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Modelling the Earth as a symmetric, spherical body (and by using the law of gravitation), we come up with the equation $$w = F_g = \frac{Gm_Em}{R_E^2}$$ How do we arrive to the equation to get the acceleration due to gravity at the earth's surface? $$g = \frac{GM_E}{R_E^2}$$ Where:

  • $G = \text{gravitational constant} = 6.67\times10^{-11}\ \mathrm{N}$
  • $M_E = \text{mass of Earth} = 5.98\times10^{24}\ \mathrm{kg}$
  • $R_E = \text{Earth's radius} = 6380\ \mathrm{km}$

One thing I know of is the fact that we use Newton's second law, but how?

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  • $\begingroup$ Oh, come on! What wrong did I do? $\endgroup$
    – weirdpanda
    Sep 9 '15 at 10:24
  • $\begingroup$ My guess: this is such an easy question that it seems you can't possibly have made any effort to figure it out yourself before posting here. That might account for the downvote. $\endgroup$
    – David Z
    Sep 9 '15 at 11:43
  • $\begingroup$ I quickly answered it the moment I realised. I posted the answer as I thought it may help someone in the future. Sadly, I am no expert in Physics. $\endgroup$
    – weirdpanda
    Sep 9 '15 at 11:48
  • $\begingroup$ Yeah, I figured that's what happened. Don't worry about it as a one-time thing. I'd just say, don't be so hasty to post a question in the future. If you have something you think would be good to post, take a little while (minutes, hours, days, depending on the question) to think about it first and try a few things yourself before posting here. $\endgroup$
    – David Z
    Sep 9 '15 at 14:15
  • $\begingroup$ @DavidZ :) Cool! I thought that it'd help someone with a similar problem, so I didn't remove the entire question! I'll keep that in mind! $\endgroup$
    – weirdpanda
    Sep 9 '15 at 16:16
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So, it turns out that I am out of coffee.

Anyway, since we already know that:

$w = m \times g$ (Newton's Second Law)

We know that $w = \frac{Gm_Em}{R_E^2}$

Dividing the entire equation by $m$, we get:

$\frac{w}{m} = \frac{mg}{m}$

Which is equal to:

$\frac{Gm_Em}{R_E^2m} = g$ (Substituting the values)

And finally:

$\frac{Gm_E}{R_E^2} = g$

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    $\begingroup$ @weirdpanda, you can use "double dollar signs" (i.e. $$ ... $$) to get your equations on a separate line. $\endgroup$
    – Danu
    Sep 9 '15 at 10:26

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