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 400 km above the Earth's surface.

Answer 1

Answer:

It's only 1.11 m/s2 weaker at 400 km above surface of Earth

Explanation:

Let Earth radius be 6371 km, or 6371000 m. At 400km above the Earth surface would be 6371 + 400 = 6771 km, or 6771000 m

We can use Newton's gravitational law to calculate difference in gravitational acceleration between point A (Earth surface) and point B (400km above Earth surface):

$g = G\frac{M}{r^2}$

where G is the gravitational constant, M is the mass of Earth and r is the distance form the center of Earth to the object

$\frac{g_B}{g_A} = \frac{GM/r^2_B}{GM/r^2_A}$

$\frac{g_B}{g_A} = \left(\frac{r_A}{r_B}\right)^2$

$\frac{g_B}{g_A} = \left(\frac{6371000}{6771000}\right)^2$

$\frac{g_B}{g_A} = 0.94^2 = 0.885$

$g_B = 0.885 g_A$

So the gravitational acceleration at 400km above surface is only 0.885 the gravitational energy at the surface, or 0.885*9.81 = 8.7 m/s2, a difference of (9.81 - 8.7) = 1.11 m/s2.