A 470-kg satellite experiences a gravitational force of 850 N.
Use M_{Earth} = 5.972 x 10^{24} kilograms
and
r_{Earth} = 6370 kilometers
What is the radius of the satellite�s orbit?
r_{orbit} = _____________kilometers
What is its altitude?
altitude = ______________kilometers
(no rounding up)
m g' at this altitude = 850
so g' = 850/470 = 1.81 m/s^2 we are way out there!
1.81/9.81 = Rearth^2/r^2
Re^2 = 6370000^2 = (6.37*10^6)^2 =40.6*10^12
so
r^2 = 9.81 * 40.6*10^12 /1.81
r^2 = 220 * 10^12
r = 14.8 * 10^6 meters = 14,800 km
r - Re = altitude = 8,430 km
To determine the radius of the satellite's orbit, we can use the following formula for gravitational force:
F = (G * M1 * M2) / r^2
Where F is the gravitational force, G is the gravitational constant, M1 and M2 are the masses of the two objects, and r is the distance between the two objects.
In this case, the satellite is experiencing a gravitational force of 850 N. Since we know the mass of the Earth (M2) and the radius of the Earth (r), we can rearrange the formula to solve for the radius of the satellite's orbit:
r^2 = (G * M2) / (F)
Substituting the given values:
r^2 = ((6.67430 x 10^-11 N*m^2/kg^2) * (5.972 x 10^24 kg)) / 850 N
Simplifying the equation:
r^2 = (4.0091424 * 10^14)
Taking the square root of both sides:
r ≈ 6.33 x 10^6 meters ≈ 6330 kilometers
Therefore, the radius of the satellite's orbit is approximately 6330 kilometers.
To find the altitude of the satellite, we subtract the radius of the Earth from the radius of the orbit:
altitude = r_{orbit} - r_{Earth}
altitude = 6330 km - 6370 km
altitude = -40 km
Note that the negative sign indicates that the satellite's altitude is below the surface of the Earth. So the satellite is approximately 40 kilometers below the surface of the Earth.