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Gravitational force is the invisible force that pulls two objects toward each other. This is the force that keeps the Earth, planets, and other celestial bodies in their orbits. It's described by Newton's law of universal gravitation. The formula used is:\[F = \frac{G M_e m}{r^2}\]where:

• F is the gravitational force
• G is the gravitational constant, \(6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2\)
• M_e is the mass of the Earth, \(5.972 \times 10^{24} \text{ kg}\)
• m is the mass of the satellite, \(220 \text{ kg}\)
• r is the distance from the center of the Earth to the satellite

This equation tells us how gravitational force decreases with the square of the distance. Simply put, the farther away the satellite is, the weaker the force. This force also provides the necessary centripetal force for an orbit.

In order for a satellite to stay in a circular orbit, a centripetal force is required to keep it moving along its curved path. This force is always directed towards the center of the orbit, which is Earth in this case. The gravitational force that we've calculated acts as this centripetal force. The formula for centripetal force is:\[F = m \frac{v^2}{r}\]Here:

• F is the centripetal force
• m is the mass of the satellite
• v is the velocity or speed of the satellite
• r is the radius of the orbit

This relationship shows the connection between velocity and the force needed to keep the satellite in a circular path. By setting the gravitational force equal to the centripetal force, one can derive the satellite's speed in orbit, ensuring it maintains its circular trajectory.

When dealing with orbital mechanics, energy conservation plays a crucial role. A satellite in orbit has both kinetic energy and potential energy. The kinetic energy depends on its speed, and the potential energy depends on its position, or distance, from the Earth.The total mechanical energy E, which is the sum of kinetic and potential energy in a stable orbit, normally remains constant. However, in this problem, the satellite loses mechanical energy by a specific amount per orbit:\[\Delta E = 1.4 \times 10^5 \text{ J}\]This energy loss must be factored in to calculate changes in the satellite's speed and altitude over multiple orbits. With each revolution, the radius of the orbit gradually decreases, affecting both potential and kinetic energy. Understanding these changes is essential to predict the satellite's future trajectory.

Angular momentum in orbital physics describes how an object moves around a fixed point, like how planets orbit the Sun or a satellite orbits Earth. It’s a crucial concept as it affects the stability of orbits.The angular momentum \(L\) for our satellite is given by:\[L = mrv\]where:

• m is the mass of the satellite
• r is the radius of its orbit
• v is the orbital speed

For a closed system without external torques, angular momentum is conserved. In this exercise, however, as the satellite loses energy, angular momentum is not maintained for the satellite itself. Yet, if we consider the Earth-satellite system as isolated, with very little external influence, the total angular momentum theoretically remains constant. This helps in understanding long-term motion of orbital bodies.