A 1010 kg geosynchronous satellite orbits a planet similar to Earth at a radius 1.97 × 105 km from the planet’s center. Its angular speed at this radius is the same as the rotational speed of the Earth, and so they appear stationary in the sky. That is, the period
of the satellite is 24 h .
What is the force acting on this satellite?
what is the force of gravity at this altitude?
Fg=GMeMs/r^2
To calculate the force acting on the satellite, we can use the equation for gravitational force:
F = G * (m1 * m2) / r^2
where:
F is the gravitational force (weight) acting on the satellite,
G is the gravitational constant (6.67430 × 10^-11 N*m^2/kg^2),
m1 is the mass of the satellite,
m2 is the mass of the planet, and
r is the distance between the satellite and the center of the planet.
Given:
m1 (mass of satellite) = 1010 kg
r (radius) = 1.97 × 10^5 km = 1.97 × 10^8 m
Now, we need to find the mass of the planet (m2). Since the satellite orbits the planet similar to Earth at a radius where it appears stationary in the sky, we can assume that its angular speed is the same as the rotational speed of the Earth. Therefore, the period of the satellite is 24 hours.
The orbital period of the satellite can be related to the radius (r) using the formula:
T = 2 * π * sqrt(r^3 / (G * m2))
Since the satellite has a period of 24 hours, we can substitute this value into the equation:
24 = 2 * π * sqrt((1.97 × 10^8)^3 / (6.67430 × 10^-11 * m2))
Now, we can solve this equation to find the mass of the planet (m2).
To determine the force acting on the geosynchronous satellite, we can use Newton's law of universal gravitation. This law states that the force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The formula for gravitational force is given by:
F = (G * m1 * m2) / r^2
Where:
F is the force of gravity between two objects
G is the gravitational constant (which is approximately 6.67430 × 10^-11 N*m^2/kg^2)
m1 and m2 are the masses of the two objects
r is the distance between their centers
In this case, the force acting on the geosynchronous satellite is the gravitational force between the satellite and the planet. The mass of the satellite, m1, is given as 1010 kg. The mass of the planet, m2, is not provided, but it is not necessary to find the force.
However, we are given the radius of the satellite's orbit, but we need to convert it to meters to be consistent with the units in the gravitational constant. The radius is given as 1.97 × 10^5 km, which can be converted to meters by multiplying by 1000.
r = 1.97 × 10^5 km * 1000 = 1.97 × 10^8 meters
Now, we can substitute the values into the formula to calculate the force acting on the satellite:
F = (G * m1 * m2) / r^2
Where:
G = 6.67430 × 10^-11 N*m^2/kg^2
m1 = 1010 kg
r = 1.97 × 10^8 meters
F = (6.67430 × 10^-11 N*m^2/kg^2 * 1010 kg * m2) / (1.97 × 10^8 meters)^2
Simplifying the equation gives us the force acting on the satellite.