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## Homework Statement

Show that the rate of change of the free-fall acceleration with vertical position near the Earth's surface is

[tex] \frac{d}{dr} [g] = -\frac{2GM_E}{R_E^3} [/tex]

Assuming

*h*is small in comparison to the radius of the Earth, show that the difference in free-fall acceleration between two points separated by vertical distance

*h*is

[tex] \Delta g = \frac{(2GM_E h)}{R_E^3} [/tex]

Show that the minimum period for a satellite in orbit around a spherical planet of uniform density ρ is

[tex] T_{min} = \sqrt{ \frac{3\pi}{Gp}} [/tex]

## Homework Equations

[tex] g=\frac{GM_E}{R_E^2} [/tex]

[tex] T^2=Kr^3 [/tex]

## The Attempt at a Solution

I did the first one easily

[tex] \frac{d}{dr} [\frac{GM_E}{R_E^2}] [/tex]

[tex] \frac{d}{dr} [GM_E R_E^{-2}] [/tex]

simple power rule

The second and third not so much, I substituted one R with (R+h) as that is what it seems like is going on and find the difference. That did not work out for me

The third I thought about p= Mass/Volume and solving for R and substitute but I'm not sure what to do with K and Kepler's 3rd observation is the only equation that I have the deals with the period

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