Question

For two identical satellites in circular motion around the Earth, which statement is true? a) The one in the lower orbit has less total energy. b) The one in the higher orbit has more kinetic energy. c) The one in the lower orbit has more total energy. d) Both have the same total energy.

Short answer

Answer: (a) The one in the lower orbit has less total energy.

Step-by-step solution

Understand total energy in orbits

Total energy of a satellite is the sum of its kinetic and potential energy. Kinetic energy, K, represents the energy a satellite has because of its motion and is given by the formula K = (1/2)mv^2, where m is the mass of the satellite and v is its velocity. Potential energy, U, represents the energy a satellite has because of its position in the Earth's gravitational field and is given by the formula U = -GMm/r, where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the Earth's center to the satellite.

Relationships in orbits

In a stable circular orbit, the force of gravity acting on the satellite provides the centripetal force needed to keep the satellite in its orbit. This centripetal force is equal to the mass of the satellite multiplied by the square of its tangential velocity divided by the radius of its orbit, Fc = mv^2/r. The force of gravity acting on the satellite is equal to GMm/r^2. From these relationships, we can further derive the following equations: v^2 = GM/r (1) and Fg = GMm/r^2 = mv^2/r (2) Equations (1) and (2) can be used to analyze the different scenarios in the exercise.

Comparing energies in different orbits

We can now analyze each statement given: a) The one in the lower orbit has less total energy. According to equation (1), for a satellite in a lower orbit (smaller r), the value of v will be larger, meaning the satellite will have more kinetic energy. However, its potential energy will be less negative due to the decrease in the radius, making its total energy more negative. Therefore, this statement is true. b) The one in the higher orbit has more kinetic energy. Using equation (1), we can see that a satellite with a larger r will have a smaller value of v. Thus, the satellite in the higher orbit will have less kinetic energy, making this statement false. c) The one in the lower orbit has more total energy. As we’ve already determined in statement (a), the one in the lower orbit has less total energy, making this statement false. d) Both have the same total energy. Both satellites have different values of potential and kinetic energy, and we already established that they have different total energy levels. Therefore, this statement is false.

Conclusion

Based on the analysis of the energy relationships in orbits, the correct answer is (a) - The one in the lower orbit has less total energy.

Key concepts

Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. For satellites orbiting Earth, this is particularly influenced by their tangential velocity as they follow their circular paths. The formula to calculate kinetic energy is given by \[K = \frac{1}{2}mv^2\]
where \(m\) is the mass of the satellite, and \(v\) is its velocity.

• More velocity means higher kinetic energy.
• In circular motion, velocity can change based on the radius of the orbit.

From the equation \(v^2 = \frac{GM}{r}\), we understand that velocity \(v\) is inversely related to the radius \(r\) of the orbit. This implies that objects closer to the Earth in lower orbits move faster, thus having greater kinetic energy compared to those in higher orbits.

Potential Energy

Potential energy in the context of satellites is the energy stored due to their position in the Earth's gravitational field. For any object situated in gravitational fields, the potential energy can be calculated using the formula:\[U = -\frac{GMm}{r}\]
where:

• \(G\) is the gravitational constant,
• \(M\) is the mass of the Earth,
• \(m\) is the mass of the satellite,
• \(r\) is the distance from the Earth's center to the satellite.

The negative sign signifies that this energy is bound by the gravitational attraction between the Earth and the satellite.
As the satellite's orbit radius increases (meaning a higher orbit), the absolute value of the potential energy decreases—becoming less negative. This means satellites further from Earth have less negative potential energy.

Gravitational Force

Gravitational force is the force of attraction between two masses. For satellites in orbit around the Earth, this force is critical in maintaining their paths in space. The gravitational force \(F_g\) exerted on a satellite can be calculated as:\[F_g = \frac{GMm}{r^2}\]
This formula shows:

• The force is directly proportional to the masses involved—both the Earth's mass \(M\) and the satellite's mass \(m\).
• It is also inversely proportional to the square of the distance \(r\) between the two.

Gravitational force acts as the centripetal force required for keeping the satellite in stable circular motion. This means for lower orbit satellites, where \(r\) is smaller, the force is stronger, requiring faster motion to counterbalance the gravitational pull.

Circular Motion

Circular motion refers to any object moving along a circular path. Satellites, when in orbit, engage in circular motion dictated by gravitational forces. These forces keep the satellite moving around the Earth and are in constant interplay with the satellite's velocity.

• The centripetal force required to sustain this motion is provided by the gravitational pull of the Earth, i.e., \(F_c = \frac{mv^2}{r}\).
• This equation is crucial because the velocity of the satellite must be just right to ensure that the centripetal force matches the gravitational force, allowing stable flight.

From our earlier derived formulas, we see:\[v^2 = \frac{GM}{r}\]This tells us that for any stable circular motion, the satellite's speed depends on the gravitational parameter \(GM\) and its orbit's radius \(r\). Essentially, the smaller the orbit (lower \(r\)), the higher the required speed, which is why satellites closer to Earth move faster.
Understanding these dynamics is key to interpreting the energy variations and movement principles of satellites in different orbits.