14.14
Energy of a Satellite in a Circular Orbit
Thousands of artificial satellites orbit the Earth every day at various distances from the Earth. Satellites that orbit the Earth below an altitude of…
Consider an artificial satellite orbiting the Earth in a perfectly circular orbit under the influence of the Earth's gravitational force.
Therefore, it orbits with critical velocity vc, which is equal to the square root of the product of the gravitational constant and the Earth's mass, divided by its total distance from the Earth's center.
Now, squaring and multiplying the critical velocity equation with half times the satellite's mass, the kinetic energy of the satellite in a circular orbit is obtained.
Recall that the potential energy of the satellite is expressed as the negative product of the gravitational constant and the two masses, divided by the distance between them.
Therefore, the potential energy of the satellite equals minus two times its kinetic energy.
Since the total energy is a sum of kinetic and potential energy, it equals the negative of its kinetic energy for the satellite in circular orbit. The negative sign in the total energy equation indicates that the satellite is bound to the Earth.
View the full transcript and gain access to JoVE Core videos
Q1: What is the relationship between kinetic and potential energy in a satellite's circular orbit?
In a circular orbit, a satellite's kinetic energy is numerically half of its potential energy. The potential energy is negative, expressed as minus the product of the gravitational constant and the two masses divided by their distance. This relationship emerges from the satellite orbiting at critical velocity, which balances gravitational force with the centripetal force required for circular motion.
Q2: Why is the total energy of a satellite in circular orbit negative?
The total energy of a satellite equals the negative of its kinetic energy because potential energy is negative and has twice the magnitude of kinetic energy. The negative total energy indicates the satellite is gravitationally bound to Earth and cannot escape without external energy input. This binding ensures the satellite remains in orbit rather than flying away.
Q3: How does critical velocity determine a satellite's kinetic energy?
Critical velocity is the square root of the gravitational constant times Earth's mass divided by the satellite's distance from Earth's center. When this velocity is squared and multiplied by half the satellite's mass, the kinetic energy is obtained. This velocity ensures the gravitational force provides exactly the centripetal acceleration needed for a stable circular orbit.
Q4: What types of orbits do artificial satellites use around Earth?
Artificial satellites occupy three main orbital regions: low-Earth orbit (LEO) below 1,600 km, used for research and observation satellites; medium-Earth orbit (MEO) between 2,000 and 36,000 km, used for navigation satellites; and geostationary orbit (GEO) at approximately 36,000 km, where communication satellites orbit with periods matching Earth's rotation.
Q5: How does total energy remain constant in elliptical orbits despite changing distances?
In elliptical orbits, a satellite's distance from Earth varies continuously, causing kinetic and potential energy to change at different points. However, the sum of kinetic and potential energy remains constant throughout the orbit. This conservation of total energy follows from Newton's law of gravitation and applies regardless of orbital shape.
Q6: What does the negative sign in a satellite's total energy equation indicate?
The negative sign in the total energy equation indicates that the satellite is gravitationally bound to Earth. A bound satellite cannot escape to infinity without additional energy. This negative total energy distinguishes bound orbits from unbound trajectories, where total energy would be zero or positive.
Q7: How is the potential energy of a satellite expressed mathematically?
The potential energy of a satellite is expressed as the negative product of the gravitational constant and the two masses (Earth and satellite) divided by the distance between their centers. This negative value reflects the attractive nature of gravity. The magnitude of potential energy decreases as the satellite moves farther from Earth.