## PhoenixFire 2 years ago 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.

1. whpalmer4

well, you don't have any units on any of this, which is problematic.

2. Diwakar

you are using 'r' is kilometers ,that's it.

3. whpalmer4

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$9.81953 \text{ N}\text {/kg}*(1 \text{ kg m s}^{-2}/1 \text{ N}) = 9.81953 \text { m}/\text{s}^{2}$

4. PhoenixFire

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