At what distance from Earth’s surface is the acceleration due to gravity 7.33 m/s2?

Question

I used the equation F = Gm1m2/d2. I'm trying to find the distance (d). I have G (7.33 m/s^2). But I also don't have F or m1 and m2. How do I find these three unknown variables?

Answer 1

You want the acceleration due to gravity, so the test mass is irrelevant. What you know is that $$GM/R^2 = g,$$ where $M$ is the mass of the earth, $R$ is the radius of the earth, and $g \approx 9.8 \text{m/sec}^2$, as this gives the acceleration at the surface of the earth.

You want to solve $GM/r^2 = 7.33$, knowing that $GM/R^2 = 9.8$. Can you do this algebra? Remember, when you've done this, that you want the distance from the surface of the earth, so you want $r-R$.

Answer 2

At the surface of the earth, you have

$$mg=\frac{Gm M}{R^2}\tag{1}$$

where $g=9.8m/s^2$ and $R$ is the radius of the earth. Similarly, at the distance $h$ from the surface,

$$mg'=\frac{Gm M}{(R+h)^2}\tag{2}$$

where $g'=7.33m/s^2$. Take the ratio of (1) and (2),

$$\frac{g'}{g} = \frac{R^2}{(R+h)^2}$$

Then, the distance $h$ is given by

$$h=R\left(\sqrt{\frac {g}{g'}}-1 \right)$$