Confused here... \[F=mg=m \frac{GM}{r^2}\] We know 'g' is an acceleration which is \(ms^{-2}\) Now to calculate 'g' we use \[g=\frac{GM}{r^2}\] \(G=6.674 \times 10^{-11}\) \(M=5.972 \times 10^{24}\) \(r=6371\) putting all this into that equation I get \(9.81 \times 10^6\) Acceleration due to gravity is definitely not 9 million. Can anyone tell where I messed up? or what I'm not understanding.

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Confused here... \[F=mg=m \frac{GM}{r^2}\] We know 'g' is an acceleration which is \(ms^{-2}\) Now to calculate 'g' we use \[g=\frac{GM}{r^2}\] \(G=6.674 \times 10^{-11}\) \(M=5.972 \times 10^{24}\) \(r=6371\) putting all this into that equation I get \(9.81 \times 10^6\) Acceleration due to gravity is definitely not 9 million. Can anyone tell where I messed up? or what I'm not understanding.

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well, you don't have any units on any of this, which is problematic.
you are using 'r' is kilometers ,that's it.
If you'd kept the units on the numbers while working the problem, you would have discovered the issue yourself. \[g = \frac{GM}{r^2} = \frac{(6.674*10^{-11} \text{ N m}^2\text{/kg} ^2)(5.972*10^{24}\text { kg})}{(6371 \text{ km})^2 } \rightarrow [num] \frac{\text{N kg }\color{red}{\text {m}^2} }{\text{ kg}^2\color{red}{\text{ km}^2 }} \]and you see right away that a conversion hasn't been applied because the units don't cancel/combine properly. Include the missing conversion factor and it comes out properly, as we see below: \[g = \frac{GM}{r^2} = \frac{(6.674*10^{-11} \text{ N m}^2\text{/kg} ^2)(5.972*10^{24}\text { kg})}{((6371 \text{ km})(1000\text{ m}/1 \text{ km}))^2 }\]\[ = 9.81953 \text{ N}\text {/kg}\]or if you prefer\[9.81953 \text{ N}\text {/kg}*(1 \text{ kg m s}^{-2}/1 \text{ N}) = 9.81953 \text { m}/\text{s}^{2}\]

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Oh wow. That was amazingly smart of me. Thank you both!

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