What is the ratio of the gravitational force on you when you are 5800 km above the Earth's surface versus when you are standing on the surface? (The Earth's radius is about 6400 km.)
small r = 6,400
big r = 12,200
little force / big force
= little r^2/big r^2 = 6.4^2/12.2^2
=.275
To find the ratio of the gravitational force on you when you are 5800 km above the Earth's surface versus when you are standing on the surface, we can use Newton's law of universal gravitation.
According to the law of universal gravitation, the gravitational force between two objects is given by the formula:
F = (G * m1 * m2) / r^2
Where F represents the gravitational force, G is the gravitational constant (approximately 6.67 x 10^-11 Nm^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.
In this case, the two objects are you and the Earth, and we want to compare the gravitational force when you are 5800 km above the Earth's surface to when you are standing on the surface.
Let's denote the gravitational force when you are 5800 km above the surface as F1 and when you are standing on the surface as F2.
To find F1, we need to calculate the distance between the center of the Earth and the point 5800 km above the surface. The total distance from the Earth's center to the point above the surface will be the sum of the Earth's radius (6400 km) and the distance above the surface (5800 km). So, the distance, r1, when you are 5800 km above the Earth's surface is:
r1 = 6400 km + 5800 km = 12200 km = 1.22 x 10^7 m
To find F2, we need to calculate the distance between the center of the Earth and the Earth's surface, which is the Earth's radius, r2:
r2 = 6400 km = 6.4 x 10^6 m
Now we can calculate the ratio of F1 to F2 by dividing the two forces:
Ratio = F1 / F2 = (G * m1 * m2 * r2^2) / (G * m1 * m2 * r1^2)
The masses of both you and the Earth cancel out, leaving us with only the ratios of the distances squared:
Ratio = (r2 / r1)^2
Substituting the values for r1 and r2, we have:
Ratio = ((6.4 x 10^6 m) / (1.22 x 10^7 m))^2
Simplifying this expression, we get:
Ratio = 0.231
Therefore, the ratio of the gravitational force on you when you are 5800 km above the Earth's surface versus when you are standing on the surface is approximately 0.231.