Answer:
A gravitational force of 6841.905 newtons is exerted on the satellite by the Earth.
Explanation:
At first we assume that Earth is represented by an uniform sphere, such that the man-made satellite rotates in a circular orbit around the planet. Hence, the following condition must be satisfied:
[tex]\left(\frac{4\pi^{2}}{T^{2}} \right)\cdot r = \frac{G\cdot M}{r^{2}}[/tex] (1)
Where:
[tex]T[/tex] - Period of rotation of the satellite, measured in seconds.
[tex]r[/tex] - Distance of the satellite with respect to the center of the planet, measured in meters.
[tex]G[/tex] - Gravitational constant, measured in newton-square meters per square kilogram.
[tex]M[/tex] - Mass of the Earth, measured in kilograms.
Now we clear the distance of the satellite with respect to the center of the planet:
[tex]r^{3} = \frac{G\cdot M\cdot T^{2}}{4\pi^{2}}[/tex]
[tex]r = \sqrt[3]{\frac{G\cdot M\cdot T^{2}}{4\pi^{2}} }[/tex] (2)
If we know that [tex]G = 6.67\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}}[/tex], [tex]M = 6.0\times 10^{24}\,kg[/tex] and [tex]T = 25800\,s[/tex], then the distance of the satellite is:
[tex]r = \sqrt[3]{\frac{\left(6.67\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}} \right)\cdot (6.0\times 10^{24}\,kg)\cdot (25800\,s)^{2}}{4\pi^{2}} }[/tex]
[tex]r \approx 18.897\times 10^{6}\,m[/tex]
The gravitational force exerted on the satellite by the Earth is determined by the Newton's Law of Gravitation:
[tex]F = \frac{G\cdot m\cdot M}{r^{2}}[/tex] (3)
Where:
[tex]m[/tex] - Mass of the satellite, measured in kilograms.
[tex]F[/tex] - Force exerted on the satellite by the Earth, measured in newtons.
If we know that [tex]G = 6.67\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}}[/tex], [tex]M = 6.0\times 10^{24}\,kg[/tex], [tex]m = 6105\,kg[/tex] and [tex]r \approx 18.897\times 10^{6}\,m[/tex], then the gravitational force is:
[tex]F = \frac{\left(6.67\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}} \right)\cdot (6105\,kg)\cdot (6\times 10^{24}\,kg)}{(18.897\times 10^{6}\,m)^{2}}[/tex]
[tex]F = 6841.905\,N[/tex]
A gravitational force of 6841.905 newtons is exerted on the satellite by the Earth.
