A satellite is placed in orbit 100 km above the surface of the earth (radius = 6370 km, mass = 5.98 x 1024 kg).

What is the value of the acceleration of gravity at that altitude as a percentage of the acceleration at the surface of the earth?

F = G Mm/r^2

g = F/m = G M/r^2

g at R = Re = G M/Re^2
g at R = Rsat = G M/Rsat^2

g at Rsat/g at Re = G M/Rsat^2/GM/Re^2

= Re^2/Rsat^2

= 6370^2/6470^2
for percent multiply by 100

To find the value of the acceleration of gravity at a certain altitude as a percentage of the acceleration at the surface of the earth, we first need to determine the gravitational acceleration at both locations.

The acceleration due to gravity can be calculated using the formula:

a = GM/r^2

Where:
a is the acceleration due to gravity,
G is the universal gravitational constant (6.67 x 10^(-11) Nm^2/kg^2),
M is the mass of the celestial object (in this case, the earth),
and r is the distance from the center of the celestial object to the point where the acceleration is measured.

First, let's calculate the acceleration due to gravity at the surface of the earth:

Given:
Radius of the Earth (r1) = 6370 km = 6,370,000 m
Mass of the Earth (M) = 5.98 x 10^24 kg

Plugging these values into the formula, we get:

a1 = GM/r1^2

a1 = (6.67 x 10^(-11) Nm^2/kg^2)(5.98 x 10^24 kg) / (6,370,000 m)^2

a1 ≈ 9.81 m/s^2

This is the acceleration due to gravity at the surface of the Earth.

Now, let's calculate the acceleration due to gravity at an altitude of 100 km above the surface of the Earth:

Radius at altitude (r2) = Earth's radius + altitude = 6,370,000 m + 100,000 m = 6,470,000 m

Using the same formula as before:

a2 = GM/r2^2

a2 = (6.67 x 10^(-11) Nm^2/kg^2)(5.98 x 10^24 kg) / (6,470,000 m)^2

a2 ≈ 9.54 m/s^2

So, the acceleration due to gravity at an altitude of 100 km above the surface of the Earth is approximately 9.54 m/s^2.

To find the percentage of this value compared to the acceleration at the surface of the Earth:

Percentage = (a2 / a1) * 100

Percentage = (9.54 m/s^2 / 9.81 m/s^2) * 100

Percentage ≈ 97.3%

Therefore, the value of the acceleration of gravity at an altitude of 100 km above the surface of the Earth is approximately 97.3% of the acceleration at the surface of the Earth.