Question

you are explaining to friends why astronauts feel weightless orbiting in the space shuttle, and they respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn't so by calculating how much weaker gravity is 250 km

km above the Earth's surface.

Δg/g=?

Answer 1

We can find the acceleration of gravity at any height above the Earth’s surface from the formula:

$$g_E = G \dfrac{M_E}{(R_E + h)^2},$$

here, $G$ is the gravitational constant, $M_E=5.98 \cdot 10^{24} kg$ is the mass of the Earth, $R_E=6.38 \cdot 10^6 m$ is the radius of the Earth and $h$ is the height above the Earth’s surface.

Let’s calculate the acceleration of gravity at $250 km$ above the Earth’s surface:

$$g_{250 km} = 6.67 \cdot 10^{-11}\ \dfrac{N \times m^2}{kg^2} \cdot \dfrac{5.98 \times 10^{24}\ kg}{(6.38 \times 10^6\ m + 2.5 \times 10^5\ m)^2} = 9.07\ \dfrac{m}{s^2}.$$

Let’s compare (in %) the acceleration of gravity at $250 km$ above the Earth’s surface to the acceleration of gravity at the Earth’s surface:

$$\dfrac{g_{250 km}}{g_{Earth's\ surface}} = \dfrac{9.07\ \dfrac{m}{s^2}}{9.8\ \dfrac{m}{s^2}} \times 100 \% = 92.5 \%.$$

Answer:

Therefore, the gravity at 250 km above the Earth's surface is 92.5% as strong as at the Earth’s surface.