with earths mass 6 x 10^24 kg and radius6.38 x 10^6, how does the inverse-square law show that in space shuttle territory, 200 km above earths surface, the force of gravity is about 94% that at Earths surface
what is (RadiusEarth/(radiusEarth+200km))^2
To understand how the inverse-square law explains the difference in gravity between the Earth's surface and an altitude of 200 km (space shuttle territory), we first need to understand how the inverse-square law works. The inverse-square law states that the strength of a force, such as gravity, is inversely proportional to the square of the distance between two objects.
The equation for the force of gravity between two objects is:
F = G * (m1 * m2) / r^2
Where:
F is the force of gravity between the objects
G is the gravitational constant (approximately 6.674 × 10^-11 N*m^2/kg^2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
In this case, we are considering the force of gravity acting on an object at the Earth's surface and at an altitude of 200 km above the surface. Let's denote the force of gravity at the Earth's surface as F1 and the force of gravity at the altitude of 200 km as F2.
At the Earth's surface:
F1 = G * (mEarth * mObject) / rEarth^2
At an altitude of 200 km:
F2 = G * (mEarth * mObject) / rShuttle^2
Since we want to compare the two forces, we can divide F2 by F1:
F2 / F1 = (G * (mEarth * mObject) / rShuttle^2) / (G * (mEarth * mObject) / rEarth^2)
Notice that the gravitational constant G and the object's mass cancel out:
F2 / F1 = (rEarth / rShuttle)^2
Given that the radius of the Earth at its surface (rEarth) is 6.38 x 10^6 meters (6.38 x 10^6 m) and the radius at an altitude of 200 km (rShuttle) is (6.38 x 10^6 m + 200,000 m), we can calculate the ratio:
F2 / F1 = (6.38 x 10^6 m / (6.38 x 10^6 m + 200,000 m))^2
Simplifying:
F2 / F1 = (6.38 x 10^6 m / 6.58 x 10^6 m)^2
F2 / F1 = (0.969)^2
F2 / F1 ≈ 0.939
Converting this to a percentage, we find that F2 is about 93.9% of F1. This means that at an altitude of 200 km above the Earth's surface, the force of gravity is approximately 93.9% that at the Earth's surface.
Therefore, using the inverse-square law, we can see that the strength of gravity decreases as we move farther away from the Earth's surface.