A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 106 m, mass = 6.0 1024 kg)
8.0
To find the minimum energy required to change the orbit of a satellite, we need to analyze the conservation of mechanical energy.
In the initial circular orbit with a period of 120 minutes, the satellite has a certain mechanical energy. Let's call this initial energy E_initial.
We can calculate the initial mechanical energy using the formula for the mechanical energy of an object in circular motion:
E_initial = -G * (m_satellite * M_earth) / (2 * r_initial)
where G is the gravitational constant (approximately 6.67 x 10^-11 N m^2 kg^-2), m_satellite is the mass of the satellite (50 kg), M_earth is the mass of the Earth (6.0 x 10^24 kg), and r_initial is the initial radius of the satellite's orbit.
Given that the radius of the Earth is 6.4 x 10^6 m, the initial radius of the satellite's orbit can be calculated by adding the radius of the Earth to the altitude of the satellite.
r_initial = (6.4 x 10^6 m) + altitude
We can find the altitude of the satellite using Kepler's third law of planetary motion, which states that the square of the period of an orbit is proportional to the cube of the semi-major axis of the orbit:
T^2 = (4 * π^2 * (r_initial + altitude)^3) / (G * M_earth)
Since we know the initial period of 120 minutes and the initial radius of the Earth, we can solve for the altitude:
(120 min)^2 = (4 * π^2 * (6.4 x 10^6 m + altitude)^3) / (G * M_earth)
Solving this equation will give us the altitude of the satellite.
Once we know the initial mechanical energy and the altitude, we can calculate the initial mechanical energy E_initial.
Now, to find the minimum energy required to change the orbit to another circular orbit with a period of 180 minutes, we follow the same steps as above but with the new period and a new radius:
T_new^2 = (4 * π^2 * (r_new + altitude)^3) / (G * M_earth)
Using the above equation, we solve for r_new and find the new radius. From there, we can calculate the new mechanical energy E_new using the formula for the mechanical energy.
Finally, the minimum energy required to change the orbit is the difference between the new mechanical energy and the initial mechanical energy:
Minimum energy = E_new - E_initial
By following these steps, you can find the minimum energy required to change the orbit of the satellite.