Find the acceleration due to gravity at a height of 3200 km above the surface of earth.

Sure, to find the acceleration due to gravity at a height above the surface of the earth, we can use the formula:

g' = g * (r/(r+h))^2

where:

• g' is the acceleration due to gravity at the height h,
• g is the acceleration due to gravity at the surface of the earth,
• r is the radius of the earth, and
• h is the height above the surface of the earth.

Given:

• g = 9.8 m/s^2 (approximate value),
• r = 6371 km (approximate value), and
• h = 3200 km.

First, we need to convert all the measurements to the same units. Here, we'll convert kilometers to meters:

r = 6371 km = 6371 * 10^3 m = 6.371 * 10^6 m, h = 3200 km = 3200 * 10^3 m = 3.2 * 10^6 m.

Now, we can substitute these values into the formula:

g' = 9.8 * (6.371 * 10^6 / (6.371 * 10^6 + 3.2 * 10^6))^2 = 9.8 * (6.371 / 9.571)^2 = 9.8 * (0.665)^2 = 9.8 * 0.442 = 4.33 m/s^2.

So, the acceleration due to gravity at a height of 3200 km above the surface of the earth is approximately 4.33 m/s^2.