A spy satellite is in circular orbit around Earth. It makes one revolution in 6.05 hours. (Radius of the Earth=6.371 106 m)
(a) How high above Earth's surface is the satellite?
What is the satellite's acceleration
To find the height above Earth's surface, we can use the formula for the circumference of a circle:
Circumference = 2πr
where r is the radius of the orbit. In this case, the radius of the orbit is equal to the sum of the radius of the Earth and the height above the Earth's surface. Let's denote the height as h.
So, the circumference of the orbit is:
Circumference = 2π(Radius of Earth + h)
We also know that the satellite completes one revolution in 6.05 hours, which is the same as the time it takes to travel the circumference of the circular orbit.
Speed = Distance / Time
The speed of the satellite is the distance it travels in one revolution divided by the time taken for that revolution. The distance traveled in one revolution is equal to the circumference of the orbit, so:
Speed = Circumference / Time
Now, since the satellite is in a circular orbit, its speed is determined by the centripetal force acting on it. This centripetal force is provided by the gravitational force:
Gravitational Force = Centripetal Force
The gravitational force can be calculated using Newton's law of universal gravitation:
Gravitational Force = (G * M * m) / r^2
where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the distance between the Earth's center and the satellite (radius of Earth + h).
Centripetal Force = (m * v^2) / r
where v is the speed of the satellite.
Equating the gravitational force and the centripetal force, we have:
(G * M * m) / r^2 = (m * v^2) / r
Canceling out the masses of the satellite, we get:
(G * M) / r = v^2
Substituting the value of v from the speed equation derived earlier:
(G * M) / r = (Circumference / Time)^2
Now, solving for r:
r = (G * M * Time^2) / (2π)^2
Given that the radius of the Earth is 6.371 * 10^6 m, we can now calculate the height above Earth's surface by subtracting the radius of the Earth from the distance between the Earth's center and the satellite (r):
Height = r - Radius of Earth
To calculate the satellite's acceleration, we can use the equation for centripetal acceleration:
Centripetal Acceleration = (v^2) / r
Substituting the value of v derived earlier:
Centripetal Acceleration = (Speed^2) / r
Again, substituting the value of r using the formula derived earlier, we can calculate the satellite's acceleration.