(a)what is the magnitude of the gravitational force of the earth on a surveillance satellite of mass 1040kg that is in a grazing circular orbit 100 km above the surface of the earth.
(b)find the magnitude of the acceleration of the satellite in such a grazing orbit.
(c)determine the speed of the satellite.
take the mass and radius of the earth as 5980000000000000000000000 kg and 6370000m respectively
The gravitational constant
G =6.67•10⁻¹¹ N•m²/kg²,
h=100000 m
m=1040 kg
Earth’s mass is M = 5.98•10²⁴kg,
Earth’s radius is R = 6.37 •10⁶ m.
(a) F =G•m •M/(R+h)²
(b) a= G• M/(R+h)²
(c) mv²/(R+h) = G•m •M/(R+h)²
v=sqrt (G•M/(R+h))
2.232667747500
To calculate the magnitude of the gravitational force on the satellite:
(a) The magnitude of the gravitational force can be found using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
where:
F is the gravitational force,
G is the gravitational constant (approximately 6.67430 × 10^-11 N*m^2/kg^2),
m1 is the mass of the satellite,
m2 is the mass of the Earth, and
r is the distance between the center of the Earth and the satellite.
In this case, the mass of the satellite (m1) is given as 1040 kg, and the distance from the satellite to the Earth's surface (r) is the sum of the Earth's radius (6370000 m) and the altitude of the satellite above the Earth's surface (100000 m).
Substituting these values into the formula:
F = (6.67430 × 10^-11 N*m^2/kg^2) * (1040 kg * 5980000000000000000000000 kg) / (6470000 m)^2
Now, calculate:
F = (6.67430 × 10^-11 N*m^2/kg^2) * (6.2248 × 10^27 kg^2) / (4.173449 × 10^13 m^2)
≈ 9.3023 × 10^15 N
Therefore, the magnitude of the gravitational force of the Earth on the satellite is approximately 9.3023 × 10^15 N.
To calculate the magnitude of the acceleration of the satellite:
(b) In a circular orbit, the gravitational force provides the necessary centripetal force to keep the satellite in motion. So, the magnitude of the acceleration (a) can be calculated using the formula:
a = F / m1
Substituting the values we found earlier:
a = (9.3023 × 10^15 N) / 1040 kg
≈ 8.9422 × 10^12 m/s^2
Therefore, the magnitude of the acceleration of the satellite in a grazing circular orbit is approximately 8.9422 × 10^12 m/s^2.
To calculate the speed of the satellite:
(c) The speed of the satellite in a circular orbit can be found using the following formula:
v = √(G * (m2/r))
Substituting the known values:
v = √((6.67430 × 10^-11 N*m^2/kg^2) * (5980000000000000000000000 kg) / (6470000 m))
≈ √(3.9299 × 10^14 m^3/s^2)
≈ 1.9825 × 10^7 m/s
Therefore, the speed of the satellite in the grazing circular orbit is approximately 1.9825 × 10^7 m/s.