Question

As one moves to a place 3200km above the earth surface the acceleration due to gravity reduces to 4/9 of its value of the earth surface calculate the radius of the earth?

Answer 1

We know that at surface of earth, where R is the radius of earth in meteres and M is the mass of earth in kg:
$g=G\frac{M}{R^2}$ ...(1)
Now, given that
g' = 4/9 g
​3200 km = 3200000 m
when R' = R + 3200000
$\frac{4}{9}g=G\frac{M}{{\left(R+3200000\right)}^2}$ ...(2)
Divide (1) and (2)
$\frac{9}{4}=\frac{{\left(R+3200000\right)}^2}{{\left(R\right)}^2}$
Take square root
$\frac{3}{2}=\frac{\left(R+3200000\right)}{\left(R\right)}$
thus
3R = 2R + 6400000
R = 6400000 m = 6400 km