To determine the height of the satellite above Earth's surface, we need to use the following concepts and equations from orbital mechanics:

1. Gravitational force is providing the necessary centripetal force for the satellite to stay in orbit.

   The gravitational force equation is: $F = \frac{G M m}{r^2}$ where:

   • $G = 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2$ is the gravitational constant,
   • $M = 5.974 \times 10^{24} \, \text{kg}$ is the mass of the Earth,
   • $m$ is the mass of the satellite,
   • $r$ is the distance from the center of the Earth to the satellite.

2. Centripetal force required to maintain the satellite in a circular orbit is given by: $F = m \frac{v^2}{r}$ where:

   • $v$ is the orbital velocity of the satellite,
   • $r$ is the distance from the center of the Earth to the satellite.

3. From these two expressions, we equate the gravitational force to the centripetal force: $\frac{G M m}{r^2} = m \frac{v^2}{r}$

   Simplifying this: $\frac{G M}{r} = v^2$

4. The satellite’s velocity is also related to the orbital period $T$ (the time it takes for one complete revolution) by: $v = \frac{2 \pi r}{T}$

5. Substituting this expression for $v$ into the previous equation: $\frac{G M}{r} = \left( \frac{2 \pi r}{T} \right)^2$

6. Rearranging for $r$, we get: $r^3 = \frac{G M T^2}{4 \pi^2}$

7. Now, we can solve for $r$ (the orbital radius from the center of the Earth) and then subtract the Earth's radius to get the satellite's height above the surface.

Let's calculate this step-by-step.The height of the satellite above Earth's surface is approximately $13,083 \, \text{km}$.

Would you like more details on the calculation, or have any further questions?

Here are 5 related questions you might explore:

1. How does changing the satellite's orbital period affect its altitude?
2. What is the satellite's orbital velocity at this altitude?
3. How would the height change if the satellite orbited a different planet?
4. What is the gravitational force acting on the satellite at this altitude?
5. How does Earth's mass influence satellite orbits?

Tip: Orbital period and altitude are linked via Kepler's third law, which states that a satellite farther from Earth takes longer to complete an orbit.