 16, 2015

The key piece of information is that the period of the orbit $T$ is the same for all orbits. The period is related to the orbital velocity and the radius for circular orbits like so:

$v = \frac{2 \pi r}{t}$

where we have to keep in mind that $t$ is constant. We can get information about the gravitational acceleration $\vec{g}$ by setting it equal to the centripetal acceleration:

$\vec{g} = \frac{v^2}{r}$

Substituting our above expression for $v$ ,

$\vec{g} = \frac{4 \pi^2 r^2}{t^2 r}$

$\vec{g} = \frac{4 \pi^2 r}{t^2}$

$\vec{g} = \frac{4 \pi^2}{t^2} r$

You can see from this last expression that $\vec{g}$ depends only on constants and a factor of $r^1$ , thus the answer is $n = 1$ .

Alien artificial gravity

The answer is 1.

1 solution

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